Properties of the set of “trajectory-control” pairs of a control systemwith subdifferential operators

We consider a control system described by an evolution equation with control constraint which is a multivalued mapping of a phase variable with closed nonconvex values. One of the evolution operators of the system is the subdifferential of a time-dependent proper, convex, and lower semicontinuous function. The other operator, acting on the derivative of the required functions, is the subdifferential of a convex continuous function. We also consider systems with the following control constraints: multivalued mappings whose values are the closed convex hulls of the values of the original constraint and multivalued mapping 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 “trajectory–control” pairs of the system with various control constraints and clarify the relations between them. An example of a parabolic system with hysteresis and diffusion phenomena is considered in detail. Bibliography: 19 titles.