Optimal Control of Partial Differential Equations

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100 CHAPTER 9. CONTROL OF A SEMILINEAR PARABOLIC EQUATION admits a unique weak solution in W (0, T ; H 1 0(Ω), H −1 (Ω)). Moreover z(t) = e tA z0 + t 0 e (t−s)A g(s) ds. Observe that if z belongs to C([0, T ]; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0(Ω)), then z belonds to L 4 (Q) (Theorem 9.6.1), and φ(z) belongs to L 2 (0, T ; H −1 (Ω)). Thus it is reasonable to consider equation (9.2.1) as a special form of equation (9.2.2) with g = f + χωu − φ(z), and to define weak solutions to equation (9.2.1) in the following manner. Definition 9.2.1 A function z ∈ C([0, T ]; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0(Ω)) is a weak solution to equation (9.2.1) if, for every ζ ∈ H 1 0(Ω), the mapping t ↦→ 〈z(t), ζ〉 belongs to H 1 (0, T ), 〈z(0), ζ〉 = 〈z0, ζ〉, and d dt 〈z(t), ζ〉 = 〈∇z(t), ∇ζ〉 + 〈f, ζ〉 + 〈χωu, ζ〉 − 〈φ(z), ζ〉. 9.3 Existence of solutions for L 2p -initial data, p > 2 9.3.1 Existence of a local solution We suppose that f ∈ L 2 (Q), and z0 ∈ L 2p (Ω) with p > 2. We want to prove that equation ∂z ∂t − ∆z + 1 2 ∂x1(z 2 ) = f in Q, z = 0 on Σ, z(x, 0) = z0 in Ω, (9.3.3) admits a solution in C([0, ¯t]; L 2p (Ω)) for ¯t small enough. Let s > 2p p−1 . Due to Theorem 9.6.3, if h ∈ Ls (0, T ; L p (Ω)) then zh, the solution to equation (9.2.2) corresponding to (g, z0) with z0 = 0, and g = ∂x1h, belongs to C([0, T ]; L 2p (Ω)), and there exists a constant C(s) such that zh C([0,T ];L 2p (Ω)) ≤ C(s)hL s (0,T ;L p (Ω)). Set R = yC([0,T ];L2p (Ω)), where y is the solution to equation (9.2.2) corresponding to (g, z0) with g = f. Let us fix s > 2p p−1 and set ¯t = (4RC(s)) −s . Let B(2R) be the closed ball in C([0, ¯t]; L2p (Ω)), centered at the origin, with radius 2R. Endowed with the distance associated with the norm · C([0,¯t];L2p (Ω)), B(2R) is a complete metric space. For z ∈ C([0, ¯t]; L2p (Ω)), denote by Ψ(z) the solution to equation (9.2.2) corresponding to (g, z0) with g = f − φ(z). Let us show that the mapping z ↦→ Ψ(z) is a contraction in B(2R). Let z ∈ B(2R), then Let z1 and z2 be in B(2R), then Ψ(z) C([0,¯t];L 2p (Ω)) ≤ y C([0,¯t];L 2p (Ω)) + C(s)z 2 L s (0,¯t;L p (Ω)) ≤ R + C(s)¯t 1/s z 2 L ∞ (0,¯t;L p (Ω)) ≤ R + C(s)¯t 1/s R 2 ≤ 2R. Ψ(z1) − Ψ(z2) C([0,¯t];L 2p (Ω)) ≤ C(s)z 2 1 − z 2 2L s (0,¯t;L p (Ω)) ≤ C(s) ¯t 1/s 2Rz1 − z2 L ∞ (0,¯t;L 2p (Ω)) ≤ 1 2 z1 − z2 L ∞ (0,¯t;L 2p (Ω)).

9.4. EXISTENCE OF A GLOBAL WEAK SOLUTION FOR L 2 -INITILAL DATA 101 9.3.2 Initial data in D((−A) α ) Set D(A) = H2 (Ω) ∩ H1 0(Ω) and Az = ∆z for z ∈ D(A). For 1 2 < α < 1, we have D((−A)α ) = H2α (Ω) ∩ H1 0(Ω) (see [18]). Suppose that f ∈ L2 (Q), and z0 ∈ D((−A) α ), with 1 < α < 1. 2 As in section 9.3.1, we can prove that equation (9.3.3) admits a unique weak solution in C([0, ˆt]; D((−A) α )) for ˆt small enough. Since D((−A) α ) ⊂ L2p (Ω) for all p ≤ ∞, this implies that the solution defined in C([0, ˆt]; D((−A) α )) is the same as the solution defined in section 9.3.1. 9.3.3 Existence of a global solution Suppose that f ∈ L 2 (Q), and z0 ∈ D((−A) α ), with 1 2 < α < 1. Let Tmax be such that the solution to equation (9.3.3) exists in C([0, τ]; L 2p (Ω)) for all p > 2 and all τ < Tmax. If Tmax = ∞, we have proved the existence of a global solution. Otherwise, we necessarily have limτ→Tmaxz C([0,τ];L 2p (Ω)) = ∞, (9.3.4) for some p > 2. Let us show that we have a contradiction. Multiplying the equation by |z| 2p−2z, and integrating on (0, τ) × Ω, we obtain 1 2p |z(τ)| 2p τ + (2p − 1)|∇z| 2 |z| 2p−2 = 1 2p |z0| 2p τ + f|z| 2p−2 z. (9.3.5) Ω 0 Ω Indeed, with an integration by parts, we get ∂x1(z 2 )|z| 2p−2 2p − 1 z = − 2 Ω Ω Ω 0 Ω |z| 2p−2 z∂x1(z 2 ). (9.3.6) Thus it yields |z| Ω 2p−2 z∂x1(z 2 ) = 0. Moreover ∇z · ∇(|z| Ω 2p−2 z) = (2p − 1)|∇z| Ω 2 |z| 2p−2 . (9.3.7) Formula (9.3.5) is established. It is clearly in contradiction with (9.3.4). Observe that calculations in (9.3.6) and (9.3.7) are justified because z is bounded, and in that case the solution to (9.3.3) belongs to Lq (0, T ; W 1,q (Ω)) for all q < ∞ (apply Theorem 9.6.3). Therefore formulas (9.3.6) and (9.3.7) are meaningful. The regularity in C([0, T ]; L2p (Ω)) is not sufficient since in that case ∂x1(z 2 )|z| 2p−2z does not belong to L1 . It is the reason why we have constructed bounded solutions to justify (9.3.6) and (9.3.7). 9.4 Existence of a global weak solution for L 2 -initilal data Theorem 9.4.1 For all z0 ∈ L 2 (Ω), all T > 0, and all f ∈ L 2 (0, T ; L 2 (Ω)), equation (9.3.3) admits a unique weak solution in C([0, T ]; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0(Ω)) in the sense of definition

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Page 120 and 121: 120 BIBLIOGRAPHY [16] I. Lasiecka,

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