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Abstract

In the past decades, the optimal control of partial differential equations governed by partial differential equations has made significant progress. This work concerns optimal control of nonlocal partial differential equations, raising the natural question as to why this type of partial differential equations are of interest and relevance. Nonlocal partial differential equations abound in modeling of various physical and biological phenomena. In contrast to classical partial differential equations, nonlocal partial differential equations take not only the local spatial or time variables into consideration, but also any possible dependence of the involved quantities on neighboring points as well as preceding times in the evolution of the process under consideration. This type of non-local reliance typically arises from interactions over a distance or from multiple conservation laws. In this thesis, our primary emphasis was placed on two significant nonlocal partial differ- ential equations: one originating from the field of physics and the other from the biology. In our first study we consider an optimal control problem for the steady-state Kirchhoff equa- tion, a prototype for nonlocal partial differential equations, different from fractional powers of closed operators. Existence and uniqueness of solutions of the state equation, existence of global optimal solutions, differentiability of the control-to-state map and first-order neces- sary optimality conditions are established. The aforementioned results require the controls to be functions in H1 and subject to pointwise lower and upper bounds. In order to obtain the Newton differentiability of the optimality conditions, we employ a Moreau-Yosida-type penalty approach to treat the control constraint and study its convergence. The first-order optimality conditions of the regularized problems are shown to be Newton differentiable, and a generalized Newton method is detailed. A discretization of the optimal control problem by piecewise linear finite elements is proposed and numerical results are presented. In our second study, we delve into an optimal control problem involving a coupled parabolic-elliptic chemotaxis system with a nonlocal logistic growth term. We establish the existence and uniqueness of the state equation. By constructing the corresponding logistic Ordinary Differential Equation (ODE), we determine the maximal existence time to prevent blow-up. Depending on the sign coefficient of the nonlinear term in the ODE, it either blows up in finite time or in infinite time. We demonstrate that the solution y of the Partial Differential Equation (PDE) is bounded above by the solution of the ODE. Subsequently, we provide an a-priori estimate for the solution of the chemotaxis system and establish the existence of an optimal solution. Finally, we demonstrate the Fréchet differentiability of the control-to-state map and derive first-order necessary conditions using the Lagrangian method.