Optimal Control of Partial Differential Equations
Optimal Control of Partial Differential Equations Optimal Control of Partial Differential Equations
92 CHAPTER 8. EQUATIONS WITH UNBOUNDED CONTROL OPERATORS with the initial condition and the boundary conditions z1(x, 0) = z01(x), z2(x, 0) = z02(x) in (0, ℓ), (8.6.23) z1(ℓ, t) = u1(t), z2(0, t) = u2(t) in (0, T ). (8.6.24) This kind of systems intervenes in heat exchangers [31]. For simplicity we suppose that the coefficients m1 > 0, m2 > 0, b11, b12, b21, b22 are constant. We also suppose that b11z 2 1 + b21z2z1 + b21z1z2 + b22z 2 2 ≥ 0 for all (z1, z2) ∈ R 2 . Before studying control problems, let us state existence results for the system (8.6.22)-(8.6.24). 8.6.1 State equation We set Z = L 2 (0, ℓ) × L 2 (0, ℓ), and we define the unbounded operator A in Z by and D(A) = {z ∈ H 1 (0, ℓ) × H 1 (0, ℓ) | z1(ℓ) = 0, z2(0) = 0} Az = ⎡ ⎢ ⎣ dz1 m1 dx − b11z1 − b12z2 dz2 −m2 dx − b21z1 − b22z2 We endow D(A) with the norm zD(A) = (z12 H1 2 (0,ℓ) + z2H 1 (0,ℓ) )1/2 . Theorem 8.6.1 For every (f, g) ∈ L 2 (0, ℓ) 2 , the system Az = (f, g) T admits a unique solution in D(A), and zD(A) ≤ C(f L 2 (0,ℓ) + g L 2 (0,ℓ)). Proof. Let A0 be the operator defined by D(A0) = D(A) and A0z = (m1 dz1 dz2 , −m2 dx dx )T . It is clear that A0 is an isomorphism from D(A0) into L2 (0, ℓ) 2 . We rewrite equation Az = (f, g) T in the form z − A −1 0 Bz = A −1 0 (f, g) T , where b11z1 + b12z2 Bz = b21z1 + b22z2 If z ∈ D(A0), then Bz ∈ (H 1 (0, ℓ)) 2 and A −1 0 Bz ∈ (H 2 (0, ℓ)) 2 ∩ D(A0). Thus the operator A −1 0 B is a compact operator in D(A0). Let us prove that I −A −1 0 B is injective. Let z ∈ D(A0) be such that (I − A −1 0 B)z = 0. Then Az = 0. Multiplying the first equation in the system Az = 0 by z1, the second equation by z2, integrating over (0, ℓ), and adding the two equalities, we obtain: . ⎤ ⎥ ⎦ . m1z1(0) 2 + m2z2(ℓ) 2 + ℓ 0 b11z 2 1 + b21z2z1 + b21z1z2 + b22z 2 2 = 0. Thus z = 0. Now the theorem follows from the Fredholm Alternative. Theorem 8.6.2 The operator (A, D(A)) is the infinitesimal generator of a strongly continuous semigroup of contractions on Z.
8.6. A FIRST ORDER HYPERBOLIC SYSTEM 93 Proof. The theorem relies the Hille-Yosida theorem. (i) The domain D(A) is dense in Z. Prove that A is a closed operator. Let (zn)n = (z1,n, z2,n)n be a sequence converging to z = (z1, z2) in Z, and such that (Azn)n converges to (f, g) in Z. We have m1 dz1 dx − b11z1 − b12z2 = f, and −m2 dz2 dx − b21z1 − b22z2 = g, because ( dz1,n dz2,n , dx dx )n ) in the sense of distributions in (0, ℓ). Due to Theorem 8.6.1, we have converges to ( dz1 dx , dz2 dx zn − zmD(A) ≤ CA(zn − zm) (L 2 (0,ℓ)) 2. Thus (zn)n is a Cauchy sequence in D(A), and z, its limit in Z, belongs to D(A). The first condition of Theorem 4.1.1 is satisfied. (ii) For λ > 0, f ∈ L2 (0, ℓ), g ∈ L2 (0, ℓ), consider the equation z ∈ D(A), z1 z1 λ − A f = g that is z2 z2 , dz1 λz1 − m1 dx + b11z1 + b12z2 = f in (0, ℓ), z1(ℓ) = 0, dz2 λz2 + m2 dx + b21z1 + b22z2 = g in (0, ℓ), z2(0) = 0. As for Theorem 8.6.1, we can prove that this equation admits a unique solution z ∈ D(A). Multiplying the first equation by z1, the second by z2, and integrating over (0, ℓ), we obtain λ ℓ 0 (z 2 1 + z 2 2) + The proof is complete. ℓ (b11z 0 2 1 + b12z2z1 + b21z1z2 + b22z 2 2) + m1z1(0) 2 + m2z2(ℓ) 2 ℓ = 0 ℓ ≤ 0 z 2 1 + ℓ 0 z 2 2 1/2 ℓ f 2 + Theorem 8.6.3 For every z0 = (z10, z20) ∈ Z, equation ∂ z1(x, t) = ∂t z2(x, t) ∂ ∂x m1z1 −m2z2 b11z1 + b12z2 − b21z1 + b22z2 with the initial condition and homogeneous boundary conditions 0 ℓ 0 g 2 1/2 . , in (0, ℓ) × (0, T ) z1(x, 0) = z01(x), z2(x, 0) = z02(x) in (0, ℓ), z1(ℓ, t) = 0, z2(0, t) = 0 in (0, T ), (fz1 + gz2) admits a unique weak solution in L 2 (0, T ; L 2 (0, ℓ)), this solution belongs to C([0, T ]; Z) and satisfies zC([0,T ];Z) ≤ z0Z.
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8.6. A FIRST ORDER HYPERBOLIC SYSTEM 93<br />
Pro<strong>of</strong>. The theorem relies the Hille-Yosida theorem.<br />
(i) The domain D(A) is dense in Z. Prove that A is a closed operator. Let (zn)n = (z1,n, z2,n)n<br />
be a sequence converging to z = (z1, z2) in Z, and such that (Azn)n converges to (f, g) in Z.<br />
We have m1 dz1<br />
dx − b11z1 − b12z2 = f, and −m2 dz2<br />
dx − b21z1 − b22z2 = g, because ( dz1,n dz2,n<br />
, dx dx )n<br />
) in the sense <strong>of</strong> distributions in (0, ℓ). Due to Theorem 8.6.1, we have<br />
converges to ( dz1<br />
dx<br />
, dz2<br />
dx<br />
zn − zmD(A) ≤ CA(zn − zm) (L 2 (0,ℓ)) 2.<br />
Thus (zn)n is a Cauchy sequence in D(A), and z, its limit in Z, belongs to D(A). The first<br />
condition <strong>of</strong> Theorem 4.1.1 is satisfied.<br />
(ii) For λ > 0, f ∈ L2 (0, ℓ), g ∈ L2 (0, ℓ), consider the equation<br />
z ∈ D(A),<br />
<br />
z1 z1<br />
λ − A<br />
<br />
f<br />
=<br />
g<br />
that is<br />
z2<br />
z2<br />
<br />
,<br />
dz1<br />
λz1 − m1<br />
dx + b11z1 + b12z2 = f in (0, ℓ), z1(ℓ) = 0,<br />
dz2<br />
λz2 + m2<br />
dx + b21z1 + b22z2 = g in (0, ℓ), z2(0) = 0.<br />
As for Theorem 8.6.1, we can prove that this equation admits a unique solution z ∈ D(A).<br />
Multiplying the first equation by z1, the second by z2, and integrating over (0, ℓ), we obtain<br />
λ<br />
ℓ<br />
0<br />
(z 2 1 + z 2 2) +<br />
The pro<strong>of</strong> is complete.<br />
ℓ<br />
(b11z<br />
0<br />
2 1 + b12z2z1 + b21z1z2 + b22z 2 2) + m1z1(0) 2 + m2z2(ℓ) 2 ℓ<br />
=<br />
0<br />
ℓ<br />
≤<br />
0<br />
z 2 1 +<br />
ℓ<br />
0<br />
z 2 2<br />
1/2 ℓ<br />
f 2 +<br />
Theorem 8.6.3 For every z0 = (z10, z20) ∈ Z, equation<br />
<br />
∂ z1(x, t)<br />
=<br />
∂t z2(x, t)<br />
∂<br />
<br />
∂x<br />
m1z1<br />
−m2z2<br />
<br />
b11z1 + b12z2<br />
−<br />
b21z1 + b22z2<br />
with the initial condition<br />
and homogeneous boundary conditions<br />
0<br />
ℓ<br />
0<br />
g 2<br />
1/2 .<br />
<br />
, in (0, ℓ) × (0, T )<br />
z1(x, 0) = z01(x), z2(x, 0) = z02(x) in (0, ℓ),<br />
z1(ℓ, t) = 0, z2(0, t) = 0 in (0, T ),<br />
(fz1 + gz2)<br />
admits a unique weak solution in L 2 (0, T ; L 2 (0, ℓ)), this solution belongs to C([0, T ]; Z) and<br />
satisfies<br />
zC([0,T ];Z) ≤ z0Z.
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