Sensitivity analysis of a Signorini-type history-dependent variational inequality

TL;DR

The paper addresses sensitivity analysis for a history-dependent Signorini-type variational inequality modeling viscoelastic contact with memory. It develops a fixed-point reformulation $\mathcal{Q}$ equivalent to the original problem $\mathcal{P}$, proves existence for an associated optimal control problem, and establishes Hadamard directional differentiability of the solution operator $S$, with the derivative characterized by a history-dependent VI. It also introduces two well-posedness concepts, proves both $\mathcal{P}$ and $\mathcal{Q}$ are well-posed under the proposed framework, and compares the strength of the two notions, providing a foundation for controllability and stability analyses of memory-influenced contact problems. The results supply rigorous tools for sensitivity, control, and stability assessments in viscoelastic contact models with normal compliance and memory effects.

Abstract

We consider a history-dependent variational inequality (P) which models the frictionless contact between a viscoelastic body and a rigid obstacle covered by a layer of soft material. The inequality is expressed in terms of the displacement field, is governed by the data f (related to the applied body forces and surface tractions) and, under appropriate assumptions, it has a unique solution, denoted by u. Our aim in this paper is to perform a sensitivity analysis of the inequality (P), including the study of the regularity of the solution operator $f \mapsto u$. To this end, we start by proving the equivalence of (P) with a fixed point problem, denoted by (Q). We then consider an associated optimal control problem, for which we present an existence result. Then, we prove the directional differentiability of the solution operator and show that the directional derivative is characterized by a history-dependent variational inequality with time-dependent constraints. Finally, we prove two well-posedness results in the study of problems (P) and (Q), respectively, and compare the two well-posedness concepts employed.