Well-posedness and porosity in optimal control without convexity assumptions

The Tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions. In 27] we considered a class of optimal control problems which is identified with the corresponding complete metric space of integrands, say . We did not impose any convexity assumptions. The main result in 27] establishes that for a generic integrand the corresponding optimal control problem is well-posed. In this paper we study the set of all integrands for which the corresponding optimal control problem is well-posed. We show that the complement of this set is not only of the first category but also a -porous set. The main result of the paper is obtained as a realization of a variational principle which can be applied to various classes of optimization problems. Received April 15, 2000 / Accepted October 10, 2000 / Published online December 8, 2000