Convergence and Optimization Results for a History-dependent Variational Problem

TL;DR

This paper establishes convergence and optimization results for a class of history-dependent variational problems in Hilbert spaces, with applications to nonlinear boundary value problems like frictional contact in viscoelastic materials.

Contribution

It provides a general framework for analyzing convergence and existence of solutions in history-dependent variational problems, extending previous results and applying to nonlinear boundary value problems.

Findings

Proved unique solvability of the variational problem.

Established a general convergence theorem using monotonicity and Mosco convergence.

Demonstrated applicability to frictional contact problems in viscoelastic materials.

Abstract

We consider a mixed variational problem in real Hilbert spaces, defined on on the unbounded interval of time and governed by a history-dependent operator. We state the unique solvability of the problem, which follows from a general existence and uniqueness result obtained in our previous paper. Then, we state and prove a general convergence result. The proof is based on arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Finally, we consider a general optimization problem for which we prove the existence of minimizers. The mathematical tools developed in this paper are useful in the analysis of a large class of nonlinear boundary value problems which, in a weak formulation, lead to history-dependent mixed variational problems. To provide an example, we illustrate our abstract results in the study of a frictional contact problem for viscoelastic materials with long memory.