Boundary optimal control of a dynamic frictional contact problem

Abstract

In this paper we study boundary optimal control of an evolutionary system governed by a history-dependent variational-hemivariational inequality. The inequality is a weak formulation of a dynamic frictional contact problem for a viscoelastic body with a multivalued normal damped response condition and a simplified version of the Coulomb law of dry friction. A continuous dependence result for the solution map is proved and the existence of optimal solutions to the control problem is established.

Optimal Control of a Quasistatic Frictional Contact Problem with History-Dependent Operators

“…Next, we introduce a result related to upper semicontinuous multivalued functions, which can be found in the appendix of [4].…”

“…According to Lemma 10 in [4], ∂ϕ 2 is upper semicontinuous as a multifunction from M w × L 2 (Γ 3,2 ; R d ) to L 2 w (Γ 3,2 ; R d ). We use Lemma 2.4 to conclude that ξ 2 (t) ∈ ∂ φ2 (µ, γ 2 u(t)), for a.e.…”

“…In [28], the authors consider the numerical solutions for the optimal control of a class of variational-hemivariational inequalities and deduce the convergence result. As for evolutional case, the reference [14] studies an optimal control for a class of subdifferential evolution inclusions involving history-dependent operators and [4] focuses on the boundary optimal control of a dynamic frictional contact problem. The works of these two papers give us a great inspiration.…”

Continuous dependence and optimal control of a dynamic elastic-viscoplastic contact problem with non-monotone boundary conditions

“…By the skills in [9], we introduce Φ(x, t) = φ(t)v(x) with φ ∈ C ∞ 0 (0, T ) and v ∈ V . Multiply (23) with Φ and integrate on (0, T ), then…”

“…[31,28,15,8,25] with elastic or piezoelectric materials. For dynamic contact models, since the discussion is more challenging, publications are limited, see [11,23,19,7] with viscoelastic or thermoviscoelastic materials. In this paper, we study the optimal control for a dynamic contact model with elastic-viscoplastic materials.…”