Investigation of regularity conditions in optimal control problems with geometric mixed constraints

Abstract

In this work, we study optimal control problems with mixed constraint of a general type R(x, u, t) ∈ C, where C is a closed set.  We undertake an effort to extend the results from [1], where, for C convex, a maximum principle was derived under a weakened regularity condi- tion. However, such a task, in its full generality, becomes rather complex, and certain obstacles appear in the proof that is not easy to overcome.  Mostly, this is due to the fact that the Lagrange multiplier associated with the mixed constraints begins to take values in the convexified normal cone, but not in the normal cone itself. Another important feature is the method we use. Upon the weakening of regularity, it restricts us to consider not all closed sets but some subclass, which, for example, includes closed semi-algebraic sets.