An Introduction to the Controllability of Partial Differential Equations

2 Controllability of Partial Differential Equationsare known today. The interested reader may learn more on this topic from thereferences above and those on the bibliography at the end of the article.When dealing with controllability problems, to begin with, one has to distinguishbetween finite-dimensional systems modelled by ODE and infinitedimensionaldistributed systems described by means of PDE. This modelling issuemay be important in practice since finite-dimensional and infinite-dimensionalsystems may have quite different properties from a control theoretical pointof view ([74]).Most of these notes deal with problems related to PDE. However, we startby an introductory chapter in which we present some of the basic problems andtools of control theory for finite-dimensional systems. The theory has evolvedtremendously in the last decades to deal with nonlinearity and uncertaintybut here we present the simplest results concerning the controllability of linearfinite-dimensional systems and focus on developing tools that will later be usefulto deal with PDE. As we shall see, in the finite-dimensional context a systemis controllable if and only if the algebraic Kalman rank condition is satisfied.According to it, when a system is controllable for some time it is controllablefor all time. But this is not longer true in the context of PDE. In particular, inthe frame of the wave equation, a model in which propagation occurs with finitevelocity, in order for controllability properties to be true the control time needsto be large enough so that the effect of the control may reach everywhere. Inthis first chapter we shall develop a variational approach to the control problem.As we shall see, whenever a system is controllable, the control can be builtby minimizing a suitable quadratic functional defined on the class of solutions ofthe adjoint system. Suitable variants of this functional allow building differenttypes of controls: those of minimal L 2 -norm turn out to be smooth whilethose of minimal L ∞ -norm are of bang-bang form. The main difficulty whenminimizing these functionals is to show that they are coercive. This turns outto be equivalent to the so called observability property of the adjoint equation,a property which is equivalent to the original control property of the stateequation.In Chapters 2 and 3 we introduce the problems of interior and boundarycontrol of the linear constant coefficient wave equation. We describe the variousvariants, namely, approximate, exact and null controllability, and its mutualrelations. Once again, the problem of exact controllability turns out to beequivalent to the observability of the adjoint system while approximate controllabilityis equivalent to a weaker uniqueness or unique continuation property.In Chapter 4 we analyze the 1 − d case by means of Fourier series expansionsand the classical Ingham’s inequality which is a very useful tool to solve controlproblems for 1 − d wave-like and beam equations.In Chapters 5 and 6 we discuss respectively the problems of interior andboundary control of the heat equation. We show that, as a consequence of

S. Micu and E. Zuazua 3Holmgren Uniqueness Theorem, the adjoint heat equation posesses the propertyof unique continuation in an arbitrarily small time. Accordingly the multidimensionalheat equation is approximately controllable in an arbitrarily smalltime and with controls supported in any open subset of the domain where theequation holds. We also show that, in one space dimension, using Fourier seriesexpansions, the null control problem, can be reduced to a problem of momentsinvolving a sequence of real exponentials. We then build a biorthogonal familyallowing to show that the system is null controllable in any time by means ofa control acting on one extreme of the space interval where the heat equationholds.As we said above these notes are not complete. The interested reader maylearn more on this topic through the survey articles [70] and [72]. For theconnections between controllability and the theory of homogenization we referto [12]. We refer to [74] for a discussion of numerical apprximation issues incontrollability of PDE.1 Controllability and stabilization of finite dimensionalsystemsThis chapter is devoted to study some basic controllability and stabilizationproperties of finite dimensional systems.The first two sections deal with the linear case. In Section 1 it is shownthat the exact controllability property may be characterized by means of theKalman’s algebraic rank condition. In Section 2 a skew-adjoint system is considered.In the absence of control, the system is conservative and generates agroup of isometries. It is shown that the system may be guaranteed to be uniformlyexponentially stable if a well chosen feedback dissipative term is addedto it. This is a particular case of the well known equivalence property betweencontrollability and stabilizability of finite-dimensional systems ([65]).1.1 Controllability of finite dimensional linear systemsLet n, m ∈ N ∗ and T > 0. We consider the following finite dimensional system:{ x ′ (t) = Ax(t) + Bu(t), t ∈ (0, T ),x(0) = x 0 .(1)In (1), A is a real n × n matrix, B is a real n × m matrix and x 0 a vector inR n . The function x : [0, T ] −→ R n represents the state and u : [0, T ] −→ R mthe control. Both are vector functions of n and m components respectivelydepending exclusively on time t. Obviously, in practice m ≤ n. The most

4 Controllability of Partial Differential Equationsdesirable goal is, of course, controlling the system by means of a minimumnumber m of controls.Given an initial datum x 0 ∈ R n and a vector function u ∈ L 2 (0, T ; R m ), system(1) has a unique solution x ∈ H 1 (0, T ; R n ) characterized by the variationof constants formula:x(t) = e At x 0 +∫ t0e A(t−s) Bu(s)ds, ∀t ∈ [0, T ]. (2)Definition 1.1 System (1) is exactly controllable in time T > 0 if givenany initial and final one x 0 , x 1 ∈ R n there exists u ∈ L 2 (0, T, R m ) such thatthe solution of (1) satisfies x(T ) = x 1 .According to this definition the aim of the control process consists in drivingthe solution x of (1) from the initial state x 0 to the final one x 1 in time T byacting on the system through the control u.Remark that m is the number of controls entering in the system, whilen stands for the number of components of the state to be controlled. Aswe mentioned before, in applications it is desirable to make the number ofcontrols m to be as small as possible. But this, of course, may affect thecontrol properties of the system. As we shall see later on, some systems witha large number of components n can be controlled with one control only (i.e. m = 1). But in order for this to be true, the control mechanism, i.e. thematrix (column vector when m = 1) B, needs to be chosen in a strategic waydepending on the matrix A. Kalman’s rank condition, that will be given insection 1.3, provides a simple characterization of controllability allowing tomake an appropriate choice of the control matrix B.Let us illustrate this with two examples. In the first one controllabilitydoes not hold because one of the components of the system is insensitive to thecontrol. In the second one both components will be controlled by means of ascalar control.Example 1. Consider the case( ) ( )1 0 1A = , B = . (3)0 1 0Then the systemcan be written asor equivalently,x ′ = Ax + Bu{ x′1 = x 1 + ux ′ 2 = x 2,{ x′1 = x 1 + ux 2 = x 0 2e t ,

S. Micu and E. Zuazua 5where x 0 = (x 0 1, x 0 2) are the initial data.This system is not controllable since the control u does not act on the second componentx 2 of the state which is completely determined by the initial data x 0 2. Hence,the system is not controllable. Nevertheless one can control the first component x 1of the state. Consequently, the system is partially controllable. □Example 2. Not all systems with two components and a scalar control (n = 2, m =1) behave so badly as in the previous example. This may be seen by analyzing thecontrolled harmonic oscillatorx ′′ + x = u, (4)which may be written as a system in the following way{x ′ = yy ′ = u − x.The matrices A and B are now respectively( ) ( )0 10A = , B = .−1 01Once again, we have at our disposal only one control u for both components x andy of the system. But, unlike in Example 1, now the control acts in the second equationwhere both components are present. Therefore, we cannot conclude immediately thatthe system is not controllable. In fact it is controllable. Indeed, given some arbitraryinitial and final data, (x 0 , y 0 ) and (x 1 , y 1 ) respectively, it is easy to construct a regularfunction z = z(t) such that{ z(0) = x 0 ,z ′ (0) = y 0 ,z(T ) = x 1 ,z ′ (T ) = y 1 .In fact, there are infinitely many ways of constructing such functions. One can,for instance, choose a cubic polynomial function z. We can then define u = z ′′ + z asbeing the control since the solution x of equation (4) with this control and initial data(x 0 , y 0 ) coincides with z, i.e. x = z, and therefore satisfies the control requirements(5).This construction provides an example of system with two components (n = 2)which is controllable with one control only (m = 1). Moreover, this example showsthat the control u is not unique. In fact there exist infinitely many controls and differentcontrolled trajectories fulfilling the control requirements. In practice, choosingthe control which is optimal (in some sense to be made precise) is an important issuethat we shall also discuss. □If we define the set of reachable statesR(T, x 0 ) = {x(T ) ∈ R n : x solution of (1) with u ∈ (L 2 (0, T )) m }, (6)the exact controllability property is equivalent to the fact that R(T, x 0 ) =R n for any x 0 ∈ R n .(5)

6 Controllability of Partial Differential EquationsRemark 1.1 In the definition of exact controllability any initial datum x 0 isrequired to be driven to any final datum x 1 . Nevertheless, in the view of thelinearity of the system, without any loss of generality, we may suppose thatx 1 = 0. Indeed, if x 1 ≠ 0 we may solve{ y ′ = Ay, t ∈ (0, T )y(T ) = x 1 (7)backward in time and define the new state z = x − y which verifies{ z ′ = Az + Buz(0) = x 0 − y(0).(8)Remark that x(T ) = x 1 if and only if z(T ) = 0. Hence, driving the solutionx of (1) from x 0 to x 1 is equivalent to leading the solution z of (8) from theinitial data z 0 = x 0 − y(0) to zero. □The previous remark motivates the following definition:Definition 1.2 System (1) is said to be null-controllable in time T > 0 ifgiven any initial data x 0 ∈ R n there exists u ∈ L 2 (0, T, R m ) such that x(T ) = 0.Null-controllability holds if and only if 0 ∈ R(x 0 , T ) for any x 0 ∈ R n .On the other hand, Remark 1.1 shows that exact controllability and nullcontrollability are equivalent properties in the case of finite dimensional linearsystems. But this is not necessarily the case for nonlinear systems, or,for strongly time irreversible infinite dimensional systems, for strongly timeirreversible ones. For instance, the heat equation is a well known example ofnull-controllable system that is not exactly controllable.1.2 Observability propertyThe exact controllability property is closely related to an inequality for thecorresponding adjoint homogeneous system. This is the so called observationor observability inequality. In this section we introduce this notion and showits relation with the exact controllability property.Let A ∗ be the adjoint matrix of A, i.e. the matrix with the property that〈Ax, y〉 = 〈x, A ∗ y〉 for all x, y ∈ R n . Consider the following homogeneous adjointsystem of (1): { −ϕ ′ = A ∗ ϕ, t ∈ (0, T )(9)ϕ(T ) = ϕ T .Remark that, for each ϕ T ∈ R, (9) may be solved backwards in time andit has a unique solution ϕ ∈ C ω ([0, T ], R n ) (the space of analytic functionsdefined in [0, T ] and with values in R n ).

S. Micu and E. Zuazua 7First of all we deduce an equivalent condition for the exact controllabilityproperty.Lemma 1.1 An initial datum x 0 ∈ R n of (1) is driven to zero in time T byusing a control u ∈ L 2 (0, T ) if and only if∫ T0〈u, B ∗ ϕ〉dt + 〈x 0 , ϕ(0)〉 = 0 (10)for any ϕ T ∈ R n , ϕ being the corresponding solution of (9).Proof: Let ϕ T be arbitrary in R n and ϕ the corresponding solution of (9).By multiplying (1) by ϕ and (9) by x we deduce thatHence,〈x ′ , ϕ〉 = 〈Ax, ϕ〉 + 〈Bu, ϕ〉; −〈x, ϕ ′ 〉 = 〈A ∗ ϕ, x〉.d〈x, ϕ〉 = 〈Bu, ϕ〉dtwhich, after integration in time, gives that〈x(T ), ϕ T 〉 − 〈x 0 , ϕ(0)〉 =∫ T0〈Bu, ϕ〉dt =∫ T0〈u, B ∗ ϕ〉dt. (11)We obtain that x(T ) = 0 if and only if (10) is verified for any ϕ T ∈ R n .□It is easy to see that (10) is in fact an optimality condition for the criticalpoints of the quadratic functional J : R n → R n ,J(ϕ T ) = 1 2∫ T0| B ∗ ϕ | 2 dt + 〈x 0 , ϕ(0)〉where ϕ is the solution of the adjoint system (9) with initial data ϕ T at timet = T .More precisely, we have the following result:Lemma 1.2 Suppose that J has a minimizer ̂ϕ T ∈ R n and let ̂ϕ be the solutionof the adjoint system (9) with initial data ̂ϕ T . Thenis a control of system (1) with initial data x 0 .u = B ∗ ̂ϕ (12)

8 Controllability of Partial Differential EquationsProof: If ̂ϕ T is a point where J achieves its minimum value, thenThis is equivalent toJ ( ̂ϕ T + hϕ T ) − J ( ̂ϕ T )lim= 0, ∀ϕ T ∈ R n .h→0h∫ T0〈B ∗ ̂ϕ, B ∗ ϕ〉dt + 〈x 0 , ϕ(0)〉 = 0, ∀ϕ T ∈ R n ,which, in view of Lemma 1.1, implies that u = B ∗ ̂ϕ is a control for (1).□Remark 1.2 Lemma 1.2 gives a variational method to obtain the control as aminimum of the functional J. This is not the unique possible functional allowingto build the control. By modifying it conveniently, other types of controls(for instance bang-bang ones) can be obtained. We shall show this in section1.4. Remark that the controls we found are of the form B ∗ ϕ, ϕ being a solutionof the homogeneous adjoint problem (9). Therefore, they are analytic functionsof time. □The following notion will play a fundamental role in solving the controlproblems.Definition 1.3 System (9) is said to be observable in time T > 0 if thereexists c > 0 such that∫ T0| B ∗ ϕ | 2 dt ≥ c | ϕ(0) | 2 , (13)for all ϕ T ∈ R n , ϕ being the corresponding solution of (9).In the sequel (13) will be called the observation or observability inequality.It guarantees that the solution of the adjoint problem at t = 0is uniquely determined by the observed quantity B ∗ ϕ(t) for 0 < t < T . Inother words, the information contained in this term completely characterizesthe solution of (9).Remark 1.3 The observation inequality (13) is equivalent to the followingone: there exists c > 0 such that∫ T0| B ∗ ϕ | 2 dt ≥ c | ϕ T | 2 , (14)for all ϕ T ∈ R n , ϕ being the solution of (9).

S. Micu and E. Zuazua 9Indeed, the equivalence follows from the fact that the map which associatesto every ϕ T ∈ R n the vector ϕ(0) ∈ R n , is a bounded linear transformation inR n with bounded inverse. We shall use the forms (13) or (14) of the observationinequality depending of the needs of each particular problem we shall deal with.□The following remark is very important in the context of finite dimensionalspaces.Proposition 1.1 Inequality (13) is equivalent to the following unique continuationprinciple:B ∗ ϕ(t) = 0, ∀t ∈ [0, T ] ⇒ ϕ T = 0. (15)Proof: One of the implications follows immediately from (14). For theother one, let us define the semi-norm in R n|ϕ T | ∗ =[ ∫ T0| B ∗ ϕ | 2 dt] 1/2.Clearly, | · | ∗ is a norm in R n if and only if (15) holds.Since all the norms in R n are equivalent, it follows that (15) is equivalentto (14). The proof ends by taking into account the previous Remark 2.3. □Remark 1.4 Let us remark that (13) and (15) will no longer be equivalentproperties in infinite dimensional spaces. They will give rise to different notionsof controllability (exact and approximate, respectively). This issue will befurther developed in the following section. □The importance of the observation inequality relies on the fact that it impliesexact controllability of (1). In this way the controllability property isreduced to the study of an inequality for the homogeneous system (9) which,at least conceptually, is a simpler problem. Let us analyze now the relationbetween the controllability and observability properties.Theorem 1.1 System (1) is exactly controllable in time T if and only if (9)is observable in time T .Proof: Let us prove first that observability implies controllability. Accordingto Lemma 1.2, the exact controllability property in time T holds if for anyx 0 ∈ R n , J has a minimum. Remark that J is continuous. Consequently, theexistence of a minimum is ensured if J is coercive too, i.e.lim J(ϕ T ) = ∞. (16)|ϕ T |→∞

10 Controllability of Partial Differential EquationsThe coercivity property (16) is a consequence of the observation propertyin time T . Indeed, from (13) we obtain thatJ(ϕ T ) ≥ c 2 | ϕ T | 2 −|〈x 0 , ϕ(0)〉|.The right hand side tends to infinity when |ϕ T | → ∞ and J satisfies (16).Reciprocally, suppose that system (1) is exactly controllable in time T . If(9) is not observable in time T , there exists a sequence (ϕ k T ) k≥1 ⊂ R n suchthat |ϕ k T | = 1 for all k ≥ 1 and∫ Tlimk→∞ 0|B ∗ ϕ k | 2 dt = 0. (17)It follows that there exists a subsequence of (ϕ k T ) k≥1, denoted in the sameway, which converges to ϕ T ∈ R n and |ϕ T | = 1. Moreover, if ϕ is the solutionof (9) with initial data ϕ T , from (17) it follows that∫ T0|B ∗ ϕ| 2 dt = 0. (18)Since (1) is controllable, Lemma 1.1 gives that, for any initial data x 0 ∈ R n ,there exists u ∈ L 2 (0, T ) such that∫ T0〈u, B ∗ ϕ k 〉dt = −〈x 0 , ϕ k (0)〉, ∀k ≥ 1. (19)By passing to the limit in (19) and by taking into account (18), we obtainthat < x 0 , ϕ(0) >= 0. Since x 0 is arbitrary in R n , it follows that ϕ(0) = 0 and,consequently, ϕ T = 0. This is in contradiction with the fact that |ϕ T | = 1.The proof of the theorem is now complete. □Remark 1.5 The usefulness of Theorem 1.1 consists on the fact that it reducesthe proof of the exact controllability to the study of the observation inequality.□1.3 Kalman’s controllability conditionThe following classical result is due to R. E. Kalman and gives a complete answerto the problem of exact controllability of finite dimensional linear systems.It shows, in particular, that the time of control is irrelevant.

S. Micu and E. Zuazua 11Theorem 1.2 ([44]) System (1) is exactly controllable in some time T if andonly ifrank [B, AB, · · · , A n−1 B] = n. (20)Consequently, if system (1) is controllable in some time T > 0 it is controllablein any time.Remark 1.6 From now on we shall simply say that (A, B) is controllable if(20) holds. The matrix [B, AB, · · · , A n−1 B] will be called the controllabilitymatrix. □Examples: In Example 1 from section 1.1 we had( ) ( )1 0 1A = , B = . (21)0 1 0Therefore( )1 1[B, AB] =0 0which has rank 1. From Theorem 1.2 it follows that the system under considerationis not controllable. Nevertheless, in Example 2,( ) ( )0 1 0A = , B =(23)−1 0 1(22)and consequently( )0 1[B, AB] =1 0which has rank 2 and the system is controllable as we have already observed.□(24)Proof of Theorem 1.2: “ ⇒” Suppose that rank([B, AB, · · · , A n−1 B]) < n.Then the rows of the controllability matrix [B, AB, · · · , A n−1 B] are linearlydependent and there exists a vector v ∈ R n , v ≠ 0 such thatv ∗ [B, AB, · · · , A n−1 B] = 0,where the coefficients of the linear combination are the components of thevector v. Since v ∗ [B, AB, · · · , A n−1 B] = [v ∗ B, v ∗ AB, · · · , v ∗ A n−1 B], v ∗ B =v ∗ AB = · · · = v ∗ A n−1 B = 0. ¿From Cayley-Hamilton Theorem we deducethat there exist constants c 1 , · · · , c n such that, A n = c 1 A n−1 + · · · + c n I andtherefore v ∗ A n B = 0, too. In fact, it follows that v ∗ A k B = 0 for all k ∈ N andconsequently v ∗ e At B = 0 for all t as well. But, from the variation of constantsformula, the solution x of (1) satisfiesx(t) = e At x 0 +∫ t0e A(t−s) Bu(s)ds. (25)

12 Controllability of Partial Differential EquationsTherefore〈v, x(T )〉 = 〈v, e AT x 0 〉 +∫ T0〈v, e A(T −s) Bu(s)〉ds = 〈v, e AT x 0 〉,where 〈 , 〉 denotes the canonical inner product in R n . Hence, 〈v, x(T )〉 =〈v, e AT x 0 〉. This shows that the projection of the solution x at time T on thevector v is independent of the value of the control u. Hence, the system is notcontrollable. □Remark 1.7 The conservation property for the quantity 〈v, x〉 we have justproved holds for any vector v for which v[B, AB, · · · , A n−1 B] = 0. Thus, if therank of the matrix [B, AB, · · · , A n−1 B] is n − k, the reachable set that x(T )runs is an affine subspace of R n of dimension n − k. □“ ⇐” Suppose now that rank([B, AB, · · · , A n−1 B]) = n. According to Theorem1.1 it is sufficient to show that system (9) is observable. By Proposition1.1, (13) holds if and only if (15) is verified. Hence, the Theorem isproved if (15) holds. ¿From B ∗ ϕ = 0 and ϕ(t) = e A∗ (T −t) ϕ T , it follows thatB ∗ e A∗ (T −t) ϕ T ≡ 0 for all 0 ≤ t ≤ T . By computing the derivatives of thisfunction in t = T we obtain thatB ∗ [A ∗ ] k ϕ T = 0 ∀k ≥ 0.But since rank( [ B, AB, · · · , A n−1 B ] ) = n we deduce thatrank( [ B ∗ , B ∗ A ∗ , · · · , B ∗ (A ∗ ) n−1] ) = nand therefore ϕ T = 0. Hence, (15) is verified and the proof of Theorem 1.2 isnow complete. □Remark 1.8 The set of controllable pairs (A, B) is open and dense. Indeed,• If (A, B) is controllable there exists ε > 0 sufficiently small such that any(A 0 , B 0 ) with | A 0 − A |< ε, | B 0 − B |< ε is also controllable. Thisis a consequence of the fact that the determinant of a matrix dependscontinuously of its entries.• On the other hand, if (A, B) is not controllable, for any ε > 0, thereexists (A 0 , B 0 ) with | A − A 0 |< ε and | B − B 0 |< ε such that (A 0 , B 0 )is controllable. This is a consequence of the fact that the determinant ofa n × n matrix depends analytically of its entries and cannot vanish in aball of R n . □

S. Micu and E. Zuazua 13The following inequality shows that the norm of the control is proportionalto the distance between e AT x 0 (the state freely attained by the system in theabsence of control, i. e. with u = 0) and the objective x 1 .Proposition 1.2 Suppose that the pair (A, B) is controllable in time T > 0and let u be the control obtained by minimizing the functional J. There existsa constant C > 0, depending on T , such that the following inequality holdsfor any initial data x 0 and final objective x 1 .‖ u ‖ L 2 (0,T )≤ C|e AT x 0 − x 1 | (26)Proof: Let us first prove (26) for the particular case x 1 = 0.Let u be the control for (1) obtained by minimizing the functional J. From(10) it follows that∫ T||u|| 2 L 2 (0,T ) = |B ∗ ̂ϕ| 2 dt = − < x 0 , ̂ϕ(0) > .0If w is the solution of{w ′ (t) = Aw(t), t ∈ (0, T ),w(0) = x 0 (27)then w(t) = e At x 0 andd< w, ϕ >= 0dtfor all ϕ T ∈ R n , ϕ being the corresponding solution of (9).In particular, by taking ϕ T = ̂ϕ T , the minimizer of J, it follows that< x 0 , ̂ϕ(0) >=< w(0), ̂ϕ(0) >=< w(T ), ̂ϕ T >=< e AT x 0 , ̂ϕ T > .We obtain that||u|| 2 L 2 (0,T ) = − < x0 , ̂ϕ(0) >= − < e AT x 0 , ̂ϕ T >≤ |e AT x 0 | | ̂ϕ T |.On the other hand, we have thatThus, the control u verifies| ̂ϕ T | ≤ c||B ∗ ̂ϕ|| L 2 (0,T ) = c||u|| L 2 (0,T ).||u|| L2 (0,T ) ≤ c|e AT x 0 |. (28)If x 1 ≠ 0, Remark 1.1 implies that a control u driving the solution from x 0to x 1 coincides with the one leading the solution from x 0 − y(0) to zero, wherey verifies (7). By using (28) we obtain thatand (26) is proved.||u|| L 2 (0,T ) ≤ c|e T A (x 0 − y(0))| = c|e T A x 0 − x 1 |□

14 Controllability of Partial Differential EquationsRemark 1.9 Linear scalar equations of any order provide examples of systemsof arbitrarily large dimension that are controllable with only one control.Indeed, the system of order kx (k) + a 1 x (k−1) + . . . + a k−1 x = uis controllable. This can be easily obtained by observing that given k initial dataand k final ones one can always find a trajectory z (in fact an infinite numberof them) joining them in any time interval. This argument was already used inExample 2 for the case k = 2.It is an interesting exercise to write down the matrices A and B in this caseand to check that the rank condition in Theorem 1.2 is fulfilled. □1.4 Bang-bang controlsLet us consider the particular caseB ∈ M n×1 , (29)i. e. m = 1, in which only one control u : [0, T ] → R is available. In order tobuild bang-bang controls, it is convenient to consider the quadratic functional:J bb (ϕ 0 ) = 1 2[ ∫ T0| B ∗ ϕ | dt] 2+ 〈x 0 , ϕ(0)〉 (30)where ϕ is the solution of the adjoint system (9) with initial data ϕ T .Note that B ∗ ∈ M 1×n and therefore B ∗ ϕ(t) : [0, T ] → R is a scalar function.It is also interesting to note that J bb differs from J in the quadratic term.Indeed, in J we took the L 2 (0, T )-norm of B ∗ ϕ while here we consider theL 1 (0, T )-norm.The same argument used in the proof of Theorem 1.2 shows that J bb is alsocontinuous and coercive. It follows that J bb attains a minimum in some point̂ϕ T ∈ R n .On the other hand, it is easy to see that⎡1limh→0 h( ∫ 2 (T∫ ) ⎤ 2 T⎣ | f + hg | dt)− | f | ⎦ == 20∫ T0∫ T| f | dt sgn(f(t))g(t)dt0if the Lebesgue measure of the set {t ∈ (0, T ) : f(t) = 0} vanishes.0(31)

S. Micu and E. Zuazua 15The sign function “sgn” is defined as a multi-valued function in the followingway⎧⎨ 1 when s > 0sgn (s) = −1 when s < 0⎩[−1, 1] when s = 0Remark that in the previous limit there is no ambiguity in the definition ofsgn(f(t)) since the set of points t ∈ [0, T ] where f = 0 is assumed to be of zeroLebesgue measure and does not affect the value of the integral.Identity (31) may be applied to the quadratic term of the functional J bbsince, taking into account that ϕ is the solution of the adjoint system (9), itis an analytic function and therefore, B ∗ ϕ changes sign finitely many times inthe interval [0, T ] except when ̂ϕ T = 0. In view of this, the Euler-Lagrangeequation associated with the critical points of the functional J bb is as follows:∫ T0| B ∗ ̂ϕ | dt∫ T0sgn(B ∗ ̂ϕ)B ∗ ψ(t)dt + 〈x 0 , ϕ(0)〉 = 0for all ϕ T ∈ R, where ϕ is the solution of the adjoint system (9) with initialdata ϕ T .Consequently, the control we are looking for is u =∫ T0| B ∗ ̂ϕ | dt sgn(B ∗ ̂ϕ)where ̂ϕ is the solution of (9) with initial data ̂ϕ T .Note that the control u is of bang-bang form. Indeed, u takes only twovalues ± ∫ T| B ∗ ̂ϕ | dt. The control switches from one value to the other0finitely many times when the function B ∗ ̂ϕ changes sign.Remark 1.10 Other types of controls can be obtained by considering functionalsof the formJ p (ϕ 0 ) = 1 2( ∫ T0| B ∗ ϕ | p dt) 2/p+ 〈x 0 , ϕ 0 〉with 1 < p < ∞. The corresponding controls areu =( ∫ T0| B ∗ ̂ϕ | p dt) (2−p)/p|B ∗ ̂ϕ| p−2 B ∗ ̂ϕwhere ̂ϕ is the solution of (9) with initial datum ̂ϕ T , the minimizer of J p .It can be shown that, as expected, the controls obtained by minimizing thisfunctionals give, in the limit when p → 1, a bang-bang control. □The following property gives an important characterization of the controlswe have studied.

16 Controllability of Partial Differential EquationsProposition 1.3 The control u 2 = B ∗ ̂ϕ obtained by minimizing the functionalJ has minimal L 2 (0, T ) norm among all possible controls. Analogously, thecontrol u ∞ = ∫ T| B ∗ ̂ϕ | dt sgn(B ∗ ̂ϕ) obtained by minimizing the functional0J bb has minimal L ∞ (0, T ) norm among all possible controls.Proof: Let u be an arbitrary control for (1). Then (10) is verified both by uand u 2 for any ϕ T . By taking ϕ T = ̂ϕ T (the minimizer of J) in (10) we obtainthatHence,∫ T0< u, B ∗ ̂ϕ > dt = − < x 0 , ̂ϕ(0) >,∫ T||u 2 || 2 L 2 (0,T ) = < u 2 , B ∗ ̂ϕ > dt = − < x 0 , ̂ϕ(0) > .0∫ T||u 2 || 2 L 2 (0,T ) = < u, B ∗ ̂ϕ > dt ≤ ||u|| L2 (0,T )||B ∗ ̂ϕ|| = ||u|| L2 (0,T )||u 2 || L2 (0,T )0and the first part of the proof is complete.For the second part a similar argument may be used. Indeed, let again ube an arbitrary control for (1). Then (10) is verified by u and u ∞ for any ϕ T .By taking ϕ T = ̂ϕ T (the minimizer of J bb ) in (10) we obtain that∫ T( ∫ 2 T||u ∞ || 2 L ∞ (0,T ) = |B ̂ϕ|dt) ∗ =Hence,00B ∗ ̂ϕudt = − < x 0 , ̂ϕ(0) >,∫ T∫ T||u ∞ || 2 L ∞ (0,T ) = B ∗ ̂ϕ udt ≤∫ T00B ∗ ̂ϕu ∞ dt = − < x 0 , ̂ϕ(0) > .≤ ||u|| L ∞ (0,T )and the proof finishes. □0|B ∗ ̂ϕ|dt = ||u|| L ∞ (0,T )||u ∞ || L ∞ (0,T )1.5 Stabilization of finite dimensional linear systemsIn this section we assume that A is a skew-adjoint matrix, i. e. A ∗ = −A. Inthis case, < Ax, x >= 0.Consider the system { x ′ = Ax + Bux(0) = x 0 (32).

S. Micu and E. Zuazua 17Remark 1.11 The harmonic oscillator, mx ′′ + kx = 0, provides the simplestexample of system with such properties. It will be studied with some detail atthe end of the section. □When u ≡ 0, the energy of the solution of (32) is conserved. Indeed, bymultiplying (32) by x, if u ≡ 0, one obtainsHence,ddt |x(t)|2 = 0. (33)|x(t)| = |x 0 |, ∀t ≥ 0. (34)The problem of stabilization can be formulated in the following way. Supposethat the pair (A, B) is controllable. We then look for a matrix L suchthat the solution of system (32) with the feedback controlu(t) = Lx(t) (35)has a uniform exponential decay, i.e. there exist c > 0 and ω > 0 suchthat|x(t)| ≤ ce −ωt |x 0 | (36)for any solution.Note that, according to the law (35), the control u is obtained in real timefrom the state x.In other words, we are looking for matrices L such that the solution of thesystemx ′ = (A + BL)x = Dx (37)has an uniform exponential decay rate.Remark that we cannot expect more than (36). Indeed, the solutions of (37)may not satisfy x(T ) = 0 in finite time T . Indeed, if it were the case, from theuniqueness of solutions of (37) with final state 0 in t = T , it would follow thatx 0 ≡ 0. On the other hand, whatever L is, the matrix D has N eigenvalues λ jwith corresponding eigenvectors e j ∈ R n . The solution x(t) = e λjt e j of (37)shows that the decay of solutions can not be faster than exponential.Theorem 1.3 If A is skew-adjoint and the pair (A, B) is controllable thenL = −B ∗ stabilizes the system, i.e. the solution of{ x ′ = Ax − BB ∗ xx(0) = x 0 (38)has an uniform exponential decay (36).

20 Controllability of Partial Differential Equationswhich indicates that an applied force, proportional to the velocity of the point-massand of opposite sign, is acting on the oscillator.It is easy to see that the solutions of this equation have an exponential decayproperties. Indeed, it is sufficient to remark that the two characteristic roots havenegative real part. Indeed,mr 2 + R + kr = 0 ⇔ r ± = −k ± √ k 2 − 4mR2mand therefore{ −kif k 2 ≤ 4mR2mRe r ± =√− k ± k 2− R if k 2 ≥ 4mR.2m 4m 2mLet us prove the exponential decay of the solutions of (50) by using Theorem 1.3.Firstly, we write (50) in the form (38). SettingX =(x√ mR x′ ),the conservative equation mx ′′ + kx = 0 corresponds to the system:√( 0R )X ′ m= AX, with A = √ .R− 0 mNote that A is a skew-adjoint matrix. On the other hand, if we choose( )0 0B =0 √ kwe obtain thatand the system( )BB ∗ 0 0=0 kX ′ = AX − BB ∗ X (51)is equivalent to (50).Now, it is easy to see that the pair (A, B) is controllable since the rank of [B, AB]is 2.It follows that the solutions of (50) have the property of exponential decay as theexplicit computation of the spectrum indicates. □If (A, B) is controllable, we have proved the uniform stability property ofthe system (32), under the hypothesis that A is skew-adjoint. However, thisproperty holds even if A is an arbitrary matrix. More precisely, we haveTheorem 1.4 If (A, B) is controllable then it is also stabilizable. Moreover, itis possible to prescribe any complex numbers λ 1 , λ 2 ,...,λ n as the eigenvalues ofthe closed loop matrix A + BL by an appropriate choice of the feedback matrixL so that the decay rate may be made arbitrarily fast.

S. Micu and E. Zuazua 21In the statement of the Theorem we use the classical term closed loop systemto refer to the system in which the control is given in feedback form.The proof of Theorem 1.4 is obtained by reducing system (32) to the socalled control canonical form (see [44] and [55]).2 Interior controllability of the wave equationIn this chapter the problem of interior controllability of the wave equation isstudied. The control is assumed to act on a subset of the domain where thesolutions are defined. The problem of boundary controllability, which is alsoimportant in applications and has attracted a lot of attention, will be consideredin the following chapter. In the later case the control acts on the boundary ofthe domain where the solutions are defined.2.1 IntroductionLet Ω be a bounded open set of R N with boundary of class C 2 and ω be an opennonempty subset of Ω. Given T > 0 consider the following non-homogeneouswave equation:⎧⎨ u ′′ − ∆u = f1 ωin (0, T ) × Ωu = 0on (0, T ) × ∂Ω(52)⎩u(0, · ) = u 0 , u ′ (0, · ) = u 1 in Ω.By ′ we denote the time derivative.In (52) u = u(t, x) is the state and f = f(t, x) is the interior control functionwith support localized in ω. We aim at changing the dynamics of the systemby acting on the subset ω of the domain Ω.It is well known that the wave equation models many physical phenomenasuch as small vibrations of elastic bodies and the propagation of sound. Forinstance (52) provides a good approximation for the small amplitude vibrationsof an elastic string or a flexible membrane occupying the region Ω at rest. Thecontrol f represents then a localized force acting on the vibrating structure.The importance of the wave equation relies not only in the fact that itmodels a large class of vibrating phenomena but also because it is the mostrelevant hyperbolic partial differential equation. As we shall see latter on,the main properties of hyperbolic equations such as time-reversibility and thelack of regularizing effects, have some very important consequences in controlproblems too.Therefore it is interesting to study the controllability of the wave equation asone of the fundamental models of continuum mechanics and, at the same time,as one of the most representative equations in the theory of partial differentialequations.

22 Controllability of Partial Differential Equations2.2 Existence and uniqueness of solutionsThe following theorem is a consequence of classical results of existence anduniqueness of solutions of nonhomogeneous evolution equations. All the detailsmay be found, for instance, in [14].Theorem 2.1 For any f ∈ L 2 ((0, T ) × ω) and (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω)equation (52) has a unique weak solution(u, u ′ ) ∈ C([0, T ], H 1 0 (Ω) × L 2 (Ω))given by the variation of constants formula(u, u ′ )(t) = S(t)(u 0 , u 1 ) +∫ t0S(t − s)(0, f(s)1 ω )ds (53)where (S(t)) t∈R is the group of isometries generated by the wave operator inH 1 0 (Ω) × L 2 (Ω).Moreover, if f ∈ W 1,1 ((0, T ); L 2 (ω)) and (u 0 , u 1 ) ∈ [H 2 (Ω) ∩ H 1 0 (Ω)] ×H 1 0 (Ω) equation (52) has a strong solution(u, u ′ ) ∈ C 1 ([0, T ], H 1 0 (Ω) × L 2 (Ω)) ∩ C([0, T ], [H 2 (Ω) ∩ H 1 0 (Ω)] × H 1 0 (Ω))and u verifies the wave equation (52) in L 2 (Ω) for all t ≥ 0.Remark 2.1 The wave equation is reversible in time. Hence, we may solve itfor t ∈ (0, T ) by considering initial data (u 0 , u 1 ) in t = 0 or final data (u 0 T , u1 T )in t = T . In the former case the solution is given by (53) and in the later oneby□∫ T(u, u ′ )(t) = S(T − t)(u 0 T , u 1 T ) + S(s − T + t)(0, f(s)1 ω )ds. (54)T −t2.3 Controllability problemsLet T > 0 and define, for any initial data (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω), the set ofreachable statesR(T ; (u 0 , u 1 )) = {(u(T ), u t (T )) : u solution of (52) with f ∈ L 2 ((0, T ) × ω)}.Remark that, for any (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω), R(T ; (u 0 , u 1 )) is an affinesubspace of H 1 0 (Ω) × L 2 (Ω).There are different notions of controllability that need to be distinguished.

S. Micu and E. Zuazua 23Definition 2.1 System (52) is approximately controllable in time T if,for every initial data (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω), the set of reachable statesR(T ; (u 0 , u 1 )) is dense in H 1 0 (Ω) × L 2 (Ω).Definition 2.2 System (52) is exactly controllable in time T if, for everyinitial data (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω), the set of reachable states R(T ; (u 0 , u 1 ))coincides with H 1 0 (Ω) × L 2 (Ω).Definition 2.3 System (52) is null controllable in time T if, for everyinitial data (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω), the set of reachable states R(T ; (u 0 , u 1 ))contains the element (0, 0).Since the only dense and convex subset of R n is R n , it follows that the approximateand exact controllability notions are equivalent in the finite-dimensionalcase. Nevertheless, for infinite dimensional systems as the wave equation,these two notions do not coincide.Remark 2.2 In the notions of approximate and exact controllability it is sufficientto consider the case (u 0 , u 1 ) ≡ 0 since R(T ; (u 0 , u 1 )) = R(T ; (0, 0)) +S(T )(u 0 , u 1 ). □In the view of the time-reversibility of the system we have:Proposition 2.1 System (52) is exactly controllable if and only if it is nullcontrollable.Proof: Evidently, exact controllability implies null controllability.Let us suppose now that (0, 0) ∈ R(T ; (u 0 , u 1 )) for any (u 0 , u 1 ) ∈ H0 1 (Ω) ×L 2 (Ω). Then any initial data in H0 1 (Ω) × L 2 (Ω) can be driven to (0, 0) in timeT . From the reversibility of the wave equation we deduce that any state inH0 1 (Ω) × L 2 (Ω) can be reached in time T by starting from (0, 0). This meansthat R(T, (0, 0)) = H0 1 (Ω) × L 2 (Ω) and the exact controllability property holdsfrom Remark 2.2. □The previous Proposition guarantees that (52) is exactly controllable if andonly if, for any (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω) there exists f ∈ L 2 ((0, T ) × ω) suchthat the corresponding solution (u, u ′ ) of (52) satisfiesu(T, · ) = u t (T, · ) = 0. (55)This is the most common form in which the exact controllability propertyfor the wave equation is formulated.Remark 2.3 The following facts indicate how the main distinguishing propertiesof wave equation affect its controllability properties:

24 Controllability of Partial Differential Equations• Since the wave equation is time-reversible and does not have any regularizingeffect, one may not exclude the exact controllability to hold. Nevertheless,as we have said before, there are situations in which the exactcontrollability property is not verified but the approximate controllabilityholds. This depends on the geometric properties of Ω and ω.• The wave equation is a prototype of equation with finite speed of propagation.Therefore, one cannot expect the previous controllability propertiesto hold unless the control time T is sufficiently large. □2.4 Variational approach and observabilityLet us first deduce a necessary and sufficient condition for the exact controllabilityproperty of (52) to hold. By 〈 · , · 〉 1,−1we denote the duality productbetween H0 1 (Ω) and its dual, H −1 (Ω).For (ϕ 0 T , ϕ1 T ) ∈ L2 (Ω) × H −1 (Ω), consider the following backward homogeneousequation⎧⎨ ϕ ′′ − ∆ϕ = 0in (0, T ) × Ωϕ = 0on (0, T ) × ∂Ω (56)⎩ϕ(T, · ) = ϕ 0 T , ϕ′ (T, · ) = ϕ 1 T in Ω.Let (ϕ, ϕ ′ ) ∈ C([0, T ], L 2 (Ω)×H −1 (Ω)) be the unique weak solution of (56).Lemma 2.1 The control f ∈ L 2 ((0, T ) × ω) drives the initial data (u 0 , u 1 ) ∈H 1 0 (Ω) × L 2 (Ω) of system (52) to zero in time T if and only if∫ T0∫ωϕfdxdt = 〈 ϕ ′ (0), u 0〉 1,−1 − ∫Ωϕ(0)u 1 dx, (57)for all (ϕ 0 T , ϕ1 T ) ∈ L2 (Ω) × H −1 (Ω) where ϕ is the corresponding solution of(56).Proof: Let us first suppose that ( u 0 , u 1) , ( ϕ 0 T , ) ϕ1 T ∈ D(Ω) × D(Ω), f ∈D((0, T ) × ω) and let u and ϕ be the (regular) solutions of (52) and (56)respectively.We recall that D(M) denotes the set of C ∞ (M) functions with compactsupport in M.By multiplying the equation of u by ϕ and by integrating by parts oneobtains∫ T ∫∫ T ∫ϕfdxdt = ϕ (u ′′ − ∆u) dxdt =0ω0Ω

S. Micu and E. Zuazua 25∫ T0∫=ΩHence,∫ω∫∫ T ∫= (ϕu ′ − ϕ ′ u) dx| T 0 + u (ϕ ′′ − ∆ϕ) dxdt =Ω0 Ω∫[ϕ(T )u ′ (T ) − ϕ ′ (T )u(T )] dx − [ϕ(0)u ′ (0) − ϕ ′ (0)u(0)] dx.∫ϕfdxdt =ΩΩ[ϕ0T u ′ (T ) − ϕ 1 T u(T ) ] ∫[dx− ϕ(0)u 1 − ϕ ′ (0)u 0] dx. (58)ΩFrom a density argument we deduce, by passing to the limit in (58), thatfor any (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω) and (ϕ 0 T , ϕ1 T ) ∈ L2 (Ω) × H −1 (Ω),= − 〈 ϕ 1 T , u(T ) 〉 1,−1 + ∫∫ T∫ϕfdxdt =0 ωϕ 0 T u ′ (T )dx + 〈 ϕ ′ (0), u 0〉 ∫1,−1 − ΩΩϕ(0)u 1 dx.(59)Now, from (59), it follows immediately that (57) holds if and only if (u 0 , u 1 )is controllable to zero and f is the corresponding control. This completes theproof. □Let us define the duality product between L 2 (Ω) × H −1 (Ω) and H0 1 (Ω) ×L 2 (Ω) by〈(ϕ 0 , ϕ 1) , ( u 0 , u 1)〉 = 〈 ϕ 1 , u 0〉 ∫− ϕ 0 u 1 dx1,−1for all ( ϕ 0 , ϕ 1) ∈ L 2 (Ω) × H −1 (Ω) and ( u 0 , u 1) ∈ H0 1 (Ω) × L 2 (Ω).Remark that the map ( ϕ 0 , ϕ 1) → 〈( ϕ 0 , ϕ 1) , ( u 0 , u 1)〉 is linear and continuousand its norm is equal to ||(u 0 , u 1 )|| H 10 ×L 2.For (ϕ 0 , ϕ 1 ) ∈ L 2 (Ω) × H −1 (Ω), consider the following homogeneous equation⎧⎨ ϕ ′′ − ∆ϕ = 0in (0, T ) × Ωϕ = 0on (0, T ) × ∂Ω(60)⎩ϕ(0, · ) = ϕ 0 , ϕ ′ (0, · ) = ϕ 1 in Ω.If (ϕ, ϕ ′ ) ∈ C([0, T ], L 2 (Ω) × H −1 (Ω)) is the unique weak solution of (60),then||ϕ|| 2 L ∞ (0,T ;L 2 (Ω)) + ||ϕ′ || 2 L ∞ (0,T ;H −1 (Ω)) ≤ ||(ϕ0 , ϕ 1 )|| 2 L 2 (Ω)×H −1 (Ω) . (61)Since the wave equation with homogeneous Dirichlet boundary conditionsgenerates a group of isometries in L 2 (Ω) × H −1 (Ω), Lemma 2.1 may be reformulatedin the following way:Ω

26 Controllability of Partial Differential EquationsLemma 2.2 The initial data (u 0 , u 1 ) ∈ H0 1 (Ω) × L 2 (Ω) may be driven to zeroin time T if and only if there exists f ∈ L 2 ((0, T ) × ω) such that∫ T ∫ϕfdxdt = 〈( ϕ 0 , ϕ 1) , ( u 0 , u 1)〉 , (62)0ωfor all (ϕ 0 , ϕ 1 ) ∈ L 2 (Ω) × H −1 (Ω) where ϕ is the corresponding solution of(60).Relation (62) may be seen as an optimality condition for the critical pointsof the functional J : L 2 (Ω) × H −1 (Ω) → R,J (ϕ 0 , ϕ 1 ) = 1 ∫ T ∫|ϕ| 2 dxdt + 〈( ϕ 0 , ϕ 1) , ( u 0 , u 1)〉 , (63)20ωwhere ϕ is the solution of (60) with initial data (ϕ 0 , ϕ 1 ) ∈ L 2 (Ω) × H −1 (Ω).We have:Theorem 2.2 Let (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω) and suppose that ( ̂ϕ 0 , ̂ϕ 1 ) ∈L 2 (Ω) × H −1 (Ω) is a minimizer of J . If ̂ϕ is the corresponding solution of(60) with initial data ( ̂ϕ 0 , ̂ϕ 1 ) thenis a control which leads (u 0 , u 1 ) to zero in time T .f = ̂ϕ |ω (64)Proof: Since J achieves its minimum at ( ̂ϕ 0 , ̂ϕ 1 ), the following relationholds1 (0 = lim J (( ̂ϕ 0 , ̂ϕ 1 ) + h(ϕ 0 , ϕ 1 )) − J ( ̂ϕ 0 , ̂ϕ 1 ) ) =h→0 h∫ T ∫∫= ̂ϕϕdxdt + u 1 ϕ 0 dx− < ϕ 1 , u 0 > 1,−10 ωΩfor any (ϕ 0 , ϕ 1 ) ∈ L 2 (Ω) × H −1 (Ω) where ϕ is the solution of (60).Lemma 2.2 shows that f = ̂ϕ |ω is a control which leads the initial data(u 0 , u 1 ) to zero in time T . □Let us now give sufficient conditions ensuring the existence of a minimizerfor J .Definition 2.4 Equation (60) is observable in time T if there exists a positiveconstant C 1 > 0 such that the following inequality is verifiedC 1 ‖ ( ϕ 0 , ϕ 1) ∫ T ∫‖ 2 L 2 (Ω)×H −1 (Ω) ≤ | ϕ | 2 dxdt, (65)for any ( ϕ 0 , ϕ 1) ∈ L 2 (Ω) × H −1 (Ω) where ϕ is the solution of (60) with initialdata (ϕ 0 , ϕ 1 ).0ω

S. Micu and E. Zuazua 27Inequality (65) is called observation or observability inequality. Itshows that the quantity ∫ T ∫0 ω | ϕ |2 (the observed one) which depends only onthe restriction of ϕ to the subset ω of Ω, uniquely determines the solution on(60).Remark 2.4 The continuous dependence (61) of solutions of (60) with respectto its initial data guarantees that there exists a constant C 2 > 0 such that∫ T ∫| ϕ | 2 dxdt ≤ C 2 ‖ ( ϕ 0 , ϕ 1) ‖ 2 L 2 (Ω)×H −1 (Ω)(66)0ωfor all (ϕ 0 , ϕ 1 ) ∈ L 2 (Ω) × H −1 (Ω) and ϕ solution of (60).□Let us show that (65) is a sufficient condition for the exact controllabilityproperty to hold. First of all let us recall the following fundamental resultin the Calculus of Variations which is a consequence of the so called DirectMethod of the Calculus of Variations.Theorem 2.3 Let H be a reflexive Banach space, K a closed convex subset ofH and ϕ : K → R a function with the following properties:1. ϕ is convex2. ϕ is lower semi-continuous3. If K is unbounded then ϕ is coercive, i. e.lim ϕ(x) = ∞. (67)||x||→∞Then ϕ attains its minimum in K, i. e. there exists x 0 ∈ K such thatFor a proof of Theorem 2.3 see [10].We have:ϕ(x 0 ) = min ϕ(x). (68)x∈KTheorem 2.4 Let (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω) and suppose that (60) is observablein time T . Then the functional J defined by (63) has an unique minimizer( ̂ϕ 0 , ̂ϕ 1 ) ∈ L 2 (Ω) × H −1 (Ω).Proof: It is easy to see that J is continuous and convex. Therefore, accordingto Theorem 2.3, the existence of a minimum is ensured if we prove that Jis also coercive i.e.lim J||(ϕ 0 ,ϕ 1 )|| L 2 ×H −1 →∞ (ϕ0 , ϕ 1 ) = ∞. (69)

28 Controllability of Partial Differential EquationsThe coercivity of functional J follows immediately from (65). Indeed,( ∫J (ϕ 0 , ϕ 1 ) ≥ 1 T ∫)|ϕ| 2 − ||(u 0 , u 1 )||2H 10 (Ω)×L (Ω)||(ϕ 0 , ϕ 1 )|| 2 L2 (Ω)×H −1 (Ω)0ω≥ C 12 ||(ϕ0 , ϕ 1 )|| 2 L 2 (Ω)×H −1 (Ω) − 1 2 ||(u0 , u 1 )|| H 10 (Ω)×L 2 (Ω)||(ϕ 0 , ϕ 1 )|| L2 (Ω)×H −1 (Ω).It follows from Theorem 2.3 that J has a minimizer ( ̂ϕ 0 , ̂ϕ 1 ) ∈ L 2 (Ω) ×H −1 (Ω).To prove the uniqueness of the minimizer it is sufficient to show that J isstrictly convex. Indeed, let (ϕ 0 , ϕ 1 ), (ψ 0 , ψ 1 ) ∈ L 2 (Ω)×H −1 (Ω) and λ ∈ (0, 1).We have thatJ (λ(ϕ 0 , ϕ 1 ) + (1 − λ)(ψ 0 , ψ 1 )) == λJ (ϕ 0 , ϕ 1 ) + (1 − λ)J (ψ 0 , ψ 1 ) −From (65) it follows that∫ T0∫ωλ(1 − λ)2∫ T0∫ω|ϕ − ψ| 2 dxdt.|ϕ − ψ| 2 dxdt ≥ C 1 ||(ϕ 0 , ϕ 1 ) − (ψ 0 , ψ 1 )|| L 2 (Ω)×H −1 (Ω).Consequently, for any (ϕ 0 , ϕ 1 ) ≠ (ψ 0 , ψ 1 ),J (λ(ϕ 0 , ϕ 1 ) + (1 − λ)(ψ 0 , ψ 1 )) < λJ (ϕ 0 , ϕ 1 ) + (1 − λ)J (ψ 0 , ψ 1 )and J is strictly convex.□Theorems 2.2 and 2.4 guarantee that, under hypothesis (65), system (52) isexactly controllable. Moreover, a control may be obtained as in (64) from thesolution of the homogeneous equation (60) with the initial data minimizing thefunctional J . Hence, the controllability problem is reduced to a minimizationproblem that may be solved by the Direct Method of the Calculus of Variations.This is very useful both from a theoretical and a numerical point of view.The following proposition shows that the control obtained by this variationalmethod is of minimal L 2 ((0, T ) × ω)-norm.Proposition 2.2 Let f = ̂ϕ |ω be the control given by minimizing the functionalJ . If g ∈ L 2 ((0, T ) × ω) is any other control driving to zero the initial data(u 0 , u 1 ) then||f|| L 2 ((0,T )×ω) ≤ ||g|| L 2 ((0,T )×ω). (70)

S. Micu and E. Zuazua 29Proof: Let ( ̂ϕ 0 , ̂ϕ 1 ) the minimizer for the functional J . Consider nowrelation (62) for the control f = ̂ϕ |ω . By taking ( ̂ϕ 0 , ̂ϕ 1 ) as test function weobtain that∫ T ∫||f|| 2 L 2 ((0,T )×ω) = | ̂ϕ| 2 dxdt = 〈 ̂ϕ 1 , u 0〉 ∫1,−1 − ̂ϕ 0 u 1 dx.0ωOn the other hand, relation (62) for the control g and test function ( ̂ϕ 0 , ̂ϕ 1 )gives∫ T ∫g ̂ϕdxdt = 〈 ̂ϕ 1 , u 0〉 ∫1,−1 − ̂ϕ 0 u 1 dx.We obtain that0ω||f|| 2 L 2 ((0,T )×ω) = 〈 ̂ϕ 1 , u 0〉 1,−1 − ∫ΩΩ̂ϕ 0 u 1 dx =∫ T0∫ωΩg ̂ϕdxdt ≤≤ ||g|| L2 ((0,T )×ω)|| ̂ϕ|| L2 ((0,T )×ω) = ||g|| L2 ((0,T )×ω)||f|| L2 ((0,T )×ω)and (70) is proved.□2.5 Approximate controllabilityUp to this point we have discussed only the exact controllability property of(52) which turns out to be equivalent to the observability property (65). Letus now address the approximate controllability one.Let ε > 0 and (u 0 , u 1 ), (z 0 , z 1 ) ∈ H0 1 (Ω) × L 2 (Ω). We are looking for acontrol function f ∈ L 2 ((0, T ) × ω) such that the corresponding solution (u, u ′ )of (52) satisfies||(u(T ), u ′ (T )) − (z 0 , z 1 )|| H 10 (Ω)×L 2 (Ω) ≤ ε. (71)Recall that (52) is approximately controllable if, for any ε > 0 and (u 0 , u 1 ),(z 0 , z 1 ) ∈ H 1 0 (Ω) × L 2 (Ω), there exists f ∈ L 2 ((0, T ) × ω) such that (71) isverified.By Remark 2.2, it is sufficient to study the case (u 0 , u 1 ) = (0, 0). From nowon we assume that (u 0 , u 1 ) = (0, 0).The variational approach considered in the previous sections may be alsovery useful for the study of the approximate controllability property. To seethis, define the functional J ε : L 2 (Ω) × H −1 (Ω) → R,= 1 2∫ T0∫ωJ ε (ϕ 0 , ϕ 1 ) =|ϕ| 2 dxdt + 〈( ϕ 0 , ϕ 1) , ( z 0 , z 1)〉 + ε||(ϕ 0 , ϕ 1 )|| L2 ×H −1, (72)

30 Controllability of Partial Differential Equationswhere ϕ is the solution of (60) with initial data (ϕ 0 , ϕ 1 ) ∈ L 2 (Ω) × H −1 (Ω).As in the exact controllability case, the existence of a minimum of thefunctional J ε implies the existence of an approximate control.Theorem 2.5 Let ε > 0 and (z 0 , z 1 ) ∈ H 1 0 (Ω) × L 2 (Ω) and suppose that( ̂ϕ 0 , ̂ϕ 1 ) ∈ L 2 (Ω) × H −1 (Ω) is a minimizer of J ε . If ̂ϕ is the correspondingsolution of (60) with initial data ( ̂ϕ 0 , ̂ϕ 1 ) thenf = ̂ϕ |ω (73)is an approximate control which leads the solution of (52) from the zero initialdata (u 0 , u 1 ) = (0, 0) to the state (u(T ), u ′ (T )) such that (71) is verified.Proof: Let ( ̂ϕ 0 , ̂ϕ 1 ) ∈ L 2 (Ω) × H −1 (Ω) be a minimizer of J ε . It followsthat, for any h > 0 and (ϕ 0 , ϕ 1 ) ∈ L 2 (Ω) × H −1 (Ω),0 ≤ 1 (Jε (( ̂ϕ 0 , ̂ϕ 1 ) + h(ϕ 0 , ϕ 1 )) − J ε ( ̂ϕ 0 , ̂ϕ 1 ) ) ≤h∫ T ∫≤ ̂ϕϕdxdt+ h ∫ T ∫|ϕ| 2 dxdt+ 〈( ϕ 0 , ϕ 1) , ( z 0 , z 1)〉 +ε||(ϕ 0 , ϕ 1 )||0 ω 2L 2 ×H −10 ωbeing ϕ the solution of (60). By making h → 0 we obtain that∫ T ∫−ε||(ϕ 0 , ϕ 1 )|| L 2 ×H −1 ≤ ̂ϕϕdxdt + 〈( ϕ 0 , ϕ 1) , ( z 0 , z 1)〉 .0A similar argument (with h < 0) leads to∫ T ∫̂ϕϕdxdt + 〈( ϕ 0 , ϕ 1) , ( z 0 , z 1)〉 ≤ ε||(ϕ 0 , ϕ 1 )|| L 2 ×H −1.0ωωHence, for any (ϕ 0 , ϕ 1 ) ∈ L 2 (Ω) × H −1 (Ω),∫ T ∫̂ϕϕdxdt + 〈( ϕ 0 , ϕ 1) , ( z 0 , z 1)〉∣ ∣ ∣∣∣≤ ε||(ϕ 0 , ϕ 1 )||∣L 2 ×H−1. (74)0ωNow, from (59) and (74) we obtain that∣ 〈( ϕ 0 , ϕ 1) , [(z 0 , z 1 ) − (u(T ), u ′ (T ))] 〉∣ ∣ ≤ ε||(ϕ 0 , ϕ 1 )|| L2 ×H −1,for any (ϕ 0 , ϕ 1 ) ∈ L 2 (Ω) × H −1 (Ω).Consequently, (71) is verified and the proof finishes.□As we have seen in the previous section, the exact controllability property of(52) is equivalent to the observation property (65) of system (60). An uniquecontinuation principle of the solutions of (60), which is a weaker version ofthe observability inequality (65), will play a similar role for the approximatecontrollability property and it will give a sufficient condition for the existenceof a minimizer of J ε . More precisely, we have

S. Micu and E. Zuazua 31Theorem 2.6 The following properties are equivalent:1. Equation (52) is approximately controllable.2. The following unique continuation principle holds for the solutions of (60)ϕ |(0,T )×ω= 0 ⇒ (ϕ 0 , ϕ 1 ) = (0, 0). (75)Proof: Let us first suppose that (52) is approximately controllable and letϕ be a solution of (60) with initial data (ϕ 0 , ϕ 1 ) ∈ L 2 (Ω) × H −1 (Ω) such thatϕ |(0,T )×ω= 0.For any ε > 0 and (z 0 , z 1 ) ∈ H 1 0 (Ω) × L 2 (Ω) there exists an approximatecontrol function f ∈ L 2 ((0, T ) × ω) such that (71) is verified.From (59) we deduce that 〈 (u(T ), u ′ (T )), (ϕ 0 , ϕ 1 ) 〉 = 0. From the controllabilityproperty and the last relation we deduce that∣ 〈 (z 0 , z 1 ), (ϕ 0 , ϕ 1 ) 〉∣ ∣ =∣ ∣〈[(z 0 , z 1 ) − (u(T ), u ′ (T ))], (ϕ 0 , ϕ 1 ) 〉∣ ∣ ≤ ε||(ϕ 0 , ϕ 1 )||.Since the last inequality is verified by any (z 0 , z 1 ) ∈ H 1 0 (Ω) × L 2 (Ω) itfollows that (ϕ 0 , ϕ 1 ) = (0, 0).Hence the unique continuation principle (75) holds.Reciprocally, suppose now that the unique continuation principle (75) isverified and let us show that (52) is approximately controllable.In order to do that we use Theorem 2.5. Let ε > 0 and (z 0 , z 1 ) ∈ H 1 0 (Ω) ×L 2 (Ω) be given and consider the functional J ε . Theorem 2.5 ensures the approximatecontrollability property of (52) under the assumption that J ε has aminimum. Let us show that this is true in our case.The functional J ε is convex and continuous in L 2 (Ω) × H −1 (Ω). Thus, theexistence of a minimum is ensured if J ε is coercive, i. e.J ε ((ϕ 0 , ϕ 1 )) → ∞ when ||(ϕ 0 , ϕ 1 )|| L2 ×H−1 → ∞. (76)In fact we shall prove thatlim inf J ε(ϕ 0 , ϕ 1 )/||(ϕ 0 , ϕ 1 )|| L2 ×H−1 ≥ ε. (77)||(ϕ 0 ,ϕ 1 )|| L 2 ×H −1 →∞Evidently, (77) implies (76) and the proof of the theorem is complete.In order to prove (77) let ( (ϕ 0 j , ϕ1 j )) j≥1 ⊂ L2 (Ω) × H −1 (Ω) be a sequenceof initial data for the adjoint system such that ‖ (ϕ 0 j , ϕ1 j ) ‖ L 2 ×H−1→ ∞. Wenormalize themso that ‖ ( ˜ϕ 0 j , ˜ϕ1 j ) ‖ L 2 ×H−1= 1.( ˜ϕ 0 j, ˜ϕ 1 j) = (ϕ 0 j, ϕ 1 j)/ ‖ (ϕ 0 j, ϕ 1 j) ‖ L 2 ×H −1,

32 Controllability of Partial Differential EquationsOn the other hand, let ( ˜ϕ j , ˜ϕ ′ j ) be the solution of (60) with initial data( ˜ϕ 0 j , ˜ϕ1 j ). ThenJ ε ((ϕ 0 j , ϕ1 j ))‖ (ϕ 0 j , ϕ1 j ) ‖ = 1 ∫ T ∫2 ‖ (ϕ0 j, ϕ 1 j) ‖ | ˜ϕ j | 2 dxdt + 〈 (z 0 , z 1 ), ( ˜ϕ 0 , ˜ϕ 1 ) 〉 + ε.0 ωThe following two cases may occur:∫ T ∫1) lim inf | ˜ϕ j | 2 > 0. In this case we obtain immediately thatj→∞0∫ T2) lim infj→∞0∫ωωJ ε ((ϕ 0 j , ϕ1 j ))‖ (ϕ 0 j , ϕ1 j ) ‖ → ∞.| ˜ϕ j | 2 = 0. In this case since ( ˜ϕ 0 j , )˜ϕ1 j j≥1is bounded inL 2 ×H −1 , by extracting a subsequence we can guarantee that ( ˜ϕ 0 j , ˜ϕ1 j ) j≥1converges weakly to (ψ 0 , ψ 1 ) in L 2 (Ω) × H −1 (Ω).Moreover, if (ψ, ψ ′ ) is the solution of (60) with the initial data (ψ 0 , ψ 1 )at t = T , then ( ˜ϕ j , ˜ϕ ′ j ) j≥1 converges weakly to (ψ, ψ ′ ) in L 2 (0, T ; L 2 (Ω)×H −1 (Ω)) ∩ H 1 (0, T ; H −1 (Ω) × [H 2 ∩ H 1 0 (Ω)] ′ ).By lower semi-continuity,∫ T0∫and therefore ψ = 0 en ω × (0, T ).ω∫ T ∫ψ 2 dxdt ≤ lim inf | ˜ϕ j | 2 dxdt = 0j→∞0 ωFrom the unique continuation principle we obtain that (ψ 0 , ψ 1 ) = (0, 0)and consequently,Hence( ˜ϕ 0 j, ˜ϕ 1 j) ⇀ (0, 0) weakly in L 2 (Ω) × H −1 (Ω).lim infj→∞J ε ((ϕ 0 j , ϕ1 j ))‖ (ϕ 0 j , ϕ1 j ) ‖ L 2 ×H −1≥ lim infj→∞ [ε + 〈 (z 0 , z 1 ), ( ˜ϕ 0 , ˜ϕ 1 ) 〉 ] = ε,and (77) follows.□When approximate controllability holds, then the following (apparentlystronger) statement also holds:

S. Micu and E. Zuazua 33Theorem 2.7 Let E be a finite-dimensional subspace of H 1 0 (Ω)×L 2 (Ω) and letus denote by π E the corresponding orthogonal projection. Then, if approximatecontrollability holds, for any ( u 0 , u 1) , ( z 0 , z 1) ∈ H 1 0 (Ω) × L 2 (Ω) and ε > 0there exists f ∈ L 2 ((0, T ) × ω) such that the solution of (52) satisfies∥ ( u(T ) − z 0 , u t (T ) − z 1)∥ ∥H 10 (Ω)×L 2 (Ω) ≤ ε; π E (u(T ), u t (T )) = π E(z 0 , z 1) .This property will be referred to as the finite-approximate controllabilityproperty. Its proof may be found in [71].2.6 CommentsIn this section we have presented some facts related with the exact and approximatecontrollability properties. The variational methods we have used allow toreduce these properties to an observation inequality and a unique continuationprinciple for the homogeneous adjoint equation respectively. The latter will bestudied for some particular cases in Chapter 4 by using nonharmonic Fourieranalysis.3 Boundary controllability of the wave equationThis chapter is devoted to study the boundary controllability problem for thewave equation. The control is assumed to act on a subset of the boundary ofthe domain where the solutions are defined.3.1 IntroductionLet Ω be a bounded open set of R N with boundary Γ of class C 2 and Γ 0be an open nonempty subset of Γ. Given T > 0 consider the following nonhomogeneouswave equation:⎧⎨ u ′′ − ∆u = 0in (0, T ) × Ωu = f1 Γ0 (x)on (0, T ) × Γ(78)⎩u(0, · ) = u 0 , u ′ (0, · ) = u 1 in Ω.In (78) u = u(t, x) is the state and f = f(t, x) is a control function whichacts on Γ 0 . We aim at changing the dynamics of the system by acting on Γ 0 .3.2 Existence and uniqueness of solutionsThe following theorem is a consequence of the classical results of existence anduniqueness of solutions of nonhomogeneous evolution equations. Full detailsmay be found in [46] and [68].

34 Controllability of Partial Differential EquationsTheorem 3.1 For any f ∈ L 2 ((0, T ) × Γ 0 ) and (u 0 , u 1 ) ∈ L 2 (Ω) × H −1 (Ω)equation (78) has a unique weak solution defined by transposition(u, u ′ ) ∈ C([0, T ], L 2 (Ω) × H −1 (Ω)).Moreover, the map {u 0 , u 1 , f} → {u, u ′ } is linear and there exists C =C(T ) > 0 such that||(u, u ′ )|| L ∞ (0,T ;L 2 (Ω)×H −1 (Ω)) ≤≤ C ( )||(u 0 , u 1 (79))|| L 2 (Ω)×H −1 (Ω) + ||f|| L 2 ((0,T )×Γ 0) .Remark 3.1 The wave equation is reversible in time. Hence, we may solve(78) for t ∈ (0, T ) by considering final data at t = T instead of initial data att = 0. □3.3 Controllability problemsLet T > 0 and define, for any initial data (u 0 , u 1 ) ∈ L 2 (Ω) × H −1 (Ω), the setof reachable statesR(T ; (u 0 , u 1 )) = {(u(T ), u ′ (T )) : u solution of (78) with f ∈ L 2 ((0, T ) × Γ 0 )}.Remark that, for any (u 0 , u 1 ) ∈ L 2 (Ω) × H −1 (Ω), R(T ; (u 0 , u 1 )) is a convexsubset of L 2 (Ω) × H −1 (Ω).As in the previous chapter, several controllability problems may be addressed.Definition 3.1 System (78) is approximately controllable in time T if,for every initial data (u 0 , u 1 ) ∈ L 2 (Ω) × H −1 (Ω), the set of reachable statesR(T ; (u 0 , u 1 )) is dense in L 2 (Ω) × H −1 (Ω).Definition 3.2 System (78) is exactly controllable in time T if, for everyinitial data (u 0 , u 1 ) ∈ L 2 (Ω)×H −1 (Ω), the set of reachable states R(T ; (u 0 , u 1 ))coincides with L 2 (Ω) × H −1 (Ω).Definition 3.3 System (78) is null controllable in time T if, for everyinitial data (u 0 , u 1 ) ∈ L 2 (Ω)×H −1 (Ω), the set of reachable states R(T ; (u 0 , u 1 ))contains the element (0, 0).Remark 3.2 In the definitions of approximate and exact controllability it issufficient to consider the case (u 0 , u 1 ) ≡ 0 sinceR(T ; (u 0 , u 1 )) = R(T ; (0, 0)) + S(T )(u 0 , u 1 ),where (S(t)) t∈R is the group of isometries generated by the wave equation inL 2 (Ω) × H −1 (Ω) with homogeneous Dirichlet boundary conditions. □

S. Micu and E. Zuazua 35Moreover, in view of the reversibility of the system we haveProposition 3.1 System (78) is exactly controllable if and only if it is nullcontrollable.Proof: Evidently, exact controllability implies null controllability.Let us suppose now that (0, 0) ∈ R(T ; (u 0 , u 1 )) for any (u 0 , u 1 ) ∈ L 2 (Ω) ×H −1 (Ω). It follows that any initial data in L 2 (Ω) × H −1 (Ω) can be driven to(0, 0) in time T . From the reversibility of the wave equation we deduce thatany state in L 2 (Ω) × H −1 (Ω) can be reached in time T by starting from (0, 0).This means that R(T, (0, 0)) = L 2 (Ω) × H −1 (Ω) and the exact controllabilityproperty holds as a consequence of Remark 3.2. □The previous Proposition guarantees that (78) is exactly controllable if andonly if, for any (u 0 , u 1 ) ∈ L 2 (Ω)×H −1 (Ω) there exists f ∈ L 2 ((0, T )×Γ 0 ) suchthat the corresponding solution (u, u ′ ) of (78) satisfiesu(T, · ) = u ′ (T, · ) = 0. (80)Remark 3.3 The following facts indicate the close connections between thecontrollability properties and some of the main features of hyperbolic equations:• Since the wave equation is time-reversible and does not have any regularizingeffect, the exact controllability property is very likely to hold. Nevertheless,as we have said before, the exact controllability property failsand the approximate controllability one holds in some situations. Thisis very closely related to the geometric properties of the subset Γ 0 of theboundary Γ where the control is applied.• The wave equation is the prototype of partial differential equation withfinite speed of propagation. Therefore, one cannot expect the previouscontrollability properties to hold unless the control time T is sufficientlylarge. □3.4 Variational approachLet us first deduce a necessary and sufficient condition for the exact controllabilityof (78). As in the previous chapter, by 〈 · , · 〉 1,−1we shall denote theduality product between H0 1 (Ω) and H −1 (Ω).For any (ϕ 0 T , ϕ1 T ) ∈ H1 0 (Ω)×L 2 (Ω) let (ϕ, ϕ ′ ) be the solution of the followingbackward wave equation⎧⎨ ϕ ′′ − ∆ϕ = 0in (0, T ) × Ωϕ | ∂Ω = 0on (0, T ) × ∂Ω (81)⎩ϕ(T, · ) = ϕ 0 T , ϕ′ (T, · ) = ϕ 1 T . in Ω.

36 Controllability of Partial Differential EquationsLemma 3.1 The initial data (u 0 , u 1 ) ∈ L 2 (Ω)×H −1 (Ω) is controllable to zeroif and only if there exists f ∈ L 2 ((0, T ) × Γ 0 ) such that∫ T0∫Γ 0∂ϕ∂n∫Ωfdσdt + u 0 ϕ ′ (0)dx − 〈 u 1 , ϕ(0) 〉 1,−1 = 0 (82)for all (ϕ 0 T , ϕ1 T ) ∈ H1 0 (Ω) × L 2 (Ω) and where (ϕ, ϕ ′ ) is the solution of thebackward wave equation (81)Proof: Let us first suppose that ( u 0 , u 1) , ( ϕ 0 T , ϕ1 T)∈ D(Ω) × D(Ω), f ∈D((0, T ) × Γ 0 ) and let u and ϕ be the (regular) solutions of (78) and (81)respectively.Multiplying the equation of u by ϕ and integrating by parts one obtainsHence,∫ T0∫Γ 00 =∫ T0∫ T∫∫ϕ (u ′′ − ∆u) dxdt = (ϕu ′ − ϕ ′ u) dx| T 0 +ΩΩ∫ (− ∂uΓ ∂n ϕ + ∂ϕ ) ∫ T ∫∂n u ∂ϕdσdt =0 Γ 0∂n udσdt+∫[ϕ(0)u ′ (0) − ϕ ′ (0)u(0)] dx+0∫+ [ϕ(T )u ′ (T ) − ϕ ′ (T )u(T )] dx −Ω∂ϕ∂n∫Ωudσdt+ [ϕ0T u ′ (T ) − ϕ 1 T u(T ) ] ∫[dx− ϕ(0)u 1 − ϕ ′ (0)u 0] dx = 0.ΩBy a density argument we deduce that for any (u 0 , u 1 ) ∈ L 2 (Ω) × H −1 (Ω)and (ϕ 0 T , ϕ1 T ) ∈ H1 0 (Ω) × L 2 (Ω),∫=Ω∫ T∫∂ϕ0 Γ 0∂n udσdt =u(T )ϕ 1 T dx + 〈 ∫u ′ (T ), ϕ 0 〉T + u 0 ϕ ′ (0)dx − 〈 u 1 , ϕ(0) 〉 (83). 1,−1 1,−1Now, from (83), it follows immediately that (82) holds if and only if (u 0 , u 1 )is controllable to zero. The proof finishes. □As in the previous chapter we introduce the duality product〈(ϕ 0 , ϕ 1) , ( u 0 , u 1)〉 ∫= u 0 ϕ 1 dx − 〈 u 1 , ϕ 1〉 1,−1for all ( ϕ 0 , ϕ 1) ∈ H 1 0 (Ω) × L 2 (Ω) and ( u 0 , u 1) ∈ L 2 (Ω) × H −1 (Ω).ΩΩΩ

S. Micu and E. Zuazua 37For any (ϕ 0 , ϕ 1 ) ∈ H0 1 (Ω) × L 2 (Ω) let (ϕ, ϕ ′ ) be the finite energy solutionof the following wave equation⎧⎨ ϕ ′′ − ∆ϕ = 0in (0, T ) × Ωϕ | ∂Ω = 0in (0, T ) × ∂Ω(84)⎩ϕ(0, · ) = ϕ 0 , ϕ ′ (0, · ) = ϕ 1 . in Ω.Since the wave equation generates a group of isometries in H 1 0 (Ω) × L 2 (Ω),Lemma 3.1 may be reformulated in the following way:Lemma 3.2 The initial data (u 0 , u 1 ) ∈ L 2 (Ω)×H −1 (Ω) is controllable to zeroif and only if there exists f ∈ L 2 ((0, T ) × Γ 0 ) such that∫ T ∫∂ϕ∂n fdσdt + 〈( ϕ 0 , ϕ 1) , ( u 0 , u 1)〉 = 0, (85)0Γ 0for all (ϕ 0 , ϕ 1 ) ∈ H 1 0 (Ω) × L 2 (Ω) and where ϕ is the solution of (84).Once again, (85) may be seen as an optimality condition for the criticalpoints of the functional J : H0 1 (Ω) × L 2 (Ω) −→ R, defined byJ (ϕ 0 , ϕ 1 ) = 1 ∫ T ∫ ∣ ∣∣∣2∂ϕ2 0 Γ 0∂n ∣ dσdt + 〈( ϕ 0 , ϕ 1) , ( u 0 , u 1)〉 , (86)where ϕ is the solution of (84) with initial data (ϕ 0 , ϕ 1 ) ∈ H 1 0 (Ω) × L 2 (Ω).We haveTheorem 3.2 Let (u 0 , u 1 ) ∈ L 2 (Ω) × H −1 (Ω) and suppose that ( ̂ϕ 0 , ̂ϕ 1 ) ∈H0 1 (Ω) × L 2 (Ω) is a minimizer of J . If ̂ϕ is the corresponding solution of (84)with initial data ( ̂ϕ 0 , ̂ϕ 1 ) then f = ∂ ̂ϕ∂n | Γ0is a control which leads (u 0 , u 1 ) tozero in time T .Proof: Since, by assumption, J achieves its minimum at ( ̂ϕ 0 , ̂ϕ 1 ), the followingrelation holds1 (0 = lim J (( ̂ϕ 0 , ̂ϕ 1 ) + h(ϕ 0 , ϕ 1 )) − J ( ̂ϕ 0 , ̂ϕ 1 ) ) =h→0 h∫ T ∫∂ ̂ϕ ∂ϕ=∂n ∂n∫Ωdσdt + u 0 ϕ 1 dx− < u 1 , ϕ 0 > 1,−10Γ 0for any (ϕ 0 , ϕ 1 ) ∈ H0 1 (Ω) × L 2 (Ω) where ϕ is the solution of (84).From Lemma 3.2 it follows that f = ∂ ̂ϕ∂n | Γ0is a control which leads theinitial data (u 0 , u 1 ) to zero in time T . □Let us now give a general condition which ensures the existence of a minimizerfor J .

38 Controllability of Partial Differential EquationsDefinition 3.4 Equation (84) is observable in time T if there exists a positiveconstant C 1 > 0 such that the following inequality is verifiedC 1 ‖ ( ϕ 0 , ϕ 1) ∫ T ∫ ∣ ∣∣∣2‖ 2 H0 1(Ω)×L2 (Ω) ≤ ∂ϕ0 Γ 0∂n ∣ dσdt, (87)for any ( ϕ 0 , ϕ 1) ∈ H 1 0 (Ω) × L 2 (Ω) where ϕ is the solution of (84) with initialdata (ϕ 0 , ϕ 1 ).Inequality (87) is called observation or observability inequality. Accordingto it, when it holds, the quantity ∫ ∣ T ∣∣ ∂ϕ0∫Γ 0 ∂n∣ 2 dσdt (the observed quantity)which depends only on the trace of ∂ϕ∂n on (0, T )×Γ 0, uniquely determinesthe solution of (84).Remark 3.4 One may show that there exists a constant C 2 > 0 such that∫ T0∫ ∣ ∣∣∣2∂ϕΓ 0∂n ∣ dσdt ≤ C 2 ‖ ( ϕ 0 , ϕ 1) ‖ 2 H0 1(Ω)×L2 (Ω)(88)for all (ϕ 0 , ϕ 1 ) ∈ H0 1 (Ω) × L 2 (Ω) and ϕ solution of (84).Inequality (88) may be obtained by multiplier techniques (see [41] or [45]).Remark that, (88) says that ∂ϕ∂n | Γ0∈ L 2 ((0, T ) × Γ 0 ) which is a “hidden” regularityresult, that may not be obtained by classical trace results. □Let us show that (87) is a sufficient condition for the exact controllabilityproperty to hold.Theorem 3.3 Suppose that (84) is observable in time T and let (u 0 , u 1 ) ∈L 2 (Ω) × H −1 (Ω). The functional J defined by (86) has an unique minimizer( ̂ϕ 0 , ̂ϕ 1 ) ∈ H 1 0 (Ω) × L 2 (Ω).Proof: It is easy to see that J is continuous and convex. The existence ofa minimum is ensured if we prove that J is also coercive i.e.lim J||(ϕ 0 ,ϕ 1 )|| H 10×L 2 →∞ (ϕ0 , ϕ 1 ) = ∞. (89)The coercivity of the functional J follows immediately from (87). Indeed,( ∫≥ 1 T ∫ ∣ ∣∣∣ ∂ϕ2 0 Γ 0∂n ∣2J (ϕ 0 , ϕ 1 ) ≥− ||(u 0 , u 1 )|| H 10 (Ω)×L 2 (Ω)||(ϕ 0 , ϕ 1 )|| L2 (Ω)×H −1 (Ω))≥

S. Micu and E. Zuazua 39≥ C 12 ||(ϕ0 , ϕ 1 )|| 2 L 2 (Ω)×H −1 (Ω) − 1 2 ||(u0 , u 1 )|| H 10 (Ω)×L 2 (Ω)||(ϕ 0 , ϕ 1 )|| L 2 (Ω)×H −1 (Ω).It follows from Theorem 2.3 that J has a minimizer ( ̂ϕ 0 , ̂ϕ 1 ) ∈ H0 1 (Ω) ×L 2 (Ω).As in the proof of Theorem 2.4, it may be shown that J is strictly convexand therefore it achieves its minimum at a unique point. □Theorems 3.2 and 3.3 guarantee that, under the hypothesis (87), system(78) is exactly controllable. Moreover, a control may be obtained from thesolution of the homogeneous system (81) with the initial data minimizing thefunctional J . Hence, the controllability is reduced to a minimization problem.This is very useful both from the theoretical and numerical point of view.As in Proposition 2.2 the control obtained by minimizing the functional Jhas minimal L 2 -norm:Proposition 3.2 Let f = ∂ ̂ϕ∂n | Γ0be the control given by minimizing the functionalJ . If g ∈ L 2 ((0, T ) × Γ 0 ) is any other control driving to zero the initialdata (u 0 , u 1 ) in time T , then||f|| L2 ((0,T )×Γ 0) ≤ ||g|| L2 ((0,T )×Γ 0). (90)Proof: It is similar to the proof of Property 2.2. We omit the details.□3.5 Approximate controllabilityLet us now briefly present and discuss the approximate controllability property.Since many aspects are similar to the interior controllability case we only givethe general ideas and let the details to the interested reader.Let ε > 0 and (u 0 , u 1 ), (z 0 , z 1 ) ∈ L 2 (Ω) × H −1 (Ω). We are looking for acontrol function f ∈ L 2 ((0, T )×Γ 0 ) such that the corresponding solution (u, u ′ )of (78) satisfies||(u(T ), u ′ (T )) − (z 0 , z 1 )|| L2 (Ω)×H −1 (Ω) ≤ ε. (91)Recall that, (78) is approximately controllable if, for any ε > 0 and (u 0 , u 1 ),(z 0 , z 1 ) ∈ L 2 (Ω) × H −1 (Ω), there exists f ∈ L 2 ((0, T ) × Γ 0 ) such that (91) isverified.By Remark 3.2, it is sufficient to study the case (u 0 , u 1 ) = (0, 0). Therefore,in this section only zero initial data (u 0 , u 1 ) will be considered.The variational approach may be also used for the study of the approximatecontrollability property.

40 Controllability of Partial Differential EquationsTo see this, define the functional J ε : H 1 0 (Ω) × L 2 (Ω) → R,= 1 2∫ T0J ε (ϕ 0 , ϕ 1 ) =∫ ∣ ∣∣∣2∂ϕΓ 0∂n ∣ dxdt + 〈( ϕ 0 , ϕ 1) , ( z 0 , z 1)〉 + ε||(ϕ 0 , ϕ 1 )|| H 10 ×L 2, (92)where ϕ is the solution of (81) with initial data (ϕ 0 , ϕ 1 ) ∈ H 1 0 (Ω) × L 2 (Ω).The following theorem shows how the functional J ε may be used to studythe approximate controllability property. In fact, as in the exact controllabilitycase, the existence of a minimum of the functional J ε implies the existence ofan approximate control.Theorem 3.4 Let ε > 0, (z 0 , z 1 )∈ L 2 (Ω) × H −1 (Ω). Suppose that ( ̂ϕ 0 , ̂ϕ 1 ) ∈H 1 0 (Ω) × L 2 (Ω) is a minimizer of J ε . If ̂ϕ is the corresponding solution of (81)with initial data ( ̂ϕ 0 , ̂ϕ 1 ) thenf = ∂ ̂ϕ∂n ∣ ∣Γ 0(93)is an approximate control which leads the solution of (78) from the zero initialdata (u 0 , u 1 ) = (0, 0) to the state (u(T ), u ′ (T )) such that (91) is verified.Proof: It is similar to the proof of Theorem 3.4.□As we have seen, the exact controllability property of (78) is related to theobservation property (65) of system (81). An unique continuation property ofthe solutions of (81) plays a similar role in the context of approximate controllabilityand guarantees the existence of a minimizer of J ε . More precisely, wehaveTheorem 3.5 The following properties are equivalent:1. Equation (78) is approximately controllable.2. The following unique continuation principle holds for the solutions of (81)∂ϕ∂n ∣ ∣(0,T )×Γ 0= 0 ⇒ (ϕ 0 , ϕ 1 ) = (0, 0). (94)Proof: The proof of the fact that the approximate controllability propertyimplies the unique continuation principle (94) is similar to the correspondingone in Theorem 2.6 and we omit it.Let us prove that, if the unique continuation principle (94) is verified, (78)is approximately controllable. By Theorem 3.4 it is sufficient to prove that J ε

S. Micu and E. Zuazua 41defined by (92) has a minimum. The functional J ε is convex and continuous inH0 1 (Ω) × L 2 (Ω). Thus, the existence of a minimum is ensured if J ε is coercive,i. e.J ε ((ϕ 0 , ϕ 1 )) → ∞ when ||(ϕ 0 , ϕ 1 )|| H 10 ×L2 → ∞. (95)In fact we shall prove thatlim inf J ε(ϕ 0 , ϕ 1 )/||(ϕ 0 , ϕ 1 )|| H 1||(ϕ 0 ,ϕ 1 )|| H 10×L 2 →∞ 0 ×L2 ≥ ε. (96)Evidently, (96) implies (95) and the proof of the theorem is complete.In order to prove (96) let ( (ϕ 0 j , ϕ1 j )) j≥1 ⊂ H1 0 (Ω) × L 2 (Ω) be a sequence ofinitial data for the adjoint system with ‖ (ϕ 0 j , ϕ1 j ) ‖ H0 1 ×L2→ ∞. We normalizethem( ˜ϕ 0 j, ˜ϕ 1 j) = (ϕ 0 j, ϕ 1 j)/ ‖ (ϕ 0 j, ϕ 1 j) ‖ H 10 ×L 2,so that‖ ( ˜ϕ 0 j, ˜ϕ 1 j) ‖ H 10 ×L2= 1.On the other hand, let ( ˜ϕ j , ˜ϕ ′ j ) be the solution of (81) with initial data( ˜ϕ 0 j , ˜ϕ1 j ). ThenJ ε ((ϕ 0 j , ϕ1 j ))‖ (ϕ 0 j , ϕ1 j ) ‖ = 1 ∫ T ∫ ∣ ∣∣∣22 ‖ (ϕ0 j, ϕ 1 ∂ ˜ϕ jj) ‖0 Γ 0∂n ∣ dσdt + 〈 (z 0 , z 1 ), ( ˜ϕ 0 , ˜ϕ 1 ) 〉 + ε.The following two cases may occur:∫ T ∫ ∣ ∣∣∣2∂ ˜ϕ j1) lim infj→∞0 Γ 0∂n ∣ > 0. In this case we obtain immediately thatJ ε ((ϕ 0 j , ϕ1 j ))‖ (ϕ 0 j , ϕ1 j ) ‖ → ∞.∫ T ∫ ∣ ∣∣∣2∂ ˜ϕ j2) lim infj→∞0 Γ 0∂n ∣ = 0. In this case, since ( ˜ϕ 0 j , )˜ϕ1 j is bounded inj≥1H0 1 × L 2 , by extracting a subsequence we can guarantee that ( ˜ϕ 0 j , ˜ϕ1 j ) j≥1converges weakly to (ψ 0 , ψ 1 ) in H0 1 (Ω) × L 2 (Ω).Moreover, if (ψ, ψ ′ ) is the solution of (81) with initial data (ψ 0 , ψ 1 ) att = T , then ( ˜ϕ j , ˜ϕ ′ j ) j≥1 converges weakly to (ψ, ψ ′ ) in L 2 (0, T ; H 1 0 (Ω) ×L 2 (Ω)) ∩ H 1 (0, T ; L 2 (Ω) × H −1 (Ω)).By lower semi-continuity,∫ T ∫ ∣ ∣∣∣2∂ψ0 Γ 0∂n ∣∫ T ∫dσdt ≤ lim infj→∞0Γ 0∣ ∣∣∣ ∂ ˜ϕ j∂n2∣ dσdt = 0

42 Controllability of Partial Differential Equationsand therefore ∂ψ∂n = 0 on Γ 0 × (0, T ).From the unique continuation principle we obtain that (ψ 0 , ψ 1 ) = (0, 0)and consequently,( ˜ϕ 0 j, ˜ϕ 1 j) ⇀ (0, 0) weakly in H 1 0 (Ω) × L 2 (Ω).Hencelim infj→∞J ε ((ϕ 0 j , ϕ1 j ))‖ (ϕ 0 j , ≥ lim infϕ1 j ) ‖ [ε + 〈 (z 0 , z 1 ), ( ˜ϕ 0 , ˜ϕ 1 ) 〉 ] = ε,j→∞and (96) follows.□As mentioned in the previous section, when approximate controllabilityholds, the following (apparently stronger) statement also holds (see [71]):Theorem 3.6 Let E be a finite-dimensional subspace of L 2 (Ω) × H −1 (Ω) andlet us denote by π E the corresponding orthogonal projection. Then, if approximatecontrollability holds, for any ( u 0 , u 1) , ( z 0 , z 1) ∈ L 2 (Ω) × H −1 (Ω) andε > 0 there exists f ∈ L 2 ((0, T ) × Γ 0 ) such that the solution of (78) satisfies∥ ( u(T ) − z 0 , u t (T ) − z 1)∥ ∥L2 (Ω)×H −1 (Ω) ≤ ε; π E (u(T ), u t (T )) = π E(z 0 , z 1) .3.6 CommentsIn the last two sections we have presented some results concerning the exactand approximate controllability of the wave equation. The variational methodswe have used allow to reduce these properties to an observation inequality anda unique continuation principle for the adjoint homogeneous equation respectively.Let us briefly make some remarks concerning the proof of the unique continuationprinciples (75) and (94).Holmgren’s Uniqueness Theorem (see [33]) may be used to show that (75)and (94) hold if T is large enough. We refer to [45], chapter 1 and [13] for adiscussion of this problem. Consequently, approximate controllability holds ifT is large enough.The same results hold for wave equations with analytic coefficients too.However, the problem is not completely solved in the frame of the wave equationwith lower order potentials a ∈ L ∞ ((0, T ) × Ω) of the formu tt − ∆u + a(x, t)u = f1 ω in (0, T ) × Ω.Once again the problem of approximate controllability of this system is equivalentto the unique continuation property of its adjoint. We refer to Alinhac[1], Tataru [62] and Robbiano-Zuilly [54] for deep results in this direction.

S. Micu and E. Zuazua 43In the following chapter we shall prove the observability inequalities (65)and (87) in some simple one dimensional cases by using Fourier expansion ofsolutions. Other tools have been successfully used to prove these observabilityinequalities. Let us mention two of them.1. Multipliers techniques: Ho in [32] proved that if one considers subsetsof Γ of the formΓ 0 = Γ(x 0 ) = { x ∈ Γ : (x − x 0 ) · n(x) > 0 }for some x 0 ∈ R N and if T > 0 is large enough, the boundary observabilityinequality (87), that is required to solve the boundary controllabilityproblem, holds. The technique used consists of multiplying equation (84)by q · ∇ϕ and integrating by parts in (0, T ) × Ω. The multiplier q is anappropriate vector field defined in Ω. More precisely, q(x) = x − x 0 forany x ∈ Ω.Later on inequality (87) was proved in [45] for any T > T (x 0 ) = 2 ‖x−x 0 ‖ L∞ (Ω). This is the optimal observability time that one may deriveby means of multipliers. More recently Osses in [51] has introduced a newmultiplier which is basically a rotation of the previous one and he hasobtained a larger class of subsets of the boundary for which observabilityholds.Concerning the interior controllability problem, one can easily prove that(87) implies (65) when ω is a neighborhood of Γ(x 0 ) in Ω, i.e. ω = Ω ∩ Θwhere Θ is a neighborhood of Γ(x 0 ) in R n , with T > 2 ‖ x − x 0 ‖ L ∞ (Ω\ω)(see in [45], vol. 1).An extensive presentation and several applications of multiplier techniquesare given in [40] and [45].2. Microlocal analysis: C. Bardos, G. Lebeau and J. Rauch [7] provedthat, in the class of C ∞ domains, the observability inequality (65) holdsif and only if (ω, T ) satisfy the following geometric control condition in Ω:Every ray of geometric optics that propagates in Ω and is reflected on itsboundary Γ enters ω in time less than T . This result was proved by meansof microlocal analysis techniques. Recently the microlocal approach hasbeen greatly simplified by N. Burq [11] by using the microlocal defectmeasures introduced by P. Gerard [30] in the context of the homogenizationand the kinetic equations. In [11] the geometric control conditionwas shown to be sufficient for exact controllability for domains Ω of classC 3 and equations with C 2 coefficients.Other methods have been developed to address the controllability problemssuch as moment problems, fundamental solutions, controllability via stabiliza-

44 Controllability of Partial Differential Equationstion, etc. We will not present them here and we refer to the survey paper byD. L. Russell [55] for the interested reader.4 Fourier techniques and the observability ofthe 1D wave equationIn Chapters 2 and 3 we have shown that the exact controllability problem maybe reduced to the corresponding observability inequality. In this chapter we developin detail some techniques based on Fourier analysis and more particularlyon Ingham type inequalities allowing to obtain several observability results forlinear 1-D wave equations. We refer to Avdonin and Ivanov [3] for a completepresentation of this approach.4.1 Ingham’s inequalitiesIn this section we present two inequalities which have been successfully used inthe study of many 1-D control problems and, more precisely, to prove observationinequalities. They generalize the classical Parseval’s equality for orthogonalsequences. Variants of these inequalities were studied in the works of Paleyand Wiener at the beginning of the past century (see [53]). The main inequalitywas proved by Ingham (see [37]) who gave a beautiful and elementary proof(see Theorems 4.1 and 4.2 below). Since then, many generalizations have beengiven (see, for instance, [6], [58], [4] and [38]).Theorem 4.1 (Ingham [37]) Let (λ n ) n∈Z be a sequence of real numbers andγ > 0 be such thatλ n+1 − λ n ≥ γ > 0, ∀n ∈ Z. (97)For any real T withT > π/γ (98)there exists a positive constant C 1 = C 1 (T, γ) > 0 such that, for any finitesequence (a n ) n∈Z ,∑C 1 | a n | 2 ≤n∈Z∫ T−T∣ ∑ ∣∣∣∣2 a n e iλnt dt. (99)∣Proof: We first reduce the problem to the case T = π and γ > 1. Indeed, if Tand γ are such that T γ > π, then∫ T−T∣ ∑ ∣∣∣∣2a∣ n e iλnt dt = T πn∫ π−πnn∈Z∣ ∑ ∣∣∣∣2a∣ n e i T λnπ s ds = T ∫ ππ−π∣ ∑ ∣∣∣∣2 a∣ n e iµns dsn

S. Micu and E. Zuazua 45where µ n = T λ n /π. It follows that µ n+1 − µ n = T (λ n+1 − λ n ) /π ≥ γ 1 :=T γ/π > 1.We prove now that there exists C 1 ′ > 0 such that∑∫ ∣πC 1′ | a n | 2 ∑ ∣∣∣∣2≤a∣ n e iµnt dt.Define the functionn∈Zh : R → R, h(t) =−πn∈Zand let us compute its Fourier transform K(ϕ),K(ϕ) =∫ π−πh(t)e itϕ dt ={ cos (t/2) if | t |≤ π0 if | t |> π∫ ∞−∞h(t)e itϕ dt =4 cos πϕ1 − 4ϕ 2 .On the other hand, since 0 ≤ h(t) ≤ 1 for any t ∈ [−π, π], we have that∣ ∑ ∣∣∣∣2 ∫ ∣πa∣ n e iµnt ∑ ∣∣∣∣2dt ≥ h(t)a∣ n e iµnt dt = ∑ a n ā m K(µ n − µ m ) =n,m∫ π−πn−πn= K(0) ∑ n| a n | 2 + ∑ n≠ma n ā m K(µ n − µ m ) ≥≥ 4 ∑ n| a n | 2 − 1 2= 4 ∑ n∑ (| an | 2 + | a m | 2) | K(µ n − µ m ) |=n≠m| a n | 2 − ∑ n| a n | 2 ∑ m≠nRemark that∑| K(µ n − µ m ) |≤ ∑ 44 | µ n − µ m | 2 −1 ≤ ∑ m≠nm≠nm≠n= 8 ∑ r≥1Hence,If we take14γ1 2r2 − 1 ≤ 8 ∑ 1γ12 4r 2 − 1 = 8 1γ 2 r≥11 2∫ π−π∣ ∑ ∣∣∣∣2 a n e iµnt dt ≥∣nC 1 = T πthe proof is concluded. □| K(µ n − µ m ) | .∑r≥1(4 − 4 ) ∑γ12 n(4 − 4 )γ12 = 4π )(T 2 − π2T γ 244γ 2 1 | n − m |2 −1 =( 12r − 1 − 1 )= 42r + 1 γ12 .| a n | 2 .

46 Controllability of Partial Differential EquationsTheorem 4.2 Let (λ n ) n∈Z be a sequence of real numbers and γ > 0 be suchthatλ n+1 − λ n ≥ γ > 0, ∀n ∈ Z. (100)For any T > 0 there exists a positive constant C 2 = C 2 (T, γ) > 0 such that,for any finite sequence (a n ) n∈Z ,∫ T−T∣ ∑ ∣∣∣∣2∑a n e iλnt dt ≤ C 2 | a n | 2 . (101)∣nProof: Let us first consider the case T γ ≥ π/2. As in the proof of the previoustheorem, we can reduce the problem to T = π/2 and γ ≥ 1. Indeed,∫ T−T∣ ∑ ∣∣∣∣2a∣ n e iλnt dt = 2T πn∫ π2− π 2n∣ ∑ ∣∣∣∣2 a∣ n e iµns dswhere µ n = 2T λ n /π. It follows that µ n+1 − µ n = 2T (λ n+1 − λ n ) /π ≥ γ 1 :=2T γ/π ≥ 1.Let h be the function introduced in the proof of Theorem 4.1. Since √ 2/2 ≤h(t) ≤ 1 for any t ∈ [−π/2, π/2] we obtain that∫ π2− π 2∫ π≤ 2−π∑∣n∣ ∣∣∣∣2 ∫ πa n e iµnt 2dt ≤ 2− π 2n∣ ∑ ∣∣∣∣2h(t)a n e iµnt dt ≤∣∣ ∑ ∣∣∣∣2h(t)a∣ n e iµnt dt = 2 ∑ a n ā m K (µ n − µ m ) =n,mnn= 8 ∑ n| a n | 2 +2 ∑ n≠ma n ā m K (µ n − µ m ) ≤≤ 8 ∑ n| a n | 2 + ∑ n≠m(| an | 2 + | a m | 2) | K (µ n − µ m ) | .As in the proof of Theorem 4.1 we obtain that∑m≠n| K(µ n − µ m ) |≤ 4γ12 .∫ π2Hence,− π 2∣ ∑ ∣∣∣∣2a∣ n e iµnt dt ≤ 8 ∑nn| a n | 2 + 8 ∑(γ12 | a n | 2 ≤ 8 1 + 1 ) ∑γ 2 n1 n| a n | 2

S. Micu and E. Zuazua 47and (101) follows with C 2 = 8 ( 4T 2 /(π 2 ) + 1/γ 2) .∫ T−TWhen T γ < π/2 we have that∣ ∑ ∣ ∫a n e iλnt ∣∣2 1 T γdt =γ −T γ∣ ∑ ∣ ∫a n e i λn γ s ∣∣2 1 π/2ds ≤γ−π/2∣ ∑ a n e i λn γs ∣ ∣∣2ds.Since λ n+1 /γ − λ n /γ ≥ 1 from the analysis of the previous case we obtainthat∫ ∣π2∑ ∣∣∣∣2a n e i λn γ s ds ≤ 16 ∑ | a n | 2 .− π ∣2 nnHence, (101) is proved withand the proof concludes. □{( 4T2C 2 = 8 maxπ 2 + 1 )γ 2 ,}2γRemark 4.1 • Inequality (101) holds for all T > 0. On the contrary,inequality (99) requires the length T of the time interval to be sufficientlylarge. Note that, when the distance between two consecutive exponentsλ n , the gap, becomes small the value of T must increase proportionally.• In the first inequality (99) T depends on the minimum γ of the distancesbetween every two consecutive exponents (gap). However, as we shall seein the next theorem, only the asymptotic distance as n → ∞ betweenconsecutive exponents really matters to determine the minimal controltime T . Note also that the constant C 1 in (99) degenerates when T goesto π/γ.• In the critical case T = π/γ inequality (99) may hold or not, dependingon the particular family of exponential functions. For instance, if λ n = nfor all n ∈ Z, (99) is verified for T = π. This may be seen immediately byusing the orthogonality property of the complex exponentials in (−π, π).Nevertheless, if λ n = n − 1/4 and λ −n = −λ n for all n > 0, (99) failsfor T = π (see, [37] or [64]). □As we have said before, the length 2T of the time interval in (99) does notdepend on the smallest distance between two consecutive exponents but on theasymptotic gap defined bylim inf | λ n+1 − λ n |= γ ∞ . (102)|n|→∞

48 Controllability of Partial Differential EquationsAn induction argument due to A. Haraux (see [31]) allows to give a resultsimilar to Theorem 4.1 above in which condition (97) for γ is replaced by asimilar one for γ ∞ .Theorem 4.3 Let (λ n ) n∈Z be an increasing sequence of real numbers such thatλ n+1 − λ n ≥ γ > 0 for any n ∈ Z and let γ ∞ > 0 be given by (102). For anyreal T withT > π/γ ∞ (103)there exist two positive constants C 1 , C 2 > 0 such that, for any finite sequence(a n ) n∈Z ,∑C 1 | a n | 2 ≤n∈Z∫ T−T∣ ∑ ∣∣∣∣2∑a n e iλnt dt ≤ C 2 | a n | 2 . (104)∣n∈ZRemark 4.2 When γ ∞ = γ, the sequence of Theorem 4.3 satisfies λ n+1 −λ n ≥γ ∞ > 0 for all n ∈ Z and we can then apply Theorems 4.1 and 4.2. However,in general, γ ∞ < γ and Theorem 4.3 gives a sharper bound on the minimaltime T needed for (104) to hold.Note that the existence of C 1 and C 2 in (104) is a consequence of Kahane’stheorem (see [40]). However, if our purpose were to have an explicit estimateof C 1 or C 2 in terms of γ, γ ∞ then we would need to use the constructiveargument below. It is important to note that these estimates depend stronglyalso on the number of eigenfrequencies λ that fail to fulfill the gap conditionwith the asymptotic gap γ ∞ .Proof of Theorem 4.3: The second inequality from (104) follows immediatelyby using Theorem 4.2. To prove the first inequality (104) we follow theinduction argument due to Haraux [31].Note that for any ε 1 > 0, there exists N = N(ε 1 ) ∈ N ∗ such thatn∈Z|λ n+1 − λ n | ≥ γ ∞ − ε 1 for any |n| > N. (105)We begin with the function f 0 (t) = ∑ |n|>N a ne iλnt and we add the missingexponentials one by one. From (105) we deduce that Theorems 4.1 and 4.2may be applied to the family ( e iλnt) |n|>N for any T > π/(γ ∞ − ε 1 )∑C 1 | a n | 2 ≤∫ Tn>N−T| f 0 (t) | 2 dt ≤ C 2∑n>N| a n | 2 . (106)Let now f 1 (t) = f 0 + a N e i λ N t = ∑ |n|>N a ne iλnt + a N e i λ N t . Without lossof generality we may suppose that λ N = 0 (since we can consider the functionf 1 (t)e −iλ N t instead of f 1 (t)).

S. Micu and E. Zuazua 49Let ε > 0 be such that T ′ = T − ε > π/γ ∞ . We have∫ ε0(f 1 (t + η) − f 1 (t)) dη = ∑ n>Na n( eiλ nε − 1iλ nApplying now (106) to the function h(t) =obtain that:∑C 1 e iλnε − 1∣ − εiλ n∣+2εn>NMoreover,:2|a n | 2 ≤∫ T′∣∫ ∣∣∣ ε−T ′ 0)− ε e iλnt , ∀t ∈ [0, T ′ ].∫ ε0(f 1 (t + η) − f 1 (t)) dη we2(f 1 (t + η) − f 1 (t)) dη∣ dt. (107)∣∣e iλnε − 1 − iλ n ε ∣ ∣ 2 = |cos(λ n ε) − 1| 2 + |sin(λ n ε) − λ n ε| 2 =⎧( )= 4sin 4 λn ε⎨+ (sin(λ n ε) − λ n ε) 2 ≥2⎩4 ( λ nε) 4π , if |λn |ε ≤ π(λ n ε) 2 , if |λ n |ε > π.Finally, taking into account that |λ n | ≥ γ, we obtain that,e iλnε − 1∣ iλ n− ε∣We return now to (107) and we get that:∫ T′0∑ε 2 C 1 |a n | 2 ≤n>N∫ T′On the other hand∣∫ ∣∣∣ ε2(f 1 (t + η) − f 1 (t)) dη∣ dt ≤−T ′02≥ cε 2 .∣∫ ∣∣∣ ε2(f 1 (t + η) − f 1 (t)) dη∣ dt. (108)−T ′∫ T′ ∫ εε−T ′ 0|f 1 (t + η) − f 1 (t)| 2 dηdt ≤∫ T′ ∫ ε≤ 2ε(|f 1 (t + η)| 2 + |f 1 (t)| 2) ∫ Tdηdt ≤ 2ε 2 |f 1 (t)| 2 dt+−T ′ 0−T ′∫ ε ∫ T′0−T ′ |f 1 (t + η)| 2 dtdη = 2ε 2 ∫ T≤ 2ε 2 ∫ T−T|f 1 (t)| 2 dt + 2ε∫ ε ∫ T0−T ′ |f 1 (t)| 2 dt+2ε−T∫ ε ∫ T ′ +η0−T ′ +η|f 1 (s)| 2 dsdη∫ T|f 1 (s)| 2 dsdη ≤ 4ε 2 |f 1 (t)| 2 dt.−T

50 Controllability of Partial Differential EquationsFrom (108) it follows that∑C 1 |a n | 2 ≤n>NOn the other hand|a N | 2 =∣ f 1(t) − ∑ ∣ ∣∣∣∣2a n e iλntn>N⎛≤ 1 ⎝T≤ 1 T∫ T−T∫ T−T= 1 ∫ T2T −T∫ T|f 1 (t)| 2 dt +−T( ∫ T≤ 1 TFrom (109) we get that−T|f 1 (t)| 2 dt + C 0 2(1 + C 2C 1) ∫ T∑C 1 |a n | 2 ≤n≥N|f 1 (t)| 2 dt. (109)∣ f 1(t) − ∑ ∣ ∣∣∣∣2a n e iλnt dt ≤n>N∣ ⎞ ∑ ∣∣∣∣2 a n e iλnt ⎠ dt ≤∣−T∫ T−Tn>N)∑|a n | 2 ≤n>N|f 1 (t)| 2 dt.|f 1 (t)| 2 dt.Repeating this argument we may add all the terms a n e iλnt , |n| ≤ N andwe obtain the desired inequalities. □4.2 Spectral analysis of the wave operatorThe aim of this section is to give the Fourier expansion of solutions of the 1-Dlinear wave equation⎧⎨⎩ϕ ′′ − ϕ xx + αϕ = 0, x ∈ (0, 1), t ∈ (0, T )ϕ(t, 0) = ϕ(t, 1) = 0, t ∈ (0, T )ϕ(0) = ϕ 0 , ϕ ′ (0) = ϕ 1 , x ∈ (0, 1)where α is a real nonnegative number.To do this let us first remark that (110) may be written as⎧⎪⎨⎪⎩ϕ ′ = zz ′ = ϕ xx − αϕϕ(t, 0) = ϕ(t, 1) = 0ϕ(0) = ϕ 0 , z(0) = ϕ 1 .(110)

S. Micu and E. Zuazua 51Nextly, denoting Φ = (ϕ, z), equation (110) is written in the followingabstract Cauchy form: { Φ ′ + AΦ = 0Φ(0) = Φ 0 (111).The differential operator A from (111) is the unbounded operator in H =L 2 (0, 1) × H −1 (0, 1), A : D(A) ⊂ H → H, defined byD(A) = H 1 0 (0, 1) × L 2 (0, 1)A(ϕ, z) = (−z, −∂ 2 xϕ + αϕ) =( )0 − 1 ϕ−∂x 2 + α 0)(z(112)where the Laplace operator −∂ 2 x is an unbounded operator defined in H −1 (0, 1)with domain H 1 0 (0, 1):−∂ 2 x : H 1 0 (0, 1) ⊂ H −1 (0, 1) → H −1 (0, 1),〈−∂ 2 xϕ, ψ〉 = ∫ 10 ϕ xψ x dx, ∀ϕ, ψ ∈ H 1 0 (0, 1).Remark 4.3 The operator A is an isomorphism from H0 1 (0, 1) × L 2 (0, 1) toL 2 (0, 1) × H −1 (0, 1). We shall consider the space H0 1 (0, 1) with the inner productdefined by(u, v) H 10 (0,1) =∫ 1which is equivalent to the usual one.0(u x )(x)v x (x)dx + α∫ 10u(x)v(x)dx (113)Lemma 4.1 The eigenvalues of A are λ n = sgn(n)πi √ n 2 + α, n ∈ Z ∗ . Thecorresponding eigenfunctions are given by( 1)Φ n =λ nsin(nπx), n ∈ Z ∗ ,−1and form an orthonormal basis in H 1 0 (0, 1) × L 2 (0, 1).Proof: Let us first determine the eigenvalues of A. If λ ∈ C and Φ = (ϕ, z) ∈H0 1 (0, 1) × L 2 (0, 1) are such that AΦ = λΦ we obtain from the definition of Athat{ −z = λϕ−∂xϕ 2 (114)+ αϕ = λz.It is easy to see that⎧⎨⎩∂ 2 xϕ − αϕ = λ 2 ϕϕ(0) = ϕ(1) = 0ϕ ∈ C 2 [0, 1].(115)

52 Controllability of Partial Differential EquationsThe solutions of (115) are given byλ n = sgn(n)πi √ n 2 + α, ϕ n = c sin(nπx), n ∈ Z ∗where c is an arbitrary complex constant.Hence, the eigenvalues of A are λ n = sgn(n)πi √ n 2 + α, n ∈ Z ∗ and thecorresponding eigenfunctions areIt is easy to see thatΦ n =( 1)λ nsin(nπx), n ∈ Z ∗ .−1(∫ 1• ‖ Φ n ‖ 2 1=H0 1×L2 (n 2 +α)π(nπ cos(nπx)) 2 dx + α2∫ 10(sin(nπx)) 2 dx = 1• (Φ n , Φ m ) = 1(α + 1)∫ 10nmπ 2 ∫ 100(nπ cos(nπx)mπ cos(mπx)) dx +(sin(nπx) sin(mπx)) dx = δ nm .∫ 10)sin 2 (nπx)dx +Hence, (Φ n ) n∈Z ∗ is an orthonormal sequence in H0 1 (0, 1) × L 2 (0, 1).The completeness of (Φ n ) n∈Z ∗ in H0 1 (0, 1) × L 2 (0, 1) is a consequence of thefact that these are all the eigenfunctions of the compact skew-adjoint operatorA −1 . It follows that (Φ n ) n∈Z ∗ is an orthonormal basis in H0 1 (0, 1) × L 2 (0, 1).□Remark 4.4 Since (Φ n ) n∈Z ∗ is an orthonormal basis in H0 1 (0, 1)×L 2 (0, 1) andA is an isomorphism from H0 1 (0, 1) × L 2 (0, 1) to L 2 (0, 1) × H −1 (0, 1) it followsimmediately that (A(Φ n )) n∈Z ∗ is an orthonormal basis in L 2 (0, 1) × H −1 (0, 1).Moreover (λ n Φ n ) n∈Z ∗ is an orthonormal basis in L 2 (0, 1)×H −1 (0, 1). We havethat• Φ = ∑ n∈Z ∗ a n Φ n ∈ H 1 0 (0, 1) × L 2 (0, 1) if and only if ∑ n∈Z ∗ |a n| 2 < ∞.• Φ = ∑ n∈Z ∗ a n Φ n ∈ L 2 (0, 1)×H −1 (0, 1) if and only if ∑ n∈Z ∗ |a n| 2|λ n| 2 < ∞.□The Fourier expansion of the solution of (111) is given in the followingLemma.

S. Micu and E. Zuazua 53Lemma 4.2 The solution of (111) with the initial dataW 0 = ∑ n∈Z ∗ a n Φ n ∈ L 2 (0, 1) × H −1 (0, 1) (116)is given byW (t) = ∑ n∈Z ∗ a n e λnt Φ n . (117)4.3 Observability for the interior controllability of the 1-D wave equationConsider an interval J ⊂ [0, 1] with | J |> 0 and a real time T > 2. We addressthe following control problem discussed in 2: given (u 0 , u 1 ) ∈ H 1 0 (0, 1)×L 2 (0, 1)to find f ∈ L 2 ((0, T ) × J) such that the solution u of the problem⎧⎨⎩u ′′ − u xx = f1 J , x ∈ (0, 1), t ∈ (0, T )u(t, 0) = u(t, 1) = 0, t ∈ (0, T )u(0) = u 0 , u ′ (0) = u 1 , x ∈ (0, 1)(118)satisfiesu(T, · ) = u ′ (T, · ) = 0. (119)According to the developments of Chapter 2, the control problem can besolved if the following inequalities hold for any (ϕ 0 , ϕ 1 ) ∈ L 2 (0, 1) × H −1 (0, 1)∫ T ∫C 1 ‖ (ϕ 0 , ϕ 1 ) ‖ 2 L 2 ×H −1≤0J| ϕ(t, x) | 2 dxdt ≤ C 2 ‖ (ϕ 0 , ϕ 1 ) ‖ 2 L 2 ×H −1(120)where ϕ is the solution of the adjoint equation (110).In this section we prove (120) by using the Fourier expansion of the solutionsof (110). Similar results can be proved for more general potentials dependingon x and t by multiplier methods and sidewiese energy estimates [73] and alsousing Carleman inequalities [66], [67].Remark 4.5 In the sequel when (120) holds , for brevity, we will denote it asfollows:∫ T ∫‖ (ϕ 0 , ϕ 1 ) ‖ 2 L 2 (0,1)×H −1 (0,1) ≍ | ϕ(t, x) | 2 dxdt. (121)Theorem 4.4 Let T ≥ 2. There exist two positive constants C 1 and C 2 suchthat (120) holds for any (ϕ 0 , ϕ 1 ) ∈ L 2 (0, 1)×H −1 (0, 1) and ϕ solution of (110).0J

54 Controllability of Partial Differential EquationsProof: Firstly, we have thatHence,‖ (ϕ 0 , ϕ 1 ) ‖ 2 L 2 ×H −1= ∥ ∥∥∥∥A −1 ( ∑n∈Z ∗ a n Φ n )∥ ∥∥∥∥2H 1 0 ×L2 =∥ ∑ 1 ∥∥∥∥2=a∥ ninπ Φn = ∑ | a n | 2 1n 2 π 2 .n∈Z ∗ H0 1×L2 n∈Z ∗‖ (ϕ 0 , ϕ 1 ) ‖ 2 L 2 ×H −1= ∑ n∈Z ∗ | a n | 2 1n 2 π 2 . (122)On the other hand, since ϕ ∈ C([0, T ], L 2 (0, 1)) ⊂ L 2 ((0, T ) × (0, 1)), weobtain from Fubini’s Theorem that∫ T ∫∫ ∫ ∣T| w(t, x) | 2 ∑dxdt =a0 JJ 0 ∣ n e inπt 1 ∣∣∣∣2nπ sin(nπx) dtdx.n∈Z ∗Let first T = 2. From the orthogonality of the exponential functions inL 2 (0, 2) we obtain that∫J∫J∫ T0∣ ∑a n e inπt 1 ∣∣∣∣2∣nπ sin(nπx) dtdx = ∑ |a n | 2n 2 π 2 sinn∈Z ∗ n∈Z∫J2 (nπx)dx.∗If T ≥ 2, it is immediate that∫ T0∑ a n e inπt2∫sin(nπx)dtdx ≥∣ nπ∣n∈Z ∗J∫ 2≥ ∑ |a n | 2n 2 π 2 sinn∈Z∫J2 (nπx)dx.∗0∑ a n e inπt2sin(nπx)dtdx ≥∣ nπ∣n∈Z ∗On the other hand, by using the 2-periodicity in time of the exponentialsand the fact that there exists p > 0 such that 2(p − 1) ≤ T < 2p, it follows that∫J∫ T0∑ a n e inπtsin(nπx)∣ nπ∣n∈Z ∗2∫dtdx ≤ pJ= p ∑ |a n | 2n 2 π 2 sinn∈Z∫J2 (nπx)dx ≤ T + 22∗∫ 20∑ a n e inπt2sin(nπx)dtdx∣ nπ∣n∈Z ∗∑ |a n | 2n 2 πn∈Z∫J2 ∗sin 2 (nπx)dx.

S. Micu and E. Zuazua 55Hence, for any T ≥ 2, we have that∫ ∫ ∣T∑aJ 0 ∣ n e inπt 1 ∣∣∣∣2nπ sin(nπx) dxdt ≍ ∑ |a n | 2n 2 π 2 sinn∈Z ∗ n∈Z∫J2 (nπx)dx. (123)∗If we denote b n = ∫ J sin2 (nπx)dx thenIndeed,∫b n =JB = infn∈Z ∗ b n > 0. (124)sin 2 (nπx)dx = | J |2∫J− cos(2nπx)≥ | J |2 2 − 12 | n | π .Since 1/[2 | n | π] tends to zero when n tends to infinity, there exists n 0 > 0such thatb n ≥ | J |2 − 12 | n | π > | J | > 0, ∀ | n |> n 0 .4It follows thatB 1 =inf b n > 0 (125)|n|>n 0and B > 0 since b n > 0 for all n.Moreover, since b n ≤ |J| for any n ∈ Z ∗ , it follows from (123) thatB ∑ n∈Z ∗ | a n | 2n 2 π 2∫ T ∫≤0J| ϕ(t, x) | 2 dxdt ≤ |J| ∑ n∈Z ∗ | a n | 2 1n 2 π 2 . (126)Finally, (120) follows immediately from (122) and (126).□As a direct consequence of Theorem 4.4 the following controllability resultholds:Theorem 4.5 Let J ⊂ [0, 1] with | J |> 0 and a real T ≥ 2. For any (u 0 , u 1 ) ∈H 1 0 (0, 1) × L 2 (0, 1) there exists f ∈ L 2 ((0, T ) × J) such that the solution u ofequation (118) satisfies (119).Remark 4.6 In order to obtain (123) for T > 2, Ingham’s Theorem 4.1 couldalso be used. Indeed, the exponents are µ n = nπ and they satisfy the uniformgap condition γ = µ n+1 − µ n = π, for all n ∈ Z ∗ . It then follows from Ingham’sTheorem 4.1 that, for any T > 2π/γ = 2, we have (123).Note that the result may not be ontained in the critical case T = 2 by usingTheorems 4.1 and 4.2. The critical time T = 2 is reached in this case becauseof the orthogonality properties of the trigonometric polynomials e iπnt . □

56 Controllability of Partial Differential EquationsConsider now the equation⎧⎨⎩u ′′ − u xx + αu = f1 J , x ∈ (0, 1), t ∈ (0, T )u(t, 0) = u(t, 1) = 0, t ∈ (0, T )u(0) = u 0 , u ′ (0) = u 1 , x ∈ (0, 1)(127)where α is a positive real number.The controllability problem may be reduced once more to the proof of thefollowing fact:∫J∫ T0∣ ∑a∣ n e λnt 1 ∣∣∣∣2nπ sin(nπx) dtdx ≍ ∑ |a n | 2|λ n | 2 sinn∈Z ∗ n∈Z∫J2 (nπx)dx (128)∗where λ n = sgn(n)πi √ n 2 + α are the eigenvalues of problem (127).Remark that,{}(2n + 1)πγ = inf{λ n+1 − λ n } = inf √(n + 1)2 + α + √ n 2 + αγ ∞ = lim inf n→∞ (λ n+1 − λ n ) = π.> π2 √ α ,(129)It follows from the generalized Ingham Theorem 4.3 that, for any T >2π/γ ∞ = 2, (128) holds. Hence, the following controllability result is obtained:Theorem 4.6 Let J ⊂ [0, 1] with | J |> 0 and T > 2. For any (u 0 , u 1 ) ∈H 1 0 (0, 1) × L 2 (0, 1) there exists f ∈ L 2 ((0, T ) × J) such that the solution u ofequation (127) satisfies (119).Remark 4.7 Note that if we had applied Theorem 4.1 the controllability timewould have been T > 2π/γ ≥ 4 √ α. But Theorem 4.3 gives a control time Tindependent of α.Note that in this case the exponential functions (e λnt ) n are not orthogonal inL 2 (0, T ). Thus we can not use the same argument as in the proof on Theorem4.4 and, accordingly, Ingham’s Theorem is needed.We have considered here the case where α is a positive constant. When α isnegative the complex exponentials entering in the Fourier expansion of solutionsmay have eigenfrequencies λ n which are not all purely real. In that case we cannot apply directly Theorem 4.3. However, its method of proof allows also todeal with the situation where a finite number of eigenfrequencies are non real.Thus,the same result holds for all real α. □

S. Micu and E. Zuazua 574.4 Boundary controllability of the 1-D wave equationIn this section we study the following boundary controllability problem: givenT > 2 and (u 0 , u 1 ) ∈ L 2 (0, 1) × H −1 (0, 1) to find a control f ∈ L 2 (0, T ) suchthat the solution u of the problem:satisfies⎧⎪⎨⎪⎩u ′′ − u xx = 0 x ∈ (0, 1), t ∈ [0, T ]u(t, 0) = 0 t ∈ [0, T ]u(t, 1) = f(t) t ∈ [0, T ]u(0) = u 0 , u ′ (0) = u 1 x ∈ (0, 1)(130)u(T, ·) = u ′ (T, ·) = 0. (131)From the developments in Chapter 3 it follows that the following inequalitiesare a necessary and sufficient condition for the controllability of (130)C 1 ‖ (ϕ 0 , ϕ 1 ) ‖ 2 H 1 0 ×L2 ≤∫ T0|ϕ x (t, 1)| 2 dt ≤ C 2 ‖ (ϕ 0 , ϕ 1 ) ‖ 2 H 1 0 ×L2 (132)for any (ϕ 0 , ϕ 1 ) ∈ H 1 0 (0, 1) × L 2 (0, 1) and ϕ solution of (110).In order to prove (132) we use the Fourier decomposition of (110) given inthe first section.Theorem 4.7 Let T ≥ 2. There exist two positive constants C 1 and C 2 suchthat (132) holds for any (ϕ 0 , ϕ 1 ) ∈ H 1 0 (0, 1) × L 2 (0, 1) and ϕ solution of (110).Proof: If (ϕ 0 , ϕ 1 ) = ∑ n∈Z a nΦ ∗ n we have that,∥ ‖ (ϕ 0 , ϕ 1 ) ‖ 2 ∑ ∥∥∥∥2H=a0 1 n Φ n = ∑ | a n | 2 . (133)×L2 ∥n∈Z ∗ H0 1×L2 n∈Z ∗On the other hand∫ T0|ϕ x (t, 1)| 2 dt =∫ T0∣ ∑∣∣∣∣2(−1) n a∣n e inπt dt.n∈Z ∗By using the orthogonality in L 2 (0, 2) of the exponentials (e inπt ) n , we getthat∫ ∣2∑∣∣∣∣2(−1) n a n e inπt dt = ∑ | a n | 2 .0 ∣n∈Z ∗ n∈Z ∗If T > 2, it is immediate that∫ ∣T∑∣∣∣∣2(−1) n a0 ∣n e inπt dt ≥n∈Z ∗∫ 20∣ ∑∣∣∣∣2(−1) n a∣n e inπt dt = ∑ | a n | 2 .n∈Z ∗ n∈Z ∗

58 Controllability of Partial Differential EquationsOn the other hand, by using the 2-periodicity in time of the exponentialsand the fact that there exists p > 0 such that 2(p − 1) ≤ T < 2p, it follows that∫ ∣T∑∣∣∣∣2 ∫ ∣2(−1) n a0 ∣n e inπt ∑∣∣∣∣2dt ≥ p(−1) n an∈Z ∗ 0 ∣n e inπt dt =n∈Z ∗= p ∑ n∈Z ∗ | a n | 2 ≤ T + 22∑n∈Z ∗ |a n | 2 .Hence, for any T ≥ 2, we have that∫ ∣T∑∣∣∣∣2(−1) n a n e inπt dt ≍ ∑ | a n | 2 . (134)0 ∣n∈Z ∗ n∈Z ∗Finally, from (133) and (134) we obtain that∫ 2and (132) is proved.0|ϕ x (t, 1)| 2 dt ≍‖ (ϕ 0 , ϕ 1 ) ‖ 2 H 1 0 (0,1)×L2 (0,1)□As a direct consequence of Theorems 4.7 the following controllability resultholds:Theorem 4.8 Let T ≥ 2. For any (u 0 , u 1 ) ∈ L 2 (0, 1) × H −1 (0, 1) there existsf ∈ L 2 (0, T ) such that the solution u of equation (130) satisfies (131).As in the context of the interior controllability problem, one may addressthe following wave equation with potential⎧⎪⎨⎪⎩u ′′ − u xx + αu = 0, x ∈ (0, 1), t ∈ (0, T )u(t, 0) = 0 t ∈ [0, T ]u(t, 1) = f(t) t ∈ [0, T ]u(0) = u 0 , u ′ (0) = u 1 , x ∈ (0, 1)(135)where α is a positive real number.The controllability problem is then reduced to the proof of the followinginequality:∫ ∣T∑(−1) n nπ ∣∣∣∣2a n e λnt dt ≍ ∑ |a n | 2 (136)0 ∣ λ nn∈Z ∗ n∈Z ∗where λ n = sgn(n)πi √ n 2 + α are the eigenvalues of problem (135).It follows from (129) and the generalized Ingham’s Theorem 4.3 that, forany T > 2π/γ ∞ = 2, (136) holds. Hence, the following controllability result isobtained:

S. Micu and E. Zuazua 59Theorem 4.9 Let T > 2. For any (u 0 , u 1 ) ∈ L 2 (0, 1) × H −1 (0, 1) there existsf ∈ L 2 (0, T ) such that the solution u of equation (135) satisfies (131).Remark 4.8 As we mentioned above, the classical Ingham inequality in (4.1)gives a suboptimal result in what concerns the time of control. □5 Interior controllability of the heat equationIn this chapter the interior controllability problem of the heat equation is studied.The control is assumed to act on a subset of the domain where the solutionsare defined. The boundary controllability problem of the heat equation will beconsidered in the following chapter.5.1 IntroductionLet Ω ⊂ R n be a bounded open set with boundary of class C 2 and ω anon empty open subset of Ω. Given T > 0 we consider the following nonhomogeneousheat equation:⎧⎨ u t − ∆u = f1 ω in (0, T ) × Ωu = 0on (0, T ) × ∂Ω(137)⎩u(x, 0) = u 0 (x) in Ω.In (137) u = u(x, t) is the state and f = f(x, t) is the control function witha support localized in ω. We aim at changing the dynamics of the system byacting on the subset ω of the domain Ω.The heat equation is a model for many diffusion phenomena. For instance(137) provides a good description of the temperature distribution and evolutionin a body occupying the region Ω. Then the control f represents a localizedsource of heat.The interest on analyzing the heat equation above relies not only in thefact that it is a model for a large class of physical phenomena but also oneof the most significant partial differential equation of parabolic type. As weshall see latter on, the main properties of parabolic equations such as timeirreversibilityand regularizing effects have some very important consequencesin control problems.5.2 Existence and uniqueness of solutionsThe following theorem is a consequence of classical results of existence anduniqueness of solutions of nonhomogeneous evolution equations. All the detailsmay be found, for instance in [14].

60 Controllability of Partial Differential EquationsTheorem 5.1 For any f ∈ L 2 ((0, T ) × ω) and u 0 ∈ L 2 (Ω) equation (137) hasa unique weak solution u ∈ C([0, T ], L 2 (Ω)) given by the variation of constantsformulau(t) = S(t)u 0 +∫ t0S(t − s)f(s)1 ω ds (138)where (S(t)) t∈R is the semigroup of contractions generated by the heat operatorin L 2 (Ω).Moreover, if f ∈ W 1,1 ((0, T ) × L 2 (ω)) and u 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω), equation(137) has a classical solution u ∈ C 1 ([0, T ], L 2 (Ω)) ∩ C([0, T ], H 2 (Ω) ∩ H 0 1 (Ω))and (137) is verified in L 2 (Ω) for all t > 0.Let us recall the classical energy estimate for the heat equation. Multiplyingin (137) by u and integrating in Ω we obtain that∫∫∫1 d| u | 2 dx + | ∇u | 2 dx = fudx ≤ 1 ∫| f | 2 dx + 1 ∫| u | 2 dx.2 dt ΩΩΩ 2 Ω2 ΩHence, the scalar function X = ∫ Ω | u |2 dx satisfies∫X ′ ≤ X + | f | 2 dxwhich, by Gronwall’s inequality, gives∫ t ∫X(t) ≤ X(0)e t + | f | 2 dxds ≤ X(0)e t +0ΩΩ∫ T0∫Ω| f | 2 dxdt.On the other hand, integrating in (137) with respect to t, it follows that∫T ∫1T ∫u 2 dx2 ∣ + | ∇u | 2 dxdt ≤ 1 ∫ T ∫f 2 dxdt + 1 ∫ T ∫u 2 dxdtΩ0 Ω2 0 Ω 2 0 Ω0From the fact that u ∈ L ∞ (0, T ; L 2 (Ω)) it follows that u ∈ L 2 (0, T ; H 1 0 (Ω)).Consequently, whenever u 0 ∈ L 2 (Ω) and f ∈ L 2 (0, T ; L 2 (ω)) the solution uverifiesu ∈ L ∞ (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0 (Ω)).5.3 Controllability problemsLet T > 0 and define, for any initial data u 0 ∈ L 2 (Ω), the set of reachablestatesR(T ; u 0 ) = {u(T ) : u solution of (137) with f ∈ L 2 ((0, T ) × ω)}. (139)By definition, any state in R(T ; u 0 ) is reachable in time T by starting fromu 0 at time t = 0 with the aid of a convenient control f.As in the case of the wave equation several notions of controllability maybe defined.

S. Micu and E. Zuazua 61Definition 5.1 System (137) is approximately controllable in time T if,for every initial data u 0 ∈ L 2 (Ω), the set of reachable states R(T ; u 0 ) is densein L 2 (Ω).Definition 5.2 System (137) is exactly controllable in time T if, for everyinitial data u 0 ∈ L 2 (Ω), the set of reachable states R(T ; u 0 ) coincides withL 2 (Ω).Definition 5.3 System (137) is null controllable in time T if, for every initialdata u 0 ∈ L 2 (Ω), the set of reachable states R(T ; u 0 ) contains the element0.Remark 5.1 Let us make the following remarks:• One of the most important properties of the heat equation is its regularizingeffect. When Ω \ ω ≠ ∅, the solutions of (137) belong to C ∞ (Ω \ ω)at time t = T . Hence, the restriction of the elements of R(T, u 0 ) to Ω \ ωare C ∞ functions. Then, the trivial case ω = Ω (i. e. when the controlacts on the entire domain Ω) being excepted, exact controllability may nothold. In this sense, the notion of exact controllability is not very relevantfor the heat equation. This is due to its strong time irreversibility of thesystem under consideration.• It is easy to see that if null controllability holds, then any initial data maybe led to any final state of the form S(T )v 0 with v 0 ∈ L 2 (Ω), i. e. to therange of the semigroup in time t = T .Indeed, let u 0 , v 0 ∈ L 2 (Ω) and remark that R(T ; u 0 − v 0 ) = R(T ; u 0 ) −S(T )v 0 . Since 0 ∈ R(T ; u 0 − v 0 ), it follows that S(T )v 0 ∈ R(T ; u 0 ).It is known that the null controllability holds for any time T > 0 andopen set ω on which the control acts (see, for instance, [29]). The nullcontrollability property holds in fact in a much more general setting ofsemilinear heat equations ([23] and [24]).• Null controllability implies approximate controllability. Indeed, we haveshown that, whenever null controllability holds, S(T )[L 2 (Ω)] ⊂ R(T ; u 0 )for all u 0 ∈ L 2 (Ω). Taking into account that all the eigenfunctions of thelaplacian belong to S(T )[L 2 (Ω)] we deduce that the set of reachable statesis dense and, consequently, that approximate controllability holds.• The problem of approximate controllability may be reduced to the caseu 0 ≡ 0. Indeed, the linearity of the system we have considered impliesthat R(T, u 0 ) = R(T, 0) + S(T )u 0 .

62 Controllability of Partial Differential Equations• Approximate controllability together with uniform estimates on the approximatecontrols as ε → 0 may led to null controllability properties.More precisely, given u 1 , we have that u 1 ∈ R(T, u 0 ) if and only if thereexists a sequence (f ε ) ε>0 of controls such that ||u(T ) − u 1 || L2 (Ω) ≤ ε and(f ε ) ε>0 is bounded in L 2 (ω × (0, T )). Indeed in this case any weak limitin L 2 (ω ×(0, T )) of the sequence (f ε ) ε>0 of controls gives an exact controlwhich makes that u(T ) = u 1 . □In this chapter we limit ourselves to study the approximate controllabilityproblem. The main ingredients we shall develop are of variational nature.The problem will be reduced to prove unique continuation properties. Nullcontrollabilitywill be addressed in the following chapter.5.4 Approximate controllability of the heat equationGiven any T > 0 and any nonempty open subset ω of Ω we analyze in thissection the approximate controllability problem for system (137).Theorem 5.2 Let ω be an open nonempty subset of Ω and T > 0. Then (137)is approximately controllable in time T .Remark 5.2 The fact that the heat equation is approximately controllable inarbitrary time T and with control in any subset of Ω is due to the infinitevelocity propagation which characterizes the heat equation.Nevertheless, the infinite velocity of propagation by itself does not allowto deduce quantitative estimates for the norm of the controls. Indeed, as itwas proved in [50], the heat equation in an infinite domain (0, ∞) of R isapproximately controllable but, in spite of the infinite velocity of propagation,it is not null-controllable. □Remark 5.3 There are several possible proofs for the approximate controllabilityproperty. We shall present here two of them. The first one is presentedbelow and uses Hahn-Banach Theorem. The second one is constructive and usesa variational technique similar to the one we have used for the wave equation.We give it in the following section. □Proof of the Theorem 5.2: As we have said before, it is sufficient to consideronly the case u 0 = 0. Thus we assume that u 0 = 0.From Hahn-Banach Theorem, R(T, u 0 ) is dense in L 2 (Ω) ∫ if the followingproperty holds: There is no ϕ T ∈ L 2 (Ω), ϕ T ≠ 0 such that u(T )ϕ T dx = 0for all u solution of (137) with f ∈ L 2 (ω × (0, T )).Ω

S. Micu and E. Zuazua 63Accordingly, the proof can be reduced to showing that, if ϕ T ∈ L 2 (Ω) issuch that ∫ Ω u(T )ϕ T dx = 0, for all solution u of (137) then ϕ T = 0.To do this we consider the adjoint equation:⎧⎨ ϕ t + ∆ϕ = 0 in (0, T ) × Ωϕ | ∂Ω = 0 on (0, T ) × ∂Ω(140)⎩ϕ(T ) = ϕ T in Ω.We multiply the equation satisfied by ϕ by u and then the equation of uby ϕ. Integrating by parts and taking into account that u 0 ≡ 0 the followingidentity is obtained∫ T0∫ω∫fϕdxdt =∫+Ωuϕdx∣Ω×(0,T )T ∫ T ∫0+0∫(u t − ∆u)ϕdxdt = − (ϕ t + ∆ϕ)udxdt+Ω×(0,T )∫∂Ω(− ∂u∂n ϕ + u∂ϕ ∂nHence, ∫ Ω u(T )ϕ T dx = 0 if and only if∫ T0)dσdt =∫ωΩfϕdxdt = 0.u(T )ϕ T dx.If the laterrelation holds for any f ∈ L 2 (ω × (0, T )), we deduce that ϕ ≡ 0 in ω × (0, T ).Let us now recall the following result whose proof may be found in [33]:Holmgren Uniqueness Theorem: Let P be a differential operator with constantcoefficients in R n . Let u be a solution of P u = 0 in Q 1 where Q 1 is anopen set of R n . Suppose that u = 0 in Q 2 where Q 2 is an open nonempty subsetof Q 1 .Then u = 0 in Q 3 , where Q 3 is the open subset of Q 1 which contains Q 2and such that any characteristic hyperplane of the operator P which intersectsQ 3 also intersects Q 1 .In our particular case P = ∂ t + ∆ x is a differential operator in R n+1 andits principal part is P p = ∆ x . A hyperplane of R n+1 is characteristic if itsnormal vector (ξ, ζ) ∈ R n+1 is a zero of P p , i. e. of P p (ξ, ζ) = |ξ| 2 . Hence,normal vectors are of the form (0, ±1) and the characteristic hyperplanes arehorizontal, parallel to the hyperplane t = 0.Consequently, for the adjoint heat equation under consideration (140), wecan apply Holmgren’s Uniqueness Theorem with Q 1 = (0, T )×Ω, Q 2 = (0, T )×ω and Q 3 = (0, T ) × Ω. Then the fact that ϕ = 0 in (0, T ) × ω implies ϕ = 0in (0, T ) × Ω. Consequently ϕ T ≡ 0 and the proof is finished. □5.5 Variational approach to approximate controllabilityIn this section we give a new proof of the approximate controllability resultTheorem 5.2. This proof has the advantage of being constructive and it allowsto compute explicitly approximate controls.

64 Controllability of Partial Differential EquationsLet us fix the control time T > 0 and the initial datum u 0 = 0. Letu 1 ∈ L 2 (Ω) be the final target and ε > 0 be given. Recall that we are lookingfor a control f such that the solution of (137) satisfiesWe define the following functional:J ε (ϕ T ) = 1 2∫ T0∫||u(T ) − u 1 || L 2 (Ω) ≤ ε. (141)ωJ ε : L 2 (Ω) → R (142)∫ϕ 2 dxdt + ε ‖ ϕ T ‖ L2 (Ω) − u 1 ϕ T dx (143)Ωwhere ϕ is the solution of the adjoint equation (140) with initial data ϕ T .The following Lemma ensures that the minimum of J ε gives a control forour problem.Lemma 5.1 If ̂ϕ T is a minimum point of J ε in L 2 (Ω) and ̂ϕ is the solutionof (140) with initial data ̂ϕ T , then f = ̂ϕ |ω is a control for (137), i. e. (141)is satisfied.Proof: In the sequel we simply denote J ε by J.Suppose that J attains its minimum value at ̂ϕ T ∈ L 2 (Ω). Then for anyψ 0 ∈ L 2 (Ω) and h ∈ R we have J( ̂ϕ T ) ≤ J ( ̂ϕ T + hψ 0 ) . On the other hand,= 1 2= 1 2∫ T0∫ T0∫∫ωωJ ( ̂ϕ T + hψ 0 ) =∫| ̂ϕ + hψ | 2 dxdt + ε ‖ ̂ϕ T + hψ 0 ‖ L 2 (Ω) −∫ TΩu 1 ( ̂ϕ T + hψ 0 )dx∫∫ T ∫| ̂ϕ | 2 dxdt + h2| ψ | 2 dxdt + h ̂ϕψdxdt +2 0 ω0 ω∫+ε ‖ ̂ϕ T + hψ 0 ‖ L2 (Ω) − u 1 ( ̂ϕ T + hψ 0 )dx.Thus0 ≤ ε [ ] h 2 ∫‖ ̂ϕ T + hψ 0 ‖ L 2 (Ω) − ‖ ̂ϕ T ‖ L 2 (Ω) +2[ ∫ T ∫∫ ]+h ̂ϕψdxdt − u 1 ψ 0 dx .0 ωΩSinceΩ(0,T )×ω‖ ̂ϕ T + hψ 0 ‖ L2 (Ω) − ‖ ̂ϕ T ‖ L2 (Ω)≤ |h| ‖ ψ 0 ‖ L2 (Ω)ψ 2 dxdt

S. Micu and E. Zuazua 65we obtain0 ≤ ε |h| ‖ ψ 0 ‖ L 2 (Ω) + h22∫ T0∫ω∫ T ∫∫ψ 2 dxdt + h ̂ϕψdxdt − h u 1 ψ 0 dx0 ωΩfor all h ∈ R and ψ 0 ∈ L 2 (Ω).Dividing by h > 0 and by passing to the limit h → 0 we obtain∫ T ∫∫0 ≤ ε ‖ ψ 0 ‖ L 2 (Ω) + ̂ϕψdxdt − u 1 ψ 0 dx. (144)0 ωΩThe same calculations with h < 0 gives that∫ T ∫∫̂ϕψdxdt − u∣1 ψ 0 dx0 ωΩ ∣ ≤ ε ‖ ψ 0 ‖ ∀ψ 0 ∈ L 2 (Ω). (145)On the other hand, if we take the control f = ̂ϕ in (137), by multiplying in(137) by ψ solution of (140) and by integrating by parts we get that∫ T0∫ω∫̂ϕψdxdt =Ωu(T )ψ 0 dx. (146)From the last two relations it follows that∫‖ (u(T ) − u 1 )ψ 0 dx‖ ≤ ε||ψ 0 || L 2 (Ω), ∀ψ 0 ∈ L 2 (Ω) (147)Ωwhich is equivalent to||u(T ) − u 1 || L 2 (Ω) ≤ ε.The proof of the Lemma is now complete.Let us now show that J attains its minimum in L 2 (Ω).Lemma 5.2 There exists ̂ϕ T ∈ L 2 (Ω) such that□J( ̂ϕ T ) = minϕ T ∈L 2 (Ω) J(ϕ T ). (148)Proof: It is easy to see that J is convex and continuous in L 2 (Ω).Theorem 2.3, the existence of a minimum is ensured if J is coercive, i. e.ByIn fact we shall prove thatJ(ϕ T ) → ∞ when ||ϕ T || L 2 (Ω) → ∞. (149)lim inf J(ϕ T )/||ϕ T || L2 (Ω) ≥ ε. (150)||ϕ T || L 2 (Ω) →∞

66 Controllability of Partial Differential EquationsEvidently, (150) implies (149) and the proof of the Lemma is complete.In order to prove (150) let (ϕ T,j ) ⊂ L 2 (Ω) be a sequence of initial data forthe adjoint system with ‖ ϕ T,j ‖ L2 (Ω)→ ∞. We normalize them˜ϕ T,j = ϕ T,j / ‖ ϕ T,j ‖ L 2 (Ω),so that ‖ ˜ϕ T,j ‖ L 2 (Ω)= 1.On the other hand, let ˜ϕ j be the solution of (140) with initial data ˜ϕ T,j .ThenJ(ϕ T,j )/ ‖ ϕ T,j ‖ L 2 (Ω)= 1 ∫ T ∫∫2 ‖ ϕ T,j ‖ L 2 (Ω) | ˜ϕ j | 2 dxdt + ε − u 1 ˜ϕ T,j dx.0 ωΩThe following two cases may occur:∫ T ∫1) lim inf | ˜ϕ j | 2 > 0. In this case we obtain immediately thatj→∞0 ωJ(ϕ T,j )/ ‖ ϕ T,j ‖ L2 (Ω)→ ∞.∫ T ∫2) lim inf | ˜ϕ j | 2 = 0. In this case since ˜ϕ T,j is bounded in L 2 (Ω),j→∞0 ωby extracting a subsequence we can guarantee that ˜ϕ T,j ⇀ ψ 0 weakly inL 2 (Ω) and ˜ϕ j ⇀ ψ weakly in L 2 (0, T ; H0 1 (Ω)) ∩ H 1 (0, T ; H −1 (Ω)), whereψ is the solution of (140) with initial data ψ 0 at t = T . Moreover, bylower semi-continuity,∫ T0∫and therefore ψ = 0 en ω × (0, T ).ω∫ T ∫ψ 2 dxdt ≤ lim inf | ˜ϕ j | 2 dxdt = 0j→∞0 ωHolmgren Uniqueness Theorem implies that ψ ≡ 0 in Ω × (0, T ) andconsequently ψ 0 = 0.Therefore, ˜ϕ T,j ⇀ 0 weakly in L 2 (Ω) and consequently ∫ Ω u 1 ˜ϕ T,j dx tendsto 0 as well.Hencelim infj→∞and (150) follows.J(ϕ T,j )≥ lim inf‖ ϕ T,j ‖ [ε − u 1 ˜ϕ T,j dx] = ε,j→∞∫Ω□

S. Micu and E. Zuazua 67Remark 5.4 Lemmas 5.1 and 5.2 give a second proof of Theorem 5.2. Thisapproach does not only guarantee the existence of a control but also provides amethod to obtain the control by minimizing a convex, continuous and coercivefunctional in L 2 (Ω).In the proof of the coercivity, the relevance of the term ε||ϕ T || L2 (Ω) is clear.Indeed, the coercivity of J depends heavily on this term. This is not only fortechnical reasons. The existence of a minimum of J with ε = 0 implies theexistence of a control which makes u(T ) = u 1 . But this is not true unless u 1 isvery regular in Ω \ ω. Therefore, for general u 1 ∈ L 2 (Ω), the term ε||ϕ T || L 2 (Ω)is needed.Note that both proofs are based on the unique continuation property whichguarantees that if ϕ is a solution of the adjoint system such that ϕ = 0 inω × (0, T ), then ϕ ≡ 0. As we have seen, this property is a consequence ofHolmgren Uniqueness Theorem. □The second proof, based on the minimization of J, with some changes onthe definition of the functional as indicated in 1, allows proving approximatecontrollability by means of other controls, for instance, of bang-bang form. Weaddress these variants in the following sections.5.6 Finite-approximate controlLet E be a subspace of L 2 (Ω) of finite dimension and Π E be the orthogonalprojection over E. As a consequence of the approximate controllability propertyin Theorem 5.2 the following stronger result may be proved: given u 0 andu 1 in L 2 (Ω) and ε > 0 there exists a control f such that the solution of (137)satisfies simultaneouslyΠ E (u(T )) = Π E (u 1 ), ‖ u(T ) − u 1 ‖ L 2 (Ω)≤ ε. (151)This property not only says that the distance between u(T ) and the targetu 1 is less that ε but also that the projection of u(T ) and u 1 over E coincide.This property, introduced in [71], will be called finite-approximate controllability.It may be proved easily by taking into account the followingproperty of Hilbert spaces: If L : E → F is linear and continuous between theHilbert spaces E and F and the range of L is dense in F , then, for any finiteset f 1 , f 2 , . . . , f N ∈ F , the set {Le : (Le, f j ) F = 0 ∀j = 1, 2, . . . , N} is densein the orthogonal of Span{f 1 , f 2 , . . . , f N }.Nevertheless, as we have said before, this result may also be proved directly,by considering a slightly modified form of the functional J used in the secondproof of Theorem 5.2. We introduceJ E (ϕ T ) = 1 ∫ T ∫∫ϕ 2 dxdt + ε ‖ (I − Π E )ϕ T ‖ L2 (Ω) − u 1 ϕ T dx.20ωΩ

68 Controllability of Partial Differential EquationsThe functional J E is again convex and continuous in L 2 (Ω). Moreover, it iscoercive. The proof of the coercivity of J E is similar to that of J. It is sufficientto note that if ̂ϕ T,j tends weakly to zero in L 2 (Ω), then Π E ( ̂ϕ T,j ) converges(strongly) to zero in L 2 (Ω).Therefore ‖ (I − Π E ) ̂ϕ T,j ‖ L2 (Ω) / ‖ ̂ϕ T,j ‖ L2 (Ω) tends to 1. According tothis, the new functional J E satisfies the coercivity property (150).It is also easy to see that the minimum of J E gives the finite-approximatecontrol we were looking for.5.7 Bang-bang controlIn the study of finite dimensional systems we have seen that one may find“bang-bang” controls which take only two values ±λ for some λ > 0.In the case of the heat equation it is also easy to construct controls of thistype. In fact a convenient change in the functional J will ensure the existenceof “bang-bang” controls. We consider:J bb (ϕ T ) = 1 2( ∫ T ∫0ω2 ∫| ϕ | dxdt)+ ε ‖ ϕ T ‖ L 2 (Ω) − u 1 ϕ T dx.ΩRemark that the only change in the definition of J bb is in the first termin which the norm of ϕ in L 2 (ω × (0, T )) has been replaced by its norm inL 1 ((0, T ) × ω).Once again we are dealing with a convex and continuous functional in L 2 (Ω).The proof of the coercivity of J bb is the same as in the case of the functionalJ. We obtain that:J bb (ϕ T )lim inf≥ ε.‖ϕ T ‖ L 2 (Ω) →∞ ‖ ϕ T ‖ L2 (Ω)Hence, J bb attains a minimum in some ̂ϕ T of L 2 (Ω). It is easy to see that,if ̂ϕ is the corresponding solution of the adjoint system with ̂ϕ T as initial data,then there exists f ∈ ∫ ∫ T| ̂ϕ|dx sgn( ̂ϕ) such that the solution of (137) withω 0this control satisfies ||u(T ) − u 1 || ≤ ε.On the other hand, since ̂ϕ is a solution of the adjoint heat equation, it isreal analytic in Ω × (0, T ). Hence, the set {t : ̂ϕ = 0} is of zero measure inΩ × (0, T ). Hence, we may consider∫f =ω∫ T0| ̂ϕ|dxdt sgn( ̂ϕ) (152)which represents a bang-bang control. Remark that the sign of the controlchanges when the sign of ̂ϕ changes. Consequently, the geometry of the setswhere the control has a given sign can be quite complex.

S. Micu and E. Zuazua 69Note also that the amplitude of the bang-bang control is ∫ ω| ̂ϕ|dxdtwhich, evidently depends of the distance from the final target u 1 to the uncontrolledfinal state S(T )u 0 and of the control time T .Remark 5.5 As it was shown in [18], the bang-bang control obtained by minimizingthe functional J bb is the one of minimal norm in L ∞ ((0, T ) × ω) amongall the admissible ones. The control obtained by minimizing the functional Jhas the minimal norm in L 2 ((0, T ) × ω). □Remark 5.6 The problem of finding bang-bang controls guaranteeing the finite-approximateproperty may also be considered. It is sufficient to take thefollowing combination of the functionals J E and J bb :( ∫J bb,E (ϕ T ) = 1 T ∫2 ∫| ϕ | dxdt)+ ε ‖ (I − Π E )ϕ T ‖ L22 (Ω) − u 1 ϕ T dx.Ω□0ω∫ T05.8 CommentsThe null controllability problem for system (137) is equivalent to the followingobservability inequality for the adjoint system (140):∫ T ∫‖ ϕ(0) ‖ 2 L 2 (Ω) ≤ C ϕ 2 dxdt, ∀ϕ 0 ∈ L 2 (Ω). (153)0ωOnce (153) is known to hold one can obtain the control with minimal L 2 -norm among the admissible ones. To do that it is sufficient to minimize thefunctionalJ(ϕ 0 ) = 1 ∫ T ∫∫ϕ 2 dxdt + ϕ(0)u 0 dx (154)2 0 ωΩover the Hilbert space∫ T ∫H = {ϕ 0 : the solution ϕ of (140) satisfies ϕ 2 dxdt < ∞}.To be more precise, H is the completion of L 2 (Ω) with respect to the norm[ ∫ T ∫0 ω ϕ2 dxdt] 1/2 . In fact, H is much larger than L 2 (Ω). We refer to [23] forprecise estimates on the nature of this space.Observe that J is convex and continuous in H. On the other hand (153)guarantees the coercivity of J and the existence of its minimizer.Due to the irreversibility of the system, (153) is not easy to prove. Forinstance, multiplier methods do not apply. Let us mention two different approachesused for the proof of (153).0ω

70 Controllability of Partial Differential Equations1. Results based on the observation of the wave or elliptic equations:In [56] it was shown that if the wave equation is exactly controllablefor some T > 0 with controls supported in ω, then the heat equation(137) is null controllable for all T > 0 with controls supported in ω. Asa consequence of this result and in view of the controllability results forthe wave equation, it follows that the heat equation (137) is null controllablefor all T > 0 provided ω satisfies the geometric control condition.However, the geometric control condition does not seem to be natural atall in the context of the heat equation.Later on, Lebeau and Robbiano [42] proved that the heat equation (137)is null controllable for every open, non-empty subset ω of Ω and T >0. This result shows, as expected, that the geometric control conditionis unnecessary in the context of the heat equation. A simplified proofof it was given in [43] where the linear system of thermoelasticity wasaddressed. The main ingredient in the proof is the following observabilityestimate for the eigenfunctions {ψ j } of the Laplace operator2∫∑√ ∑a j ψ j (x)dx ≥ C 1 e −C2 µ| a j | 2 (155)ω ∣∣λ j≤µλ j≤µwhich holds for any {a j } ∈ l 2 and for all µ > 0 and where C 1 , C 2 > 0 aretwo positive constants.This result was implicitly used in [42] and it was proved in [43] by meansof Carleman’s inequalities for elliptic equations.2. Carleman inequalities for parabolic equations: The null controllabilityof the heat equation with variable coefficients and lower ordertime-dependent terms has been studied by Fursikov and Imanuvilov (seefor instance [16], [26], [27], [28], [34] and [35]). Their approach is basedon the use of the Carleman inequalities for parabolic equations and is differentto the one we have presented above. In [29], Carleman estimatesare systematically applied to solve observability problem for linearizedparabolic equations.In [21] the boundary null controllability of the heat equation was proved inone space dimension using moment problems and classical results on the linearindependence in L 2 (0, T ) of families of real exponentials. We shall describe thismethod in the next chapter.

S. Micu and E. Zuazua 716 Boundary controllability of the heat equationIn this chapter the boundary null-controllability problem of the heat equationis studied. We do it by reducing the control problem to an equivalent problemof moments. The latter is solved with the aid of a biorthogonal sequence toa family of real exponential functions. This technique was used in the studyof several control problems (the heat equation being one of the most relevantexamples of application) in the late 60’s and early 70’s by R. D. Russell andH. O. Fattorini (see, for instance, [21] and [22]).6.1 IntroductionGiven T > 0 arbitrary, u 0 ∈ L 2 (0, 1) and f ∈ L 2 (0, T ) we consider the followingnon-homogeneous 1-D problem:⎧⎨⎩u t − u xx = 0 x ∈ (0, 1), t ∈ (0, T )u(t, 0) = 0, u(t, 1) = f(t) t ∈ (0, T )u(0, x) = u 0 (x) x ∈ (0, 1).(156)In (156) u = u(x, t) is the state and f = f(t) is the control function whichacts on the extreme x = 1. We aim at changing the dynamics of the system byacting on the boundary of the domain (0, 1).6.2 Existence and uniqueness of solutionsThe following theorem is a consequence of classical results of existence anduniqueness of solutions of nonhomogeneous evolution equations. All the detailsmay be found, for instance in [47].Theorem 6.1 For any f ∈ L 2 (0, T ) and u 0 ∈ L 2 (Ω) equation (156) has aunique weak solution u ∈ C([0, T ], H −1 (Ω)).Moreover, the map {u 0 , f} → {u} is linear and there exists C = C(T ) > 0such that||u|| L ∞ (0,T ;H −1 (Ω)) ≤ C ( ||u 0 || L 2 (Ω) + ||f|| L 2 (0,T )). (157)6.3 Controllability and problem of momentsIn this section we introduce several notions of controllability.Let T > 0 and define, for any initial data u 0 ∈ L 2 (Ω), the set of reachablestatesR(T ; u 0 ) = {u(T ) : u solution of (156) with f ∈ L 2 (0, T )}. (158)An element of R(T, u 0 ) is a state of (156) reachable in time T by startingfrom u 0 with the aid of a control f.As in the previous chapter, several notions of controllability may be defined.

72 Controllability of Partial Differential EquationsDefinition 6.1 System (156) is approximately controllable in time T if,for every initial data u 0 ∈ L 2 (Ω), the set of reachable states R(T ; u 0 ) is densein L 2 (Ω).Definition 6.2 System (156) is exactly controllable in time T if, for everyinitial data u 0 ∈ L 2 (Ω), the set of reachable states R(T ; u 0 ) coincides withL 2 (Ω).Definition 6.3 System (156) is null controllable in time T if, for every initialdata u 0 ∈ L 2 (Ω), the set of reachable states R(T ; u 0 ) contains the element0.Remark 6.1 Note that the regularity of solutions stated above does not guaranteethat u(T ) belongs to L 2 (Ω). In view of this it could seem that the definitionsabove do not make sense. Note however that, due to the regularizing effect ofthe heat equation, if the control f vanishes in an arbitrarily small neighborhoodof t = T then u(T ) is in C ∞ and in particular in L 2 (Ω). According to this,the above definitions make sense by introducing this minor restrictions on thecontrols under consideration. □Remark 6.2 Let us make the following remarks, which are very close to thosewe did in the context of interior control:• The linearity of the system under consideration implies that R(T, u 0 ) =R(T, 0) + S(T )u 0 and, consequently, without loss of generality one mayassume that u 0 = 0.• Due to the regularizing effect the solutions of (156) are in C ∞ far awayfrom the boundary at time t = T . Hence, the elements of R(T, u 0 ) areC ∞ functions in [0, 1). Then, exact controllability may not hold.• It is easy to see that if null controllability holds, then any initial data maybe led to any final state of the form S(T )v 0 with v 0 ∈ L 2 (Ω).Indeed, let u 0 , v 0 ∈ L 2 (Ω) and remark that R(T ; u 0 − v 0 ) = R(T ; u 0 ) −S(T )v 0 . Since 0 ∈ R(T ; u 0 − v 0 ), it follows that S(T )v 0 ∈ R(T ; u 0 ).• Null controllability implies approximate controllability. Indeed we havethat S(T )[L 2 (Ω)] ⊂ R(T ; u 0 ) and S(T )[L 2 (Ω)] is dense in L 2 (Ω).• Note that u 1 ∈ R(T, u 0 ) if and only if there exists a sequence (f ε ) ε>0of controls such that ||u(T ) − u 1 || L2 (Ω) ≤ ε and (f ε ) ε>0 is bounded inL 2 (0, T ). Indeed, in this case, any weak limit in L 2 (0, T ) of the sequence(f ε ) ε>0 gives an exact control which makes that u(T ) = u 1 . □

S. Micu and E. Zuazua 73Remark 6.3 As we shall see, null controllability of the heat equation holdsin an arbitrarily small time. This is due to the infinity speed of propagation.It is important to underline, however, that, despite of the infinite speed ofpropagation, the null controllability of the heat equation does not hold in aninfinite domain. We refer to [50] for a further discussion of this issue.The techniques we shall develop in this section do not apply in unboundeddomains. Although, as shown in [50], using the similarity variables, one canfind a spectral decomposition of solutions of the heat equation on the whole orhalf line, the spectrum is too dense and biorthogonal families do not exist. □In this chapter the null-controllability problem will be considered. Let usfirst give the following characterization of the null-controllability property of(156).Lemma 6.1 Equation (156) is null-controllable in time T > 0 if and only if,for any u 0 ∈ L 2 (0, 1) there exists f ∈ L 2 (0, T ) such that the following relationholds∫ T0f(t)ϕ x (t, 1)dt =∫ 10u 0 (x)ϕ(0, x)dx, (159)for any ϕ T ∈ L 2 (0, 1), where ϕ(t, x) is the solution of the backward adjointproblem ⎧⎨ ϕ t + ϕ xx = 0 x ∈ (0, 1), t ∈ (0, T )ϕ(t, 0) = ϕ(t, 1) = 0 t ∈ (0, T )(160)⎩ϕ(T, x) = ϕ T (x) x ∈ (0, 1).Proof: Let f ∈ L 2 (0, T ) be arbitrary and u the solution of (156). If ϕ T ∈L 2 (0, 1) and ϕ is the solution of (160) then, by multiplying (156) by ϕ and byintegrating by parts we obtain that0 =∫ T ∫ 10∫ T+00∫ 10Consequently∫ T0(u t − u xx )ϕdxdt =∫ 1u(−ϕ t − ϕ xx )dxdt =f(t)ϕ x (t, 1)dt =∫ 100uϕdx∣∫ 10T0+uϕdx∣u 0 (x)ϕ(0, x)dx −T0∫ 10∫ T01(−u x ϕ + uϕ x )dt+∣∫ T+ f(t)ϕ x (t, 1)dt.0u(T, x)ϕ T (x)dx. (161)Now, if (159) is verified, it follows that ∫ 10 u(T, x)ϕ T (x)dx = 0, for allϕ 1 ∈ L 2 (0, 1) and u(T ) = 0.0

74 Controllability of Partial Differential EquationsHence, the solution is controllable to zero and f is a control for (156).Reciprocally, if f is a control for (156), we have that u(T ) = 0. From (161)it follows that (159) holds and the proof finishes. □From the previous Lemma we deduce the following result:Proposition 6.1 Equation (156) is null-controllable in time T > 0 if and onlyif for any u 0 ∈ L 2 (0, 1), with Fourier expansionu 0 (x) = ∑ n≥1a n sin(πnx),there exists a function w ∈ L 2 (0, T ) such that,∫ T0w(t)e −n2 π 2t dt = (−1) n a n2nπ e−n2 π 2T , n = 1, 2, .... (162)Remark 6.4 Problem (162) is usually refered to as problem of moments.Proof: From the previous Lemma we know that f ∈ L 2 (0, T ) is a controlfor (156) if and only if it satisfies (159). But, since (sin(nπx)) n≥1 forms anorthogonal basis in L 2 (0, 1), (159) is verified if and only if it is verified byϕ 1 n = sin(nπx), n = 1, 2, ....If ϕ 1 n = sin(nπx) then the corresponding solution of (160) is ϕ(t, x) =e −n2 π 2 (T −t) sin(nπx) and from (159) we obtain that∫ T0f(t)(−1) n nπe −n2 π 2 (T −t) = a n2 e−n2 π 2T .The proof ends by taking w(t) = f(T − t). □The control property has been reduced to the problem of moments (162).The latter will be solved by using biorthogonal techniques. The main ideas aredue to R.D. Russell and H.O. Fattorini (see, for instance, [21] and [22]).( The ) eigenvalues of the heat equation are λ n = n 2 π 2 , n ≥ 1. Let Λ =e−λ ntbe the family of the corresponding real exponential functions.n≥1Definition 6.4 (θ m ) m≥1 is a biorthogonal sequence to Λ in L 2 (0, T ) if andonly if∫ T0e −λnt θ m (t)dt = δ nm , ∀n, m = 1, 2, ....If there exists a biorthogonal sequence (θ m ) m≥1 , the problem of moments(162) may be solved immediately by settingw(t) = ∑ (−1) m a m π 2T θ2mπ e−m2 m (t). (163)m≥1

S. Micu and E. Zuazua 75As soon as the series converges in L 2 (0, T ), this provides the solution to(162).We have the following controllability result:Theorem 6.2 Given T > 0, suppose that there exists a biorthogonal sequence(θ m ) m≥1 to Λ in L 2 (0, T ) such thatwhere M and ω are two positive constants.Then (156) is null-controllable in time T .||θ m || L2 (0,T ) ≤ Me ωm , ∀m ≥ 1 (164)Proof: From Proposition 6.1 it follows that it is sufficient to show that forany u 0 ∈ L 2 (0, 1) with Fourier expansionu 0 = ∑ n≥1a n sin(nπx),there exists a function w ∈ L 2 (0, T ) which verifies (162).Considerw(t) = ∑ (−1) m a m π 2T θ2mπ e−m2 m (t). (165)m≥1Note that the series which defines w is convergent in L 2 (0, T ). Indeed,∑∣∣∣(−1) m a ∣∣ m π 2T ∣∣ ∣∣Lθ2mπ e−m2 m = ∑ |a m | π 2T ||θm≥12 (0,T ) 2mπ e−m2 m || L2 (0,T ) ≤m≥1≤ M ∑ m≥1|a m |2mπ e−m2 π 2 T +ωm < ∞where we have used the estimates (164) of the norm of the biorthogonal sequence(θ m ).On the other hand, (165) implies that w satisfies (162) and the proof finishes.□Theorem 6.2 shows that, the null-controllability problem (156) is solved ifwe prove the existence of a biorthogonal sequence (θ m ) m≥1 to Λ in L 2 (0, T )which verifies (164). The following sections are devoted to accomplish this task.6.4 Existence of a biorthogonal sequenceThe existence of a biorthogonal sequence to the family Λ is a consequence ofthe following Theorem (see, for instance, [57]).

76 Controllability of Partial Differential EquationsTheorem 6.3 (Münz) Let 0 < µ 1 ≤ µ 2 ≤ ... ≤ µ n ≤ ... be a sequence ofreal numbers. The family of exponential functions (e −µnt ) n≥1is complete inL 2 (0, T ) if and only if∑ 1= ∞. (166)µ nn≥1Given any T > 0, from Münz’s Theorem we obtain that the space generatedby the family Λ is a proper space of L 2 (0, T ) since∑ 1= ∑ 1λ n n 2 π 2 < ∞.n≥1n≥1Let E(Λ, T ) be the space generated by Λ in L 2 (0, T ) and E(m, Λ, T ) be thesubspace generated by ( e −λnt) n≥1 in L 2 (0, T ).n≠mWe also introduce the notation p n (t) = e −λnt .Theorem 6.4 Given any T > 0, there exists a unique sequence (θ m (T, · )) m≥1 ,biorthogonal to the family Λ, such that(θ m (T, · )) m≥1 ⊂ E(Λ, T ).Moreover, this biorthogonal sequence has minimal L 2 (0, T )-norm.Proof: Since Λ is not complete in L 2 (0, T ), it is also minimal. Thus, p m /∈E(m, Λ, T ), for each m ∈ I.Let r m be the orthogonal projection p m over the space E(m, Λ, T ) anddefinep m − r mθ m (T, · ) =||p m − r m || 2 . (167)L 2 (0,T )From the projection properties (see [10], pp. 79-80), it follows that1. r m ∈ E(m, Λ, T ) verifies ||p m (t) − r m (t)|| L2 (0,T ) = min r∈E(m,Λ,T ) ||p m −r|| L 2 (0,T )2. (p m − r m ) ⊥ E(m, Λ, T )3. (p m − r m ) ⊥ p n ∈ E(m, Λ, T ), ∀n ≠ m4. (p m − r m ) ⊥ r m ∈ E(m, Λ, T ).From the previous properties and (167) we deduce that1. ∫ T0 θ m(T, t)p n (t)dt = δ m,n2. θ m (T, · ) = pm−rm||p m−r m|| 2 ∈ E(Λ, T ).

S. Micu and E. Zuazua 77Thus, (167) gives a biorthogonal sequence (θ m (T, · )) m≥1⊂ E(Λ, T ) to thefamily Λ.The uniqueness of the biorthogonal sequence is obtained immediately. Indeed,if (θ m) ′ m≥1⊂ E(Λ, T ) is another biorthogonal sequence to the family Λ,then(θ m − θ ′ m) ∈ E(Λ, T )p n ⊥ (θ m − θ ′ m), ∀n ≥ 1}⇒ θ m − θ m ′ = 0where we have taken into account that (p m ) m≥1 is complete in E(Λ, T ).To prove the minimality of the norm of (θ m (T, · )) m≥1, let us consider anyother biorthogonal sequence (θ ′ m) m≥1⊂ L 2 (0, T ).E(Λ, T ) being closed in L 2 (0, T ), its orthogonal complement, E(Λ, T ) ⊥ , iswell defined. Thus, for any m ≥ 1, there exists a unique q m ∈ E(Λ, T ) ⊥ suchthat θ ′ m = θ m + q m .Finally,||θ ′ m|| 2 = ||θ m + q m || 2 = ||θ m || 2 + ||q m || 2 ≥ ||θ m || 2and the proof ends.□Remark 6.5 The previous Theorem gives a biorthogonal sequence of minimalnorm. This property is important since the convergence of the series of (163)depends directly of these norms. □The existence of a biorthogonal sequence (θ m ) m≥1 to the family Λ beingproved, the next step is to evaluate its L 2 (0, T )-norm. This will be done in twosteps. First for the case T = ∞ and next for T < ∞.6.5 Estimate of the norm of the biorthogonal sequence:T = ∞Theorem 6.5 There exist two positive constants M and ω such that the biorthogonalof minimal norm (θ m (∞, · ) m≥1 given by Theorem 6.4 satisfies thefollowing estimate||θ m (∞, · )|| L2 (0,∞) ≤ Mπe ωm , ∀m ≥ 1. (168)Proof: Let us introduce the following notations: E n := E n (Λ, ∞) is thesubspace generated by Λ n := ( e −λ kt ) 1≤k≤n in L2 (0, T ) and E n m := E 2 (m, Λ, ∞)is the subspace generated by ( e ) −λ kt1≤k≤n in L 2 (0, T ).k≠mRemark that E n and Em n are finite dimensional spaces andE(Λ, ∞) = ∪ n≥1 E n , E(m, Λ, ∞) = ∪ n≥1 E n m.

78 Controllability of Partial Differential EquationsWe have that, for each n ≥ 1, there exists a unique biorthogonal family(θm) n 1≤m≤n ⊂ E n , to the family of exponentials ( e ) −λ kt. More precisely,1≤k≤nθm n p m − rmn =||p m − rm|| n 2 , (169)L 2 (0,∞)where rm n is the orthogonal projection of p m over Em.nIfn∑θm n = c m k p k (170)k=1then, by multiplying (170) by p l and by integrating in (0, ∞), it follows thatδ m,l = ∑ k≥1c m k∫ T0p l (t)p k (t)dt, 1 ≤ m, l ≤ n. (171)Moreover, by multiplying in (170) by θ n m and by integrating in (0, ∞), weobtain that||θ n m|| 2 L 2 (0,∞) = cm m. (172)If G denotes the Gramm matrix of the family Λ, i. e. the matrix of elementsg l k =∫ ∞0p k (t)p l (t)dt,1 ≤ k, l ≤ nwe deduce from (171) that c m krule implies thatare the elements of the inverse of G. Cramer’sc m m = |G m||G|(173)where |G| is the determinant of matrix G and |G m | is the determinant of thematrix G m obtained by changing the m−th column of G by the m−th vectorof the canonical basis.It follows that√||θm|| n |G m |L 2 (0,∞) =|G| . (174)The elements of G may be computed explicitlyg n k =∫ ∞0p k (t)p n (t) =∫ ∞0e −(n2 +k 2 )π 2t dt =1n 2 π 2 + k 2 π 2 .Remark 6.6 A formula, similar to (174), may be obtained for any T > 0.Nevertheless, the determinants may be estimated only in the case T = ∞. □

S. Micu and E. Zuazua 79To compute the determinats |G| and |G m | we use the following lemma (see[17]):Lemma 6.2 If C = (c ij ) 1≤i,j≤n is a matrix of coefficients c ij = 1a i+b jthen∏1≤i

80 Controllability of Partial Differential EquationsConsequently, the limit lim n→∞ ||θ n m|| L 2 (0,∞) = L ≥ 1 exists. The proofends if we prove thatlimn→∞ ||θn m|| L 2 (0,∞) = ||θ m || L 2 (0,∞). (179)Identity (169) implies that lim n→∞ ||p m − r n m|| L2 (0,∞) = 1/L and (179) isequivalent tolimn→∞ ||p m − r n m|| L2 (0,∞) = ||p m − r m || L2 (0,∞). (180)Let now ε > 0 be arbitrary. Since r m ∈ E(m, Λ, ∞) it follows that thereexist n(ε) ∈ N ∗ and rm ε ∈ Emn(ε) withFor any n ≥ n(ε) we have that||p m − r m || =||r m − r ε m|| L 2 (0,∞) < ε.min ||p m − r|| ≤ ||p m − rm|| n = min ||p m − r|| ≤r∈E(m,Λ,∞) r∈Emn≤ ||p m − r ε m|| ≤ ||p m − r m || + ||r m − r ε m|| < ||p m − r m || + ε.Thus, (180) holds and Lemma 6.3 is proved.Finally, to evaluate θ m (∞, · ) we use the following estimateLemma 6.4 There exist two positive constants M and ω such that for anym ≥ 1,∞∏′ m2 + k 2|m 2 − k 2 | ≤ Meωm . (181)∏ ′kk=1Proof: Remark thatm 2 + k 2 [ ∑′( m 2|m 2 − k 2 | = exp ln + k 2 )]k |m 2 − k 2 |□[ ∑′)]≤ exp(1 ln + 2m2k |m 2 − k 2 .|Now+∑)′(1 ln + 2m2k |m 2 − k 2 ≤|∫ 2mmln(1 + 2m2[∫ 1= m ln0x 2 − m 2 )dx +∫ m(1 + 21 − x 2 )dx+1∫ ∞)ln(1 + 2m2m 2 − x 2 dx+2m∫ 21ln(1 + 2m2)x 2 − m 2 dx =(ln 1 + 2 )x 2 dx+− 1

S. Micu and E. Zuazua 81+∫ ∞2(ln 1 + 2 ) ]x 2 dx = m (I 1 + I 2 + I 3 ) .− 1We evaluate now each one of these integrals.I 1 =∫ 10∫ 10(ln 1 + 2 )1 − x 2 dx =(ln 1 + 2 )dx == −1 − x(== − (1 − x) ln 1 + 21 − xI 2 =∫ 21(ln 1 + 2=∫ 2(= − (x − 1) ln 1 +I 3 =∫ ∞21x 2 − 1(ln 1 + 2≤(x − 1) ′ ln∫ 10∫ 10)∣ ∣∣∣1)dx ≤)∣2 ∣∣∣1(x − 1) 2x 2 − 1∫ ∞2(1 +0)dx ≤()2ln 1 +dx ≤(1 − x)(1 + x)((1 − x) ′ ln 1 + 2∫ 1+00∫ 21ln1 − x)dx =23 − x dx = c 1 < ∞,()21 +(x − 1) 2 dx =)2(x − 1) 2 dx =∫ 2+1∫ ∞22(x − 1) 2 dx = c 3 < ∞.The proof finishes by taking ω = c 1 + c 2 + c 3 .22 + (x − 1) 2 dx = c 2 < ∞.()2ln 1 +(x − 1) 2 dx ≤The proof of Theorem 6.5 ends by taking into account relation (178) andLemma 6.4. □□6.6 Estimate of the norm of the biorthogonal sequence:T < ∞We consider now T < ∞. To evaluate the norm of the biorthogonal sequence(θ m (T, · )) m≥1 in L 2 (0, T ) the following result is necessary. The first version ofthis result may be found in [57] (see also [21] and [58]).Theorem 6.6 Let Λ be the family of exponential functions ( e −λnt) and letn≥1T be arbitrary in (0, ∞). The restriction operatorR T : E(Λ, ∞) → E(Λ, T ), R T (v) = v |[0,T ]

82 Controllability of Partial Differential Equationsis invertible and there exists a constant C > 0, which only depends on T , suchthat||R −1T|| ≤ C. (182)Proof: Suppose that, on the contrary, for some T > 0 there exists a sequenceof exponential polynomialssuch thatandP k (t) =N(k)∑n=1a kn e −λnt ⊂ E(Λ, T )lim ||P k|| Lk→∞ 2 (0,T ) = 0 (183)||P k || L2 (0,∞) = 1, ∀k ≥ 1. (184)By using the estimates from Theorem 6.5 we obtain that∫ ∞|a mn | =∣ P k (t)θ m (∞, t)dt∣ ≤ ||P k|| L 2 (0,∞)||θ m (∞, · )|| L 2 (0,∞) ≤ Me ωm .Thus0|P k (z)| ≤N(k)∑n=1|a kn | ∣ ∣ ∑ ∞e−λ nz ≤ M e ωn−n2 π 2 Re(z) . (185)If r > 0 is given let ∆ r = {z ∈ C : Re(z) > r}. For all z ∈ ∆ r , we havethat∞∑|P k (z)| ≤ M e ωn−n2 π 2r ≤ M(ω, r). (186)n=1Hence, the family (P k ) k≥1 consists of uniformly bounded entire functions.From Montel’s Theorem (see [15]) it follows that there exists a subsequence,denoted in the same way, which converges uniformly on compact sets of ∆ r toan analytic function P .Choose r < T . From (183) it follows that lim k→∞ ||P k || L2 (r,T ) = 0 andtherefore P (t) = 0 for all t ∈ (r, T ). Since P is analytic in ∆ r , P must beidentically zero in ∆ r .Hence, (P k ) k≥1 converges uniformly to zero on compact sets of ∆ r .Let us now return to (185). There exists r 0 > 0 such thatIndeed, there exists r 0 > 0 such thatn=1|P k (z)| ≤ Me −Re(z) , ∀z ∈ ∆ r0 . (187)ωn − n 2 π 2 Re(z) ≤ −Re(z) − n,∀z ∈ ∆ r0

S. Micu and E. Zuazua 83and therefore, for any z ∈ ∆ r0 ,|P k (z)| ≤ M ∑ n≥1Lebesgue’s Theorem implies thate ωn−n2 π 2 Re(z) ≤ Me −Re(z) ∑ n≥1e −n =Me − 1 e−Re(z) .lim ||P k|| Lk→∞ 2 (r,∞) = 0and consequentlylim ||P k|| Lk→∞ 2 (0,r) = 1.If we take r < T the last relation contradicts (184) and the proof ends.□We can now evaluate the norm of the biorthogonal sequence.Theorem 6.7 There exist two positive constants M and ω with the propertythat||θ m (T, · )|| L 2 (0,T ) ≤ Me ωm , ∀m ≥ 1 (188)where (θ m (T, · )) m≥1 is the biorthogonal sequence to the family Λ in L 2 (0, T )which belongs to E(Λ, T ) and it is given in Theorem 6.4.Proof: Let (R −1T)∗ : E(Λ, ∞) → E(Λ, T ) be the adjoint of the boundedoperator R −1T. We have thatδ kj =∫ ∞0p k (t)θ j (∞, t)dt ==∫ T0∫ ∞0(R −1T R T )(p k (t))θ j (∞, t)dt =R T (p k (t))(R −1T)∗ (θ j (∞, t))dt.Since (R −1T)∗ (θ j (∞, · )) ∈ E(Λ, T ), from the uniqueness of the biorthogonalsequence in E(Λ, T ), we finally obtain thatHence||θ j (T, · )|| L 2 (0,T ) = ∣ ∣ ∣ ∣ (R−1T(R −1T )∗ (θ j (∞, · )) = θ j (T, · ), ∀j ≥ 1.)∗ (θ j (∞, · )) ∣ ∣ ≤ L 2 (0,T ) ||R−1 T|| ||θ j(∞, · )|| L 2 (0,∞),since ||(R −1T)∗ || = ||R −1T ||.The proof finishes by taking into account the estimates from Theorem 6.5.□Remark 6.7 From the proof of Theorem 6.7 it follows that the constant ω doesnot depend of T . □

84 Controllability of Partial Differential EquationsReferences[1] S. Alinhac, Non unicité du problème de Cauchy, Annals of Mathematics,117 (1983), 77-108.[2] B. Allibert, Analytic controllability of the wave equation over cylinder,ESAIM: COCV, 4 (1999), 177-207.[3] S. A. Avdonin and S. A. Ivanov, Families of exponentials. Themethod of moments in controllability problems for distributedparameter systems, Cambridge Univ. Press, 1995.[4] C. Baiocchi, V. Komornik and P. Loreti, Ingham-Beurling type theoremswith weakened gap condition, Acta Math. Hungar., 97, (2002), 55-95.[5] J. M. Ball, Strongly Continuous Semigroups, Weak Solutions and Variationof Constant Formula, Proc. AMS, 63, 2 (1977), 370-373.[6] J. M. Ball and M. Slemrod: Nonharmonic Fourier Series and the Stabilizationof Distributed Semi-Linear Control Systems, Comm. Pure Appl.Math., XXXII (1979), 555-587.[7] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for theobservation, control and stabilization of waves from the boundary, SIAMJ. Cont. Optim., 30 (1992), 1024-1065.[8] A. Beurling and P. Malliavin, On the closure of characters and the zerosof entire functions, Acta Math., 118 (1967), 79-93.[9] R. P. Boas, Entire Functions, Academic Press, New York, 1954.[10] H. Brezis, Analyse fonctionnelle. Théorie et applications. Masson,Paris, 1983.[11] N. Burq, Contrôle de l’équation des ondes dans des ouverts peu réguliers,Asymptotic Analysis, 14 (1997), 157-191.[12] C. Castro and E. Zuazua, Some topics on the control and homogenizationof parabolic Partial Differential Equations, in “Homogenization 2001.Proceedings of the First HMS2000 International School andConference on Homogenization”, Naples, June 2001, L. Carboneand R. De Arcangelis Eds., GAKUTO Internat. Ser. Math. Sci. Appl. 18,Gakkotosho, Tokyo (2003), pp. 45-94.[13] T. Cazenave, On the propagation of confined waves along the geodesics, J.Math. Anal. Appl., 146 (1990), 591-603.

S. Micu and E. Zuazua 85[14] T. Cazenave and A. Haraux, Introduction aux problèmes d’évolutionsemi-linéaires, Mathématiques et Applications, 1, Ellipses,Paris, 1990.[15] A. Chabat, Introduction à l’analyse complexe. Tome 1, “Mir”,Moscow, 1990.[16] D. Chae, O. Yu. Imanuvilov and S. M. Kim, Exact controllability forsemilinear parabolic equations with Neumann boundary conditions, J. ofDynamical and Control Systems, 2 (1996),449-483.[17] E. W. Cheney, Introduction to approximation theory, MacGraw-Hill, New York, 1966.[18] C. Fabre, J. P. Puel and E. Zuazua, Approximate controllability of thesemilinear heat equation, Proc. Royal Soc. Edinburgh, 125 A (1995), 31-61.[19] C. Fabre, J. P. Puel and E. Zuazua, Contrôlabilité approchée de l’équationde la chaleur linéaire avec des contrôles de norme L ∞ minimale, C. R.Acad. Sci. Paris, 316 (1993), 679-684.[20] H. O. Fattorini, Estimates for sequences biorthogonal to certain exponentialsand boundary control of the wave equation, Lecture Notes in Controland Inform. Sci., 2, (1979), 111-124.[21] H. O. Fattorini and D. L. Russell, Exact controllability theorems for linearparabolic equation in one space dimension, Arch. Rat. Mech. Anal., 43(1971), 272-292.[22] H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonalfunctions for real exponentials and applications to the control theory ofparabolic equations, 32 (1974), Quart. Appl. Math., 45-69.[23] E. Fernández-Cara and E. Zuazua, The cost of approximate controllabilityfor heat equations: The linear case, Adv. Differential Equations, 5 (2000),465-514.[24] E. Fernández-Cara and E. Zuazua, Null and approximate controllability forweakly blowing-up semilinear heat equations, Annales Inst. Henri Poincaré,Analyse non-linéaire, 17(2000), 583-616.[25] E. Fernández-Cara and E. Zuazua, Control Theory: History, mathematicalachievements and perspectives. Boletín SEMA (Sociedad Española deMatemática Aplicada), 26, 2003, 79-140.

86 Controllability of Partial Differential Equations[26] A. V. Fursikov, Exact boundary zero controllability of three dimensionalNavier-Stokes equations, Journal of Dynamical and Control Systems, 1(3) (1995), 325-350.[27] A. V. Fursikov and O. Yu. Imanuvilov, Local exact controllability of theBoussinesqu equation, SIAM J. Cont. Opt., 36 (1998), 391-421.[28] A. V. Fursikov and O. Yu. Imanuvilov, On exact boundary zero-controllabilityof two-dimensional Navier-Stokes equations, Acta ApplicandaeMathematicae, 37 (1994), 67-76.[29] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of evolutionequations, Lecture Notes Series # 34, Research Institute of Mathematics,Global Analysis Research Center, Seoul National University, 1996.[30] P. Gérard, Microlocal defect measures, Comm. P.D.E., 16 (1991), 1761-1794.[31] A. Haraux, Systèmes dynamiques dissipatifs et applications, RMA17, Masson, Paris, 1990.[32] L. F. Ho, Observabilité frontière de l’équation des ondes, C. R. Acad. Sci.Paris, 302 (1986), 443-446.[33] L. Hörmander, Linear Partial Differential Equations, SpringerVerlag, 1969.[34] O. Yu. Imanuvilov, Boundary controllability of parabolic equations, RussianAcad. Sci. Sb. Math., 186 (1995), 109-132 (in Russian).[35] O. Yu Imanuvilov and M. Yamamoto, Carleman estimate for a parabolicequation in Sobolev spaces of negative order and its applications, in Controlof Nonlinear Distributed Parameter Systems, G. Chen etal. eds., Marcel-Dekker, 2000, pp. 113-137.[36] J.-A. Infante and E. Zuazua, Boundary observability for the space semidiscretizationof the 1-D wave equation, M2AN, 33 (1999), 407-438.[37] A. E. Ingham, Some trigonometrical inequalities with applications to thetheory of series, Math. Z., 41(1936), 367-369.[38] S. Jaffard and S. Micu, Estimates of the constants in generalized Ingham’sinequality and applications to the control of the wave equation, AsymptoticAnalysis, 28 (2001), pp. 181-214.[39] S. Kaczmarz and H. Steinhaus, Theorie der orthogonalreihen,Monografje Matematyczne, Tom VI., Warsaw-Lwow, 1935.

S. Micu and E. Zuazua 87[40] J. P. Kahane, Pseudo-Périodicité et Séries de Fourier Lacunaires, Ann.Scient. Ec. Norm. Sup., 37 (1962), 93-95.[41] V. Komornik, Exact controllability and stabilization. The multipliermethod., RAM: Research in Applied Mathematics, Masson,Paris, 1994.[42] G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur,Comm. P.D.E., 20 (1995), 335-356.[43] G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity,Archive Rat. Mech. Anal, 141(4)(1998), 297-329.[44] E. B. Lee and L. Markus, Foundations of Optimal Control Theory,John Wiley & Sons, 1967.[45] J. L. Lions, Contrôlabilité exacte, perturbations et stabilisationde systèmes distribués, Vol. 1 & 2, Masson, RMA, Paris, 1988.[46] J. L. Lions and E. Magenes, Problèmes aux Limites non Homogèneset Applications, Vol. 1, 2, Dunod, 1968.[47] A. López and E. Zuazua, Uniform null controllability for the one dimensionalheat equation with rapidly oscillating periodic density, Annales IHP.Analyse non linéaire, 19 (5) (2002), 543-580.[48] C. Morawetz, Notes on time decay and scatterig for some hyperbolicproblems, Regional conference series in applied mathematics, 19, SIAM,Philadelphia, 1975.[49] S. Micu, Uniform boundary controllability of a semi-discrete 1-D waveequation, Numer. Math., 91 (2002), 723-768.[50] S. Micu and E. Zuazua, On the lack of null-controllability of the heatequation on the half-line, TAMS, 353 (2001), 1635-1659.[51] A. Osses, A rotated multiplier applied to the controllability of waves, elasticity,and tangential Stokes control, SIAM J. Control Optim., 40 (2001),777-800.[52] A. Osses and J. P. Puel, On the controllability of the Lapalce equation onan interior curve, Revista Matemática Complutense, 11 (2)(1998).[53] R. E. A. C. Paley and N. Wiener, Fourier Transforms in ComplexDomains, AMS Colloq. Publ., 19, 1934.[54] L. Robbiano and C. Zuily, Uniqueness in the Cauchy problem for operatorswith partially holomorphic coefficients, Invent. math., 131 (1998), 493-539.

88 Controllability of Partial Differential Equations[55] D. L. Russell, Controllability and stabilizability theory for linear partialdifferential equations. Recent progress and open questions, SIAM Rev., 20(1978), 639-739.[56] D. L. Russell, A unified boundary controllability theory for hyperbolic andparabolic partial differential equations, Studies in Appl. Math., 52 (1973),189-221.[57] L. Schwartz, Etude des sommes d’exponentielles, Hermann Paris,1959.[58] S. Hansen, Bounds on functions biorthogonal to a set of complex expoenentials:Control of damped elastic system, J. Math. Anal. Appl., 158 (1991),487-508.[59] SIAM, Future Directions in Control Theory, Report of the Panelof Future Directions in Control Theory, SIAM Report on Issues in MathematicalSciences, Philadelphia, 1988.[60] E. D. Sontag, Mathematical control theory. Deterministic finite-dimensionalsystems, Second edition, Texts in Applied Mathematics,6. Springer-Verlag, New York, 1998.[61] D. Tataru, A priori estimates of Carleman’s type in domains with boundary,J. Math. Pures Appl., 73 (1994), 355-387.[62] D. Tataru, Unique continuation for solutions to PDE’s: between Hörmander’stheorem and Holmgren’s theorem, Comm. PDE, 20 (6-7)(1996), 855-884.[63] M. Tucsnak, Regularity and exact controllability for a beam with piezoelectricactuators, SIAM J. Cont. Optim., 34 (1996), 922-930.[64] R. Young, An Introduction to Nonharmonic Fourier Series, AcademicPress, 1980.[65] J. Zabczyk, Mathematical Control Theory: An Introduction,Birkhäuser, Basel, 1992.[66] X. Zhang, Explicit observability estimate for the wave equation with potentialand its application, Proceedings of The Royal Society of London A,456 (2000), 1101-1115.[67] X. Zhang, Explicit observability inequalities for the wave equation withlower order terms by means of Carleman inequalities, SIAM J. Cont. Optim.,39 (2001), 812-834.

S. Micu and E. Zuazua 89[68] E. Zuazua, Controlabilidad exacta y estabilización la ecuaciónde ondas, Textos de métodos matemáticos 23, Universidad Federal doRio de Janeiro, 1990.[69] E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math.Pures Appl., 74 (1995), 303-346.[70] E. Zuazua, Some problems and results on the controllability of PartialDifferential Equations, Proceedings of the Second European Conference ofMathematics, Budapest, July 1996, Progress in Mathematics, 169, 1998,Birkhäuser Verlag Basel/Switzerland, pp. 276–311.[71] E. Zuazua, Finite dimensional null controllability for the semilinear heatequation, J. Math. Pures Appl., 76 (1997), 570-594.[72] E. Zuazua, Controllability of Partial Differential Equations and its Semi-Discrete Approximations, Discrete and Continuous Dynamical Systems, 8(2) (2002), 469-513.[73] E. Zuazua, Exact controllability for the semilinear wave equation in onespace dimension, Ann. IHP. Analyse non linéaire, 10 (1993), 109-129.[74] E. Zuazua, Propagation, Observation, Control and Finite-Difference NumericalApproximation of Waves, preprint, 2004.S. M. E. Z.Facultatea de Matematica, Departamento de Matemáticas,Universitatea din Craiova, Facultad de Ciencias,Al. I. Cuza, 13,Universidad Autónoma, Cantoblanco,Craiova, 1100, Romania. 28049, Madrid, Spain.sd micu@yahoo.comenrique.zuazua@uam.es