Nonconvex Optimization Problems Depending on a Parameter

Abstract

This paper deals with infinite-dimensional nonconvex optimization problems (or non-convex control problems), which usually admit no solutions. The problem is perturbed by adding to the cost functional certain expressions depending on a parameter. The main result is that for “almost” all values of the parameter the optimization problem possesses at least one solution. After deriving the general result with the use of convex analysis and the properties of normed spaces, we present an example of a control problem for a system governed by a partial differential equation with boundary conditions of the Dirichlet type.