Some variational convergence results for a class of evolution inclusions of second order using Young measures

The paper is essentially a continuation of previous research of the same authors and A.Jofre [see 17, 15, 16 in the references] dealing with control problems where the dynamics are given by ordinary differential equations and evolution inclusions, governed by nonconvex sweeping process and m-accretive operators via the fiber product of Young measures. We give here new results on variational convergence for both ordinary differential equations and evolution inclusions of second order. The paper is divided into two parts. In the first part we deal with existence and uniqueness for weak solutions (in W^(2,1)_E) of a second order differential equation with two boundary points conditions, in a finite dimensional space E, governed by controls which are measures on a compact metric space. We also discuss the dependence on the controls and the variational properties of the value function V_h, associated with a bounded lower semicontinuous function h. In the second part, we discuss the limiting behaviour of a sequence of dynamics governed by second order evolution inclusions, with two boundary points conditions. We prove that (up to extracted sequences) the solutions “stably” converge to a Young measure and we show that the limit measure satisfies a Fatou-type lemma, with variational-type inclusion property. The study of value function and of Fatou-type lemmas has interesting application in Mathematical Economics.