Optimal control and partial differential equations

Abstract

In this work, some type of optimal control problems with equality constraints given by Partial Differential Equations (PDE) and convex inequality constraints are considered, obtaining their corresponding first order necessary optimality conditions by means of Dubovitskii-Milyutin (DM) method. Firstly, we consider problems with one objective functional (or scalar problems) but non-well posed equality constraints, where existence and uniqueness of state in function on control is not true (either one has existence but not uniqueness of state, or one has not existence of state for any control). In both cases, the classical Lions argument (re-writing the problem as an optimal control problem for the control without equality constraints, see for instance Lions, J. L. – Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1970) can not be applied. Afterwards, we consider multiobjective problems (or vectorial problems), considering three different concepts of solution: Pareto, Nash and Stackelberg. In all cases, an adequate abstract DM method is developed followed by an example.