# Contributions to the Control Theory of Some Partial Functional Integrodifferential Equations In Banach Spaces

Thesis

This thesis is a contribution to Control Theory of some Partial Functional Integrodifferential Equations in Banach spaces. It is made up of two parts: controllability and existence of optimal controls. In the first part, we consider the dynamical control systems given by the following models that arise in the analysis of heat conduction in materials with memory, and viscoelasticity, and take the form of a: • Partial functional integrodifferential equation subject to a nonlocal initial condition in a Banach space (X, · ) : t B(t − s)x(s)ds + f (t, x(t)) + Cu(t) x (t) = Ax(t) += 0for t ∈ I = [0, b], x(0) = x0 + g(x),(0.0.1) where x0 ∈ X, g : C(I, X) → X and f : I × X → X are functions satisfying some conditions; A : D(A) → X is the infinitesimal generator of a C0 -semigroup T (t) t≥0 on X; for t ≥ 0, B(t) is a closed linear operator with domain D(B(t)) ⊃ D(A). The control u belongs to L2 (I, U ) which is a Banach space of admissible controls, where U is a Banach space. • Partial functional integrodifferential equation with finite delay in a Ba-nach space (X, · ) : t x (t) = Ax(t) + B(t − s)x(s)ds + f (t, xt ) + Cu(t) , 0 for t ∈ I = [0, b], x0 = φ ∈ C = C([−r, 0]; X), viii (0.0.2) where f : I × C → X is a function satisfying some conditions; A : D(A) → X is the infinitesimal generator of a C0 -semigroup T (t) t≥0 on X; for t ≥ 0, B(t) is a closed linear operator with domain D(B(t)) ⊃ D(A). The control u belongs to L2 (I, U ) which is a Banach space of admissible controls, where U is a Banach space, and xt denotes the history function of C of the state from the time t − r up to the present time t, and is defined by xt (θ) = x(t + θ) for −r ≤ θ ≤ 0. • Partial functional integrodifferential equation with infinite delay in a Banach space (X, · ) : t x (t) = Ax(t) + γ(t − s)x(s)ds + f (t, xt ) + Cu(t), 0 for t ∈ I = [0, b] (0.0.3) x0 = φ ∈ B, where A : D(A) → X is the infinitesimal generator of a C0 -semigroup T (t) t≥0 on a Banach space X; for t ≥ 0, γ(t) is a closed linear operator with domain D(γ(t)) ⊃ D(A). The control u takes values from another Banach space U . The operator C(t) belongs to L(U, X) which is the Banach space of bounded linear operators from U into X, and the phase space B is a linear space of functions mapping ]−∞, 0] into X satisfying axioms which will be described later, for every t ≥ 0, xt denotes the history function of B defined by xt (θ) = x(t + θ) for − ∞ ≤ θ ≤ 0, f : I × B → X is a continuous function satisfying some conditions. We give sufficient conditions that ensure the controllability of the systems without assuming the compactness of the semigroup, by supposing that their linear homogeneous and undelayed parts admit a resolvent operator in the sense of Grimmer and by making use of the Hausdorff measure of noncom- pactness. In the second part, we consider equations (0.0.1), (0.0.2) and (0.0.3), in the case where the operator C = C(t) (time dependent), the function g ≡= 0, the Banach spaces X and U are separable and reflexive. Using techniques of convex optimization, a priori estimation, and applying Balder’s Theorem, we establish the existence of optimal controls for the following Lagrange optimal control problem associated to each of the equations: Find a control u0 ∈ Uad such that (LP) J (u0 ) ≤ J (u) for all u ∈ Uad , where T L t, xu , xu (t), u(t) dt, t J (u) := 0 L is some functional, x denotes the mild solution corresponding to the control u ∈ Uad , and Uad denotes the set of admissible controls.

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