Some variational convergence results for a class of evolution inclusions of second order using Young measures
Résumé
This paper has two main parts. In the first part, we discuss the existence and uniqueness of the $W^ {2, 1}_E$-solution $u_{\mu, \nu}$ of a second order differential equation with two boundary points conditions in a finite dimensional space, governed by controls $\mu,\nu$ which are measures on a compact metric space. We also discuss the dependence on the controls and the variational properties of the value function $V_h(t,\mu) :=\sup_{\nu\in {\mathcal R}} h( u_{\mu, \nu}(t)),$ associated with a bounded lower semicontinuous function $h$. In the second main 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 $\nu$ and we show that the limit measure $\nu$ satisfies a Fatou-type lemma in Mathematical Economics with variational-type inclusion property.