ON SOLUTION SETS OF NONCONVEX DARBOUX PROBLEMS AND APPLICATIONS TO OPTIMAL-CONTROL WITH END-POINT CONSTRAINTS

We prove a continuous version of a relaxation theorem for the nonconve x Darboux problem x(t tau) is an element of F(t, tau, x, x(t), x(tau), ). This result allows us to use Warga's open mapping theorem for deriv ing necessary conditions in the form of a maximum principle for optimi zation problems with endpoint constraints. Neither constraint qualific ation nor regularity assumption is supposed.