Controllability and extremality in nonconvex differential inclusions

Abstract

LetF:[0, T]×R n →2 R n be a set-valued map with compact values; let η:R n →R m be a locally Lipschitzian map,z(t) a given trajectory, andR the reachable set atT of the differential inclusion $$\dot x(t) \in F(t,x(t))$$ . We prove sufficient conditions for η(z(T))∈intR and establish necessary conditions in maximum principle form for η(z(T))∈(R). As a consequence of these results, we show that every boundary trajectory is simultaneously a Pontryagin extremal, Lagrangian extremal, and relaxed Lagrangian extremal.