Numerical analysis of PDE constrained optimal control problems with pointwise inequality constraints on the state and the control

Abstract

The purpose of this thesis is the numerical analysis of optimal control problems with partial di erential equations (PDEs), whose control or state is subject to pointwise inequality constraints. We are speci cally interested in nonconvex problems with semilinear state equation. It is intrinsic to the considered problem class that solutions can often only be found by numerical methods. Consequently, one is interested in estimating the error between a (local) solution of the continuous problem and an associated discrete local solution. A basic requirement for this purpose is a thorough discussion of the continuous problem. In the presence of pointwise state constraints this leads to speci c di culties of analytical and numerical nature. These di culties have to be approached either directly or by means of regularization. With this thesis we make several new contributions to the discussion of optimal control problems with pointwise state constraints but also those with pure control constraints. After a short introduction into the eld of research and providing some basic theoretical results we discuss in Chapter 3 a semiin nite elliptic optimal control problem. Known results on necessary and su cient optimality conditions, the comparably high regularity of solutions to elliptic PDEs, as well as the nite dimensional control space allows to address a priori discretization error estimates for this problem directly, i. e. without further regularization. Our main result in this chapter is an a priori error bound of order O(h2j lnhj) for local solutions of a nite element discretization with mesh size h of this optimal control problem in two space dimensions. For that, we rely on certain assumptions on the structure of the active set. In Chapter 4 we address a parabolic optimal control problem with L bounds on the control functions, semilinear state equation, and pointwise state constraints in the whole space-time-domain. In contrast to the elliptic problem discussed in Chapter 3, second order su cient conditions are only available for one-dimensional spatial domains. Moreover, the use of control functions instead of nitely many control parameters does not allow any a priori assumptions on the structure of the active sets. Therefore, we use a Lavrentiev regularization as suggested originally by Meyer, Rosch and Troltzsch. Consequently, we can make use of available higher regularity results for the Lagrange multipliers that allow for a deeper analysis. We prove a convergence result for locally optimal solutions of the regularized problem and show local uniqueness of regularized local solutions. In Chapter 5 we analyze the nite element discretization of a control-constrained parabolic optimal control problem with semilinear state equation. We prove error estimates for discrete local solutions in the L-norm. We extend known results for linear-quadratic problems to the nonconvex setting. In particular, we have to prove boundedness results for the semidiscrete and discrete state functions in the L-norm that hold independently of the discretization parameters. Moreover, the discussion of local solutions requires convergence results and quadratic growth conditions to be considered in the same spaces.