Properties of the set of admissible “state-control” pairs for first-order evolution control systems

We consider a control system described by a non-linear first-order evolution equation on an evolution triple of Banach spaces (a “Gelfand triple”) with a mixed multivalued control constraint whose values are non-convex closed sets in the control space. Besides the original system, we consider systems with the following control constraints: the constraint whose values are the closed convex hulls of the values of the original constraint, and the constraint whose values are the extreme points of the convexified constraint that belong to the original one. We study topological properties of the sets of admissible “state-control” pairs for the same system with various constraints and consider the relations between them. An example of a non-linear parabolic control system is worked out in detail.