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

  • C. Castaing
  • M. D. P. Monteiro Marques
  • P. Raynaud de Fitte
Chapter
Part of the Advances in Mathematical Economics book series (MATHECON, volume 22)

Abstract

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
$$\displaystyle 0\in \ddot u(t) + A(t) \dot u(t) + f(t, u(t)), \hskip 2pt t\in [0, T] $$
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
$$\displaystyle 0= \ddot u(t) + A(t) \dot u(t) + \nabla \varphi (u(t)), \hskip 2pt t\in [0, T] $$
where A(t) is a time dependent with Lipschitz variation single-valued maximal monotone operator and ∇φ is the gradient of a smooth Lipschitz function φ are stated. Some more general inclusion of the form
$$\displaystyle 0\in \ddot u(t) + A(t) \dot u(t) + \partial \Phi (u(t)), \hskip 2pt t\in [0, T] $$
where  Φ(u(t)) denotes the subdifferential of a proper lower semicontinuous convex function Φ 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 : [0, T] × H × H are stated. Second- order evolution inclusion with perturbation f and Young measure control νt
$$\displaystyle \left \{ \begin {array}{lll} 0\in \ddot u_{x, y, \nu }(t) + A(t) \dot u_{x, y, \nu }(t) + f(t, u_{x, y, \nu }(t))+ \operatorname {{\mathrm {bar}}}(\nu _t), \hskip 2pt t \in [0, T] \\ u_{x, y, \nu }(0) = x, \dot u_{x, y, \nu } (0) =y \in D(A(0)) \end {array} \right . $$
where \( \operatorname {{\mathrm {bar}}}(\nu _t)\) denotes the barycenter of the Young measure νt is considered, and applications to optimal control are presented. Some variational limit theorems related to convex sweeping process are provided.

Keywords

Bolza control problem Lipschitz mapping Maximal monotone operators Pseudo-distance Subdifferential Viscosity Young measures 

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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • C. Castaing
    • 1
  • M. D. P. Monteiro Marques
    • 2
  • P. Raynaud de Fitte
    • 3
  1. 1.IMAGUniversité de MontpellierMontpellierFrance
  2. 2.CMAF-CIO, Departamento de MatemáticaFaculdade de Ciências da Universidade de LisboaLisbonPortugal
  3. 3.Laboratoire de Mathématiques Raphaël SalemNormandie UniversitéRouenFrance

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