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An analytical and numerical approach to a bilateral contact problem with nonmonotone friction

Barboteu M., Bartosz K., Kalita P.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 2

263--276

EN

We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is bilateral, i.e., there is no loss of contact. The friction is modeled with a nonmotonone law. The purpose of this work is to provide an error estimate for the Galerkin method as well as to present and compare two numerical methods for solving the resulting nonsmooth and nonconvex frictional contact problem. The first approach is based on the nonconvex proximal bundle method, whereas the second one deals with the approximation of a nonconvex problem by a sequence of nonsmooth convex programming problems. Some numerical experiments are realized to compare the two numerical approaches.

2

Optimal control for a class of mechanical thermoviscoelastic frictional contact problems

Denkowski Z., Migórski S., Ochal A.

Control and Cybernetics

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2007

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Vol. 36, no 3

611-632

EN

We study optimal control of systems governed by a coupled system of hemivariational inequalities, modeling a dynamic thermoviscoelastic problem, which describes frictional contact between a body and a foundation. We employ the Kelvin-Voigt vis-coelastic law, include the thermal effects and consider the general nonmonotone and multivalued subdifferential boundary conditions. We consider optimal control problem for boundary and distributed parameter control systems, time optimal control problem and maximum stay control problem. We deliver conditions that guarantee the existence of optimal solutions.

3

Sensitivity of optimal solutions to control problems for systems described by hemivariational inequalities

Denkowski Z., Migórski S.

Control and Cybernetics

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2004

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Vol. 33, no 2

211-236

EN

In this paper the sensitivity of optimal solutions to control problems for the systems described by stationary and evolution heinivariational inequalities (HVIs) under perturbations of state relations and of cost functionals is investigated. First, basing on the theory of sequential [Gamma]-convergence we recall the abstract scheme concerning convergence of minimal values and minimizers. The abstract scheme works provided we can establish two properties: the Kuratowski convergence of solution sets for HVIs (state relations) and some complementary [Gamma]-convergence of the cost functionals. Then these two properties are implemented in each considered case.

4

Hemivariational inequalities governed by the p-Laplacian -Dirichlet problem

Naniewicz Z.

Control and Cybernetics

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2004

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Vol. 33, no 2

181-210

EN

A hemivariational inequality involving p-Laplacian is studied under the hypothesis that the nonlinear part fulfills the unilateral growth condition (Naniewicz, 1994). The existence of solutions for problems with Dirichlet boundary conditions is established by making use of Chang's version of the critical point theory for non-smooth locally Lipschitz functionals (Chang, 1981), combined with the Galerkin method. A class of problems with nonlinear potentials fulfilling the classical growth hypothesis without Ainbrosetti-Rabinowitz type assumption is discussed. The approach is based on the recession technique introduced in Naniewicz (2003).

5

A topological approach to hemivariational inequalities with unilateral growth condition

Motreanu D., Naniewicz Z.

Journal of Applied Analysis

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2001

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Vol. 7, nr 1

23-41

EN

The present paper is devoted to the study of the existence solution problem for a hemivariational inequality on vector-valued function space in the case when the nonlinear nonconvex part satisfies the unilateral growth condition. The critical point theory combined with the Galerkin approximation method have been used to establish the result.