Sensitivity analysis and optimal control for a contact problem with friction in the linear elastic model
Résumé
This paper investigates, without any regularization procedure, the sensitivity analysis of a
mechanical contact 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.
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