Second-Order Evolution Problems with Time-Dependent Maximal Monotone Operator and Applications

Charles Castaing (1) , Paul Raynaud de Fitte (2) , Manuel D.P. Monteiro Marques

1 UM2 - Université Montpellier 2 - Sciences et Techniques
2 LMRS - Laboratoire de Mathématiques Raphaël Salem

We consider at first the existence and uniqueness of solution for a general second-order evolution inclusion in a separable Hilbert space of the form 0∈{\"u}(t)+A(t)u̇(t)+f(t,u(t)),t∈[0,T]{\$}{\$}{\backslash}displaystyle 0{\backslash}in {\backslash}ddot u(t) + A(t) {\backslash}dot u(t) + f(t, u(t)), {\backslash}hskip 2pt t{\backslash}in [0, T] {\$}{\$}where A(t) is a time dependent with Lipschitz variation maximal monotone operator and the perturbation f(t, .) is boundedly Lipschitz. Several new results are presented in the sense that these second-order evolution inclusions deal with time-dependent maximal monotone operators by contrast with the classical case dealing with some special fixed operators. In particular, the existence and uniqueness of solution to 0={\"u}(t)+A(t)u̇(t)+∇$\phi$(u(t)),t∈[0,T]{\$}{\$}{\backslash}displaystyle 0= {\backslash}ddot u(t) + A(t) {\backslash}dot u(t) + {\backslash}nabla {\backslash}varphi (u(t)), {\backslash}hskip 2pt t{\backslash}in [0, T] {\$}{\$}where A(t) is a time dependent with Lipschitz variation single-valued maximal monotone operator and ∇$\phi$ is the gradient of a smooth Lipschitz function $\phi$ are stated. Some more general inclusion of the form 0∈{\"u}(t)+A(t)u̇(t)+∂$\Phi$(u(t)),t∈[0,T]{\$}{\$}{\backslash}displaystyle 0{\backslash}in {\backslash}ddot u(t) + A(t) {\backslash}dot u(t) + {\backslash}partial {\backslash}Phi (u(t)), {\backslash}hskip 2pt t{\backslash}in [0, T] {\$}{\$}where ∂{\thinspace}$\Phi$(u(t)) denotes the subdifferential of a proper lower semicontinuous convex function $\Phi$ at the point u(t) is provided via a variational approach. Further results in second-order problems involving both absolutely continuous in variation maximal monotone operator and bounded in variation maximal monotone operator, A(t), with perturbation f{\thinspace}:{\thinspace}[0, T]{\thinspace}{\texttimes}{\thinspace}H{\thinspace}{\texttimes}{\thinspace}H are stated. Second- order evolution inclusion with perturbation f and Young measure control $\nu$t 0∈{\"u}x,y,$\nu$(t)+A(t)u̇x,y,$\nu$(t)+f(t,ux,y,$\nu$(t))+bar($\nu$t),t∈[0,T]ux,y,$\nu$(0)=x,u̇x,y,$\nu$(0)=y∈D(A(0)){\$}{\$}{\backslash}displaystyle {\backslash}left {\backslash}{\{} {\backslash}begin {\{}array{\}}{\{}lll{\}} 0{\backslash}in {\backslash}ddot u{\_}{\{}x, y, {\backslash}nu {\}}(t) + A(t) {\backslash}dot u{\_}{\{}x, y, {\backslash}nu {\}}(t) + f(t, u{\_}{\{}x, y, {\backslash}nu {\}}(t))+ {\backslash}operatorname {\{}{\{}{\backslash}mathrm {\{}bar{\}}{\}}{\}}({\backslash}nu {\_}t), {\backslash}hskip 2pt t {\backslash}in [0, T] {\backslash}{\backslash} u{\_}{\{}x, y, {\backslash}nu {\}}(0) = x, {\backslash}dot u{\_}{\{}x, y, {\backslash}nu {\}} (0) =y {\backslash}in D(A(0)) {\backslash}end {\{}array{\}} {\backslash}right . {\$}{\$}where bar($\nu$t){\$}{\$} {\backslash}operatorname {\{}{\{}{\backslash}mathrm {\{}bar{\}}{\}}{\}}({\backslash}nu {\_}t){\$}{\$}denotes the barycenter of the Young measure $\nu$t is considered, and applications to optimal control are presented. Some variational limit theorems related to convex sweeping process are provided.

Mots clés: Bolza control problem; Lipschitz mapping; maximal monotone operators; pseudo- distance; subdifferential; viscosity; Young measures

Domaines: Mathématiques [math]

Charles Castaing, Paul Raynaud de Fitte, Manuel D.P. Monteiro Marques. Second-Order Evolution Problems with Time-Dependent Maximal Monotone Operator and Applications. Kusuoka, Shigeo and Maruyama, Toru. Advances in Mathematical Economics: Volume 22, 22, Springer Singapore, pp.25-77, 2018, 978-981-13-0605-1. ⟨10.1007/978-981-13-0605-1_2⟩. ⟨hal-01784844⟩