UC Berkeley

In this thesis, we consider a class of optimal control problems known as coefficient control problems.Such problems are constrained by uniformly elliptic PDE in which the controls appear as some of the coefficients in the differential operator. We begin with a brief review of standard optimal control theory and show that it does not apply to control coefficient problems generally by means of two counterexamples. We then present existence results for the solution to such problems under two different assumptions regarding the controls: Lipschitz continuity with a bounded Lipschitz constant and the case in which the admissible set of controls is closed under H-convergence. We then present a new maximum principle for the latter class of problems which we subsequently use to characterize the nature and behavior of optimal solutions.