Generic existence of solutions of nonconvex optimal control problems

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 1996, the author obtained a generic existence and uniqueness result (with respect to variations of the integrand of the integral functional) without the convexity condition for a class of optimal control problems satisfying the Cesari growth condition. In this paper, we survey this result and its recent extensions, and establish several new results in this direction.