Abstract
In this paper, we consider optimal control problems associated with a class of quasilinear parabolic equations, where the coefficients of the elliptic part of the operator depend on the state function. We prove existence, uniqueness and regularity for the solution of the state equation. Then, we analyze the control problem. The goal is to get first and second order optimality conditions. To this aim we prove the necessary differentiability properties of the relation control-to-state and of the cost functional.
Highlights
In this paper, we analyze the following optimal control problem (P) min J(u) α≤u(x,t)≤β with 1 J(u) = (yu(x, Q t) − yd(x, t))2 dx dt + ν 2u2(x, t) dx dt, yu being the solution of the following quasilinear partial differential equation∂y ∂t − divx [a(x, t, y(x, t))∇xy] + a0(x, t, y(x, t)) = u in Q = Ω × (0, T ), (1)y(x, t) = 0 on Σ = Γ × (0, T ), y(x, 0) = y0(x) in Ω.Above Ω ⊂ Rn, 1 ≤ n ≤ 3, is a bounded open set with a C1,1 boundary Γ ([33, Definition 1.2.1.1]), T > 0, −∞ < α < β < +∞, and ν > 0 are given numbers
We have proved the existence of a solution y of (1) with y ∈ L∞(Q) ∩ W (0, T )
Since we have that F = (0, 0), the theorem follows from the implicit function theorem if we prove that
Summary
We analyze the following optimal control problem (P) min J(u) α≤u(x,t)≤β with. We prove existence, uniqueness and regularity of. To prove the existence of a solution for this problem we use the Schauder’s fixed point theorem as follows. Let F : L2(Q) −→ L2(Q) be the functional associating with each element w ∈ L2(Q) the unique solution F (w) = yw ∈ W (0, T ) of the linear problem. Since the embedding W (0, T ) ⊂ L2(Q) is compact, we apply the Schauder’s fixed point theorem to deduce the existence of at least one fixed point yM of the functional F This fixed point is a solution of (11). There exists a unique solution y ∈ Wp2,,q1(Q) of (9) and its norm is estimated by the norms of ∇xb, f and y0 in their corresponding spaces. For these domains the operator ∆ is an isomorphism between W 2,q(Ω) ∩ W0q(Ω) and Lq(Ω) with 2 ≤ q < q∗ for some q∗ > 2, and between W01,p(Ω) and W −1,p(Ω) for every 1 < p < ∞; see [33, Chapter 4] and [27]
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