Adv. Math. Econ. 8, 33-73 (2006)

Some variational convergence results with applications to evolution inclusions

Charles Castaing^{1}, Paul Raynaud de Fitte^{2} and Anna Salvadori^{3}

1 Département de Mathématiques, Université Montpellier II, 34095 Montpellier Cedex 5, France

2 Laboratoire Raphaël Salem, UMR CNRS 6085, UFR Sciences, Université de Rouen,Avenue de l’UniversitéBP.12, 76801 Saint Etienne du Rouvary,France

3 Dipartimento di Matematica, Università di Perugia, via Vanvitelli 1, 06123 Perugia, Italy

Received: April 27, 2005 Revised: September 16, 2005

JEL classification: C61

Mathematical Subject Classification (2000): 49J40, 49J45, 46N10, 49J24, 49L25

Abstract. We study variational convergence for integral functionals defined on L_{ℍ}^{∞}([0, 1]; dt) × Y([0, 1]; 𝕐) where ℍ is a separable Hilbert space, 𝕐 is a Polish space and Y([0, 1]; 𝕐) is the space of Young measures on [0, 1] × 𝕐, and we investigate its applications to evolution inclusions. We prove the dependence of solutions with respect to the control Young measures 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 HJB equation. Some limiting properties for nonconvex integral functionals in proximal analysis are also investigated.

Key words: Young measure, relaxed control, semicontinuity, integral functional, subdifferential, proximal analysis, viscosity subsolution