The Tonelli existence theorem in the calculus of variations and its subsequent modi cations were established for integrands f which satisfy convexity and growth conditions. In A. J. Zaslavski [Nonlinear Anal., to appear], a generic existence and uniqueness result ( with respect to variations of the integrand of the integral functional) without the convexity condition was established for a class of optimal control problems satisfying the Cesari growth condition. In this paper we extend the generic existence and uniqueness result in A. J. Zaslavski [ Nonlinear Anal., to appear], to a class of optimal control problems in which constraint maps are also subject to variations. The main result of the paper is obtained as a realization of a variational principle extending the variational principle introduced in A. D. Ioffe and A. J. Zaslavski [SIAM J. Control Optim., 38 ( 2000), pp. 566-581].