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

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.