On solutions of an evolution control system depending on parameters

Abstract

A control system governed by a non-linear first-order evolution equation with mixed non-convex control constraints is examined. The system depends on parameters entering all its data, including the non-linear evolution operator and the control constraints. The system with convexified control constraints is also considered. The general concept of -convergence of operators is used for the proof of the existence of selectors continuously dependent on the parameters with values in the solution set of the original system; a continuous version of the selector relaxation theorem is also proved, which concerns the approximation of the continuous solution selectors with convexified constraints by continuous solution selectors of the original system. An example of a parabolic control system is discussed.