CONTROLLABILITY AND EXTREMALITY IN NONCONVEX DIFFERENTIAL-INCLUSIONS

Let F: [0, T] x R(n) --> 2(Rn) be a set-valued map with compact values ; let eta: R(n) --> R(m) be a locally Lipschitzian map, z(t) a given t rajectory, and R the reachable set at T of the differential inclusion x over dot (t)epsilon F(t, x(t)). We prove sufficient conditions for e ta(z(T)) epsilon int eta(R) and establish necessary conditions in maxi mum principle form for eta(z(T)) epsilon partial derivative eta(R). As a consequence of these results, we show that every boundary trajector y is simultaneously a Pontryagin extremal, Lagrangian extremal, and re laxed Lagrangian extremal.