Properties of solutions to second order evolution control systems. I

We consider a control system described by a nonlinear second order evolution equation defined on an evolution triple of Banach spaces (Gelfand triple) with a mixed multivalued control constraint whose values are nonconvex closed sets. Alongside the original system we consider a system with the following control constraints: a constraint whose values are the closed convex hull of the values of the original constraint and a constraint whose values are extreme points of the constraint which belong simultaneously to the original constraint. By a solution to the system we mean an admissible “trajectory-control” pair. In this part of the article we study existence questions for solutions to the control system with various constraints and density of the solution set with nonconvex constraints in the solution set with convexified constraints.