Controllability and extremality in nonconvex differential inclusions

LetF:[0, T]×Rⁿ2^Rn be a set-valued map with compact values; let :RⁿR^m be a locally Lipschitzian map,z(t) a given trajectory, andR the reachable set atT of the differential inclusion. We prove sufficient conditions for (z(T))intR and establish necessary conditions in maximum principle form for (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.The author is grateful to an anonymous referee for his valuable remarks and comments which have helped to improve the paper.The paper was written while the author was visiting the laboratory of Prof. S. Suzuki, Department of Mechanical Engineering, Sophia University, Tokyo, Japan.