Optimal Control of Partial Differential Equations

In this chapter, we present an introduction to the optimal control of partial differential equations. After explaining what an optimal control problem is and the goals of the analysis of these problems, we focus the study on a model example. We consider an optimal control problem governed by a semilinear elliptic equation, the control being subject to bound constraints. Then we explain the methods to prove the existence of a solution; to derive the first and second order optimality conditions; to approximate the control problem by discrete problems; to prove the convergence of the discretization and to get some error estimates. Finally we present a numerical algorithm to solve the discrete problem and we provide some numerical results. Though the whole analysis is done for an elliptic control problem, with distributed controls, some other control problems are formulated, which show the scope of the field of control theory and the variety of mathematical methods necessary for the analysis. Among these problems, we consider the case of evolution equations, Neumann or Dirichlet boundary controls, and state constraints.