Charles Castaing

Evolution equations governed by the sweeping process

This paper is concerned with variants of the sweeping process introduced by J.J. Moreau in 1971. In Section 4, perturbations of the sweeping process are studied. The equation has the formX′(t) ∈ -NC(t) (X(t)) +F(t, X(t)). The dimension is finite andF is a bounded closed convex valued multifunction. WhenC(t) is the complementary of a convex set,F is globally measurable andF(t, ·) is upper semicontinuous, existence is proved (Th. 4.1). The Lipschitz constants of the solutions receive particular attention. This point is also examined for the perturbed version of the classical convex sweeping process in Th. 4.1′. In Sections 5 and 6, a second-order sweeping process is considered:X″ (t) ∈ -NC(X(t)) (X′(t)). HereC is a bounded Lipschitzean closed convex valued multifunction defined on an open subset of a Hilbert space. Existence is proved whenC is dissipative (Th. 5.1) or when allC(x) are contained in a compact setK (Th. 5.2). In Section 6, the second-order sweeping process is solved in finite dimension whenC is continuous.

BV periodic solutions of an evolution problem associated with continuous moving convex sets

This paper is concerned with BV periodic solutions for multivalued perturbations of an evolution equation governed by the sweeping process (or Moreaus process). The perturbed equation has the form −Du∈NC(t)(u(t))+F(t,u(t)), whereC is a closed convex valued continuousT-periodic multifunction from [0,T] to ℝd,NC(t)(u(t)) is the normal cone ofC(t) atu(t),F: [0,T]×ℝd→ℝd is a compact convex valued multifunction and Du is the differential measure of the periodic BV solutionu. Several existence results for this differential inclusion are stated under various assumptions on the perturbationF.

Some various convergence results for multivalued martingales

We prove various convergence results for multivalued martingales, sub- supermartingales and mils with respect to the Mosco topology and the linear topology both in Bochner integration and Pettis integration. We also state some existence theorems of Pettis conditional expectation for multivalued Pettis-integrable multifunctions.

Convergences in a dual space with applications to Fatou lemma

We present new convergence results and new versions of Fatou lemma in Mathematical Economics based on various tightness conditions and the existence of scalarly integrable selections theorems for the (sequential)-weak-star upper limit of a sequence of measurable multifunctions taking values in the dual Eof a separable Banach space E. Existence of conditional expectation of weakly-star closed random sets in a non norm separable dual space is also provided.

Some variational convergence results with applications to evolution inclusions

We study variational convergence for integral functionals defined on L H ∞ ([0, 1];dt) × y([0,1]; \( \mathbb{Y} \) ) where ℍ is a separable Hilbert space, \( \mathbb{D} \) is a Polish space and y[0,1]; \( \mathbb{D} \) ) is the space of Young measures on [0,1] × \( \mathbb{D} \) , 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.

Weak compactness and convergences in L E’ 1 [E]

Suppose that (Ω, ℱ, μ) is a complete probability space, E is a Banach space, E’ is the topological dual of E and ρ is a lifting in ℒ ∞ (μ). We state several convergences and weak compactness results in the Banach space (L E’ 1 , [E], N 1) of weak*-scalarly integrable E’-valued functions via the Banach space (L E’ 1,ρ , [E], N 1,ρ) associated to the lifting ρ.) Applications to Young measures, Mathematical Economics, Minimization problems and Set-valued integration are also presented.

Multivalued differential equations on closed convex sets in Banach spaces

New derivation results for integrands and multifunctions via the Lipschitzean approximations are obtained. Applications to multivalued differential equations on closed convex sets are presented.

Compacité faible dans l'espaceL E 1 et dans l'espace des multifonctions intégralement bornées, et minimisation (*).

SummarySome compactness results in the space of Bochner integrable functions in Banach spaces and in the space of integrably bounded multifunction with non empty convex weakly compact values are presented. Applications to minimization problems are given.

Law of large numbers and Ergodic Theorem for convex weak star compact valued Gelfand-integrable mappings

We prove several results in the integration of convex weak star (resp. norm compact) valued random sets with application to weak star Kuratowski convergence in the law of large numbers for convex norm compact valued Gelfand-integrable mappings in the dual of a separable Banach space. We also establish several weak star Kuratowski convergence in the law of large numbers and ergodic theorem involving the subdifferential operators of Lipschitzean functions defined on a separable Banach space, and also provide an application to a closure type result arisen in evolution inclusions.

Tightness conditions and integrability of the sequential weak upper limit of a sequence of multifunctions

Various notions of tightness for measurable multifunctions are introduced and compared. They are used to derive results on the existence of integrable selections for the sequential weak upper limit of a sequence of multifunctions. Similar questions are examined for multifunctions with values in a dual space. Some results are particularized in the single-valued case, and applications to the multidimensional Fatou Lemma, both in the primal and in the dual space, are derived. This is achieved under conditions weaker than or noncomparable to L 1-boundedness.