Optimal control and inverse problems for partial differential equations and variational inequalities

Abstract

This dissertation addresses optimal control problems for nonlinear evolutionary variational inequalities involving Volterra-type operators and inverse problems for the Dirac system on finite metric graphs. The first part presents the historical background, novelty, and motivation behind the research studies. In the second part, we focus on solving the initial value problem for nonlinear evolutionary variational inequalities with Volterra-type operators, proving the existence of a unique solution using the Banach fixed-point theorem. The third part explores an optimal control problem for these inequalities, establishing the existence of a solution under specific assumptions on the given data. In the last part of the dissertation, we examine the inverse dynamic problem for the Dirac system on finite metric tree graphs, as well as a graph with a single cycle (a ring with two attached edges). First, we solve the forward problem for this system on general graphs using a novel dynamic algorithm and then address the inverse problem for the same system on finite metric tree graphs. We recover unknown data such as the topology (connectivity) of a tree, edge lengths, and matrix potential functions associated with each edge. This is achieved using the dynamic response operator as the inverse data and the leaf peeling method. We also determine the minimum time required to uniquely identify the unknown data. Finally, we demonstrate the solution to the inverse problem for the Dirac system on a ring with two attached edges, establishing the minimum time needed to uniquely determine the unknown parameters for this graph.