Some variational convergence results with applications to evolution inclusions

The study of variational convergence for integral functionals via the convergence of Young measure with application to Control problems and some classes of evolution inclusions of second order was developed in previous research by the same authors, A.Jofre and M.Valadier [see 8, 9, 14, 13 in the references]. See also H.Attouch, A.Cabot, P.Redont [1 in the references] and the reference herein. In the same vein we present here a more general study for integral functionals defined on L^(infinity)_HxY, where H is a separable Hilbert space and Y is the space of Young measures from [0,1] to a Polish space K. We investigate the application of the variational convergence to evolution inclusions. We prove the dependence of solutions with respect to the control Young measure and apply it to the study of the value function associated with these control problems. In this framework we then prove that the value function is a viscosity subsolution of the associated Hamilton-Jacobi-Bellman equation. Some limiting properties for nonconvex integral functionals in proximal analysis are also investigated.