Abstract
In this paper, we consider extremal solutions of multivalued differential equations, i.e., solutions that steer to the boundary of the attainable set. Multivalued differential equations arise in a natural way from control systems governed by ordinary differential equations that have a variable control-constraint set. Extremal solutions of multi-valued differential equations are important in the study of the optimal control of such systems. We give conditions under which extremality of a solution at a certain time implies extremality of the solution at all previous times where it is defined. Necessary conditions for extremality are also obtained. We treat both the time-dependent case and the time-independent case.
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