Second-Order Evolution Problems with Time-Dependent Maximal Monotone Operator and Applications
| Autor: | M. D. P. Monteiro Marques, P. Raynaud de Fitte, Charles Castaing |
|---|---|
| Přispěvatelé: | Université Montpellier 2 - Sciences et Techniques (UM2), Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Kusuoka, Shigeo and Maruyama, Toru |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Physics
021103 operations research Lipschitz mapping 0211 other engineering and technologies subdifferential MSC: 34A60 34B15 Monotonic function 02 engineering and technology Subderivative Bolza control problem Lipschitz continuity 01 natural sciences 010101 applied mathematics Combinatorics Monotone polygon Bounded function Young measures viscosity Uniqueness Nabla symbol maximal monotone operators 0101 mathematics [MATH]Mathematics [math] Convex function pseudo- distance |
| Zdroj: | Advances in Mathematical Economics: Volume 22 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⟩ Advances in Mathematical Economics ISBN: 9789811306044 |
| DOI: | 10.1007/978-981-13-0605-1_2⟩ |
| Popis: | International audience; 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. |
| Databáze: | OpenAIRE |
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