Simultaneous Exact/Approximate Boundary Controllability of Thermo-Elastic Plates with Variable Thermal Coefficient and Moment Control1

Model of Non-homogeneous Problem (3.2) This is given explicitly in [E-L-T.2]. 4. CONSEQUENCE OF THE STRUCTURAL DECOMPOSITION: A STRATEGY FOR THE CONTROLLABILITY PROBLEM 4.0. Preliminaries [L-T.7 Sect. 6] presents a strategy, essentially already used in [LT.3, pp. 119–120], to obtain an exact controllability (surjectivity) result. In the present case of thermo-elastic plates, its applicability is based on the structural decomposition Theorem 3.2, combined with a soft argument as in [E-L-T.2 Appendix D, A-L.1]. This is amply elaborated in [L-T.7, Sect. 6.2]. With reference to the thermo-elastic plate (1.1), we take henceforth zero initial condition y0 = w0 w1 θ0 = 0 and boundary controls u = u1 u2 u3 in (1.2), of the same class as specified in Theorem 1.1: i.e., u1 ∈ C∞ 0 1 u3 ∈ C∞ 0 3 , and u2 ∈ L2 2 . We then define the input-solution operator T at the terminal time T by Tu = w T wt T θ T = ∫ T 0 e γ T−t u t dt (4.0.1a) continuous → Xγ ≡ Y1 γ ×H1 0 (4.0.1b) Xγ ≡ Y1 γ ×H1 0 ≡ H2 ∩H1 0 ×H1 0 ×H1 0 ⊂ Yγ (4.0.1c) (see (2.3)), where the asserted regularity in (4.0.1b) follows (mostly) from Proposition 2.2, where u1 ≡ u3 ≡ 0, u2 ∈ L2 . The precise form of the boundary → interior operator is given in [E-L-T.2, Appendix B]. Let &m be the projection (see (2.3)) Yγ → Y1 γ,: v1 v2 v3 → v1 v2 onto the mechanical state space and let &m v1 v2 → v1 v2 0 be its adjoint Y1 γ → Yγ. The strategy for controllability, as stated in Theorem 1.1, hinges on the following two steps. Step 1. Show exact controllability on the space Y1 γ in (4.0.1b,c) or (2.3) from the origin at time t = T of the thermo-elastic plate problem (1.1), (1.2) in the mechanical variables; in symbols, with reference to (4.0.1), show that &m T surjective onto Y1 γ (4.0.2) controllability of thermo-elastic plates 463 where is the preassigned space of controls u = u1 u2 u3 , specified by Theorem 1.1. Step 2. Show approximate controllability on the space Xγ in (4.0.1b,c) from the origin at time t = T of the thermo-elastic plate (1.1), (1.2): in symbols, show that the range of T is dense in Xγ = Y1 γ × Y2, Y2 = H1 0 : T ≡ T = Xγ = range (4.0.3) Once Steps 1 and 2 are accomplished, a soft argument as in [E-L-T.2, Appendix D, Theorem D.1, A-L.1], where T is continuous → Xγ, as noted in (4.0.1a), then shows the following Desired Conclusion Steps 1 and 2 imply exact controllability on the space Y1 γ from the origin at time t = T of the thermo-elastic plate (1.1), (1.2) in the mechanical variable and, simultaneously, approximate controllability on the space H1 0 from the origin at time t = T in the thermal variable, i.e., precisely, the statement of Theorem 1.1. 4.1. Implementation of Step 1: Exact Controllability of the Thermo-Elastic Plate Problem in the Mechanical Variables As explained in [L-T.7, Sect. 6], it is at the level of implementing Step 1 that the structural decomposition of the thermo-elastic semigroup as in Theorem 3.2 is critically used. The key is the following simple result, essentially already used in [L-T.3, pp. 119–120], from approximate to exact controllability. Proposition 4.1.1 [L-T.7, Proposition 6.1.1]. Let J = S + Q, and let X be a Hilbert space, where: (i) J is a closed operator ⊂ J → X with dense range J = X (approximate controllability), equivalently, with trivial null space of the adjoint J∗ J∗ = 0 ; (ii) S is a closed, surjective operator: ⊂ S onto X, where S = J ; (iii) Q is a compact operator: → X. Then, J is surjective ⊂ J onto X (exact controllability). To implement Step 1 to our problem, and with reference to the decomposition (3.8) of Theorem 3.2, we return to (4.1) and take in 464 eller, lasiecka, and triggiani agreement with Theorem 3.2(b), Eq. (3.9), (3.10):