THE DUBOIS-REYMOND DIFFERENTIAL INCLUSION FOR AUTONOMOUS OPTIMAL CONTROL PROBLEMS WITH POINTWISE-CONSTRAINED DERIVATIVES

摘要

We prove validity of the classical DuBois-Reymond differential inclusion for the minimizers y (·)of the integral whose velocities are not a.e.constrained by the domain boundary. Thus we do not ask (as preceding results do) the free-velocity times to have "full measure"; on the contrary,"positive measure" of T~free suffices here to guarantee the above necessary condition. One main feature of our result is that L (S, ξ) = ∞ is freely allowed, hence the domains domL(S,·) may be e.g. compact and (*) can be seen as the variational reformulation of general state-and-velocity constrained optimal control problems. Another main feature is the clean generality of our assumptions on L (·) : any Borel-measurable function L : R～n×R~n→[0,∞] having L(·,0) lsc and L(S,·) convex Isc VS. The nonconvex case is also considered, for L (S, ·) almost convex lsc VS.