Well-posedness and porosity in optimal control without convexity assumptions

Abstract.

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 $\mathcal{F}$. We did not impose any convexity assumptions. The main result in [27] establishes that for a generic integrand $f \in\mathcal{F}$ the corresponding optimal control problem is well-posed. In this paper we study the set of all integrands $f \in \mathcal{F}$ 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 $\sigma$-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.