Optimal Control of Partial Differential Equations

More documents

Recommendations

Info

98 CHAPTER 8. EQUATIONS WITH UNBOUNDED CONTROL OPERATORS and D(A) = {y | y1 ∈ H 2 ∩H 1 0(0, L), y2 ∈ H 1 0(0, L), y3 ∈ H 2 (0, L) such that y3x(0) = y3x(L) = 0}. We endow Y = H 1 0(0, L) × L 2 (0, L) × L 2 (0, L) with the scalar product (y, w) = L 0 ( dy1 dx dw1 dx + y2w2 + γ1 y3w3). 1 - Prove that (A, D(A)) is the infinitesimal generator of a strongly continuous semigroup on Y . 2 - We suppose that z0 ∈ H 1 0(0, L), z1 ∈ L 2 (0, L), θ ∈ L 2 (0, L), u1 ∈ L 2 (0, T ), u2 ∈ L 2 (0, T ). Prove that system (8.7.32)-(8.7.34) admits a unique solution (z, zt, θ) in C([0, T ]; H 1 0(0, L)) × C([0, T ]; L 2 (0, L)) × C([0, T ]; L 2 (0, L)). 3 - Consider the control problem (P2) inf{J2(z, θ, u) | (z, zt, θ, u) ∈ C([0, T ]; Y ) × L 2 (0, T ) 2 , (z, zt, θ, u) satisfies (8.7.32) − (8.7.34)}, where J2(z, θ, u) = 1 T 2 0 L 0 (|z| 2 + |θ| 2 ) + β 2 γ2 T 0 (u 2 1 + u 2 2), and β > 0. Prove that (P2) admits a unique solution. Characterize this solution by establishing first order optimality conditions.
Chapter 9 Control of a semilinear parabolic equation 9.1 Introduction In this chapter we study control problems for a semilinear parabolic equation of Burgers’ type in dimension 2. For a L 2 -distributed control, we prove the existence of a unique solution to the state equation in C([0, T ]; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0(Ω)). Following the approach of the previous chapters we use the semigroup theory. We first prove the existence of a local solution for initial data in L 2p (Ω) for p > 2, and next the existence of a global solution by establishing an energy estimate. The existence of a (uinque) solution for L 2 -initial data is obtained by approximation. The classical method to study this kind of equation is the variational method (also called the Faedo-Galerkin method). This approach is treated in exercise 9.7.2. This chapter can be considered as an introduction to the optimal control of the Navier-Stokes equations [27]. Indeed the proof of optimality conditions is very similar in both cases. To study the state equation and the control problem, we need additional regularity results on parabolic equations. These results are stated in Appendix (section 9.6). 9.2 Distributed control Let Ω be a bounded domain in R 2 , with a regular boundary Γ. Let T > 0, set Q = Ω × (0, T ) and Σ = Γ × (0, T ). We consider the equation ∂z ∂t − ∆z + Φ(z) = f + χωu in Q, z = 0 on Σ, z(x, 0) = z0 in Ω, (9.2.1) with f ∈ L2 (Q), u ∈ L2 (0, T ; L2 (ω)), z0 ∈ L2 (Ω). The function f is a given source term, χω is the characteristic function of ω, ω is an open subset of Ω, and the function u is a control variable. The nonlinear term φ is defined by φ(z) = 2z∂x1z = ∂x1(z 2 ). Any other combination of first order partial derivatives may be considered, for example we can as well consider φ(z) = Σ 2 i=1z∂xi z. We first want to prove the existence of a unique weak solution (in a sense to be precised) to equation (9.2.1). Recall that, if z0 ∈ L 2 (Ω) and g ∈ L 2 (0, T ; H −1 (Ω)), equation ∂z ∂t − ∆z = g in Q, z = 0 on Σ, z(x, 0) = z0 in Ω, (9.2.2) 99
Page 1:
Université Paul Sabatier Optimal C
Page 4 and 5:
4 CONTENTS 4.3 Weak solutions in L
Page 6 and 7:
6 CONTENTS
Page 8 and 9:
8 CHAPTER 1. EXAMPLES OF CONTROL PR
Page 10 and 11:
10 CHAPTER 1. EXAMPLES OF CONTROL P
Page 12 and 13:
12 CHAPTER 1. EXAMPLES OF CONTROL P
Page 14 and 15:
14 CHAPTER 2. CONTROL OF ELLIPTIC E
Page 16 and 17:
Page 18 and 19:
Page 20 and 21:
Page 22 and 23:
Page 24 and 25:
Page 26 and 27:
Page 28 and 29:
28 CHAPTER 3. CONTROL OF SEMILINEAR
Page 30 and 31:
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
Page 38 and 39:
Page 40 and 41:
Page 42 and 43:
42 CHAPTER 4. EVOLUTION EQUATIONS 4
Page 44 and 45:
44 CHAPTER 4. EVOLUTION EQUATIONS w
Page 46 and 47:
46 CHAPTER 4. EVOLUTION EQUATIONS D
Page 48 and 49: 48 CHAPTER 5. CONTROL OF THE HEAT E
Page 64 and 65: 64 CHAPTER 6. CONTROL OF THE WAVE E
Page 76 and 77: 76 CHAPTER 7. EQUATIONS WITH BOUNDE
Page 78 and 79: 78 CHAPTER 7. EQUATIONS WITH BOUNDE
Page 80 and 81: 80 CHAPTER 8. EQUATIONS WITH UNBOUN
Page 100 and 101: 100 CHAPTER 9. CONTROL OF A SEMILIN
Page 110 and 111: 110 CHAPTER 10. ALGORITHMS FOR SOLV
Page 120 and 121: 120 BIBLIOGRAPHY [16] I. Lasiecka,
show all

Optimal Control of Partial Differential Equations

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?