Sensitivity analysis and optimal control for a friction problem in the linear elastic model

TL;DR

This paper conducts a sensitivity analysis of a nonsmooth Tresca friction problem in linear elasticity without regularization, deriving derivatives via convex analysis and applying results to optimal control.

Contribution

It introduces a novel sensitivity analysis method for Tresca friction problems using convex analysis, and applies it to optimal control without regularization.

Findings

Derived the solution via proximal operators for Tresca friction

Proved differentiability of the solution with respect to parameters

Applied the analysis to solve an optimal control problem

Abstract

This paper investigates, without any regularization procedure, the sensitivity analysis of a mechanical friction problem involving the (nonsmooth) Tresca friction law in the linear elastic model. To this aim a recent methodology based on advanced tools from convex and variational analyses is used. Precisely we express the solution to the so-called Tresca friction problem thanks to the proximal operator associated with the corresponding Tresca friction functional. Then, using an extended version of twice epi-differentiability, we prove the differentiability of the solution to the parameterized Tresca friction problem, characterizing its derivative as the solution to a boundary value problem involving tangential Signorini's unilateral conditions. Finally our result is used to investigate and numerically solve an optimal control problem associated with the Tresca friction model.