Existence of optimal controls for semilinear elliptic equations without Cesari-type conditions

Abstract

Optimal control problems governed by semilinear elliptic partial differential equations are considered. No Cesari-type conditions are assumed. By proving an existence theorem and the Pontryagin maximum principle of optimal "state-control" pairs for the corresponding relaxed problems, we establish an existence theorem of optimal pairs for the original problem.

On singular sets of local solutions to p-Laplace equations

“…It is an important problem in control theory to study the existence of a solution when Cesari condition does not hold. In the following, we will use Theorem 1.1 and the method we used in [5] to prove that for Problem 3.1, there exists at least one solution. For simplicity, the equation (3.3) and cost functional (3.2) are special, but the method we used here contains some basic ideas to study the existence of minimizers for such kind of optimal control problems.…”