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