Anna Ochal

Optimal Control of Parabolic Hemivariational Inequalities

In this paper we study the optimal control of systems driven by parabolic hemivariational inequalities. First, we establish the existence of solutions to a parabolic hemivariational inequality which contains nonlinear evolution operator. Introducing a control variable in the second member and in the multivalued term, we prove the upper semicontinuity property of the solution set of the inequality. Then we use this result and the direct method of the calculus of variations to show the existence of optimal admissible state–control pairs.

Quasi-Static Hemivariational Inequality via Vanishing Acceleration Approach

We study an abstract second order nonlinear evolution inclusion in a framework of evolution triple of spaces. We consider time-dependent possibly nonconvex nonsmooth functions and their Clarke subdifferentials operating on the unknown function. First we prove the existence of a weak solution. Then we study the asymptotic behavior of a sequence of solutions when a small parameter in the inertial term tends to zero. We prove that the limit function is a solution of a parabolic hemivariational inequality. Finally, we give an application of the abstract theorem to a quasi-static viscoelastic contact problem.

INTEGRODIFFERENTIAL HEMIVARIATIONAL INEQUALITIES WITH APPLICATIONS TO VISCOELASTIC FRICTIONAL CONTACT

We consider a class of abstract second-order evolutionary inclusions involving a Volterra-type integral term, for which we prove an existence and uniqueness result. The proof is based on arguments of evolutionary inclusions with monotone operators and the Banach fixed point theorem. Next, we apply this result to prove the solvability of a class of second-order integrodifferential hemivariational inequalities and, under an additional assumption, its unique solvability. Then we consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is dynamic, the material behavior is described with a viscoelastic constitutive law involving a long memory term and the contact is modelled with subdifferential boundary conditions. We derive the variational formulation of the problem which is of the form of an integrodifferential hemivariational inequality for the displacement field. Then we use our abstract results to prove the existence of a unique weak solution to the frictional contact model.

Existence of solutions for second order evolution inclusions with application to mechanical contact problems

In this article, we study nonlinear second order evolution inclusions defined on an evolution triple of spaces. First, we prove the existence of solutions. We introduce the integral operator, reduce the problem to the Cauchy one for a first order evolution inclusion and exploit a surjectivity result for operators which are pseudomonotone with respect to the domain of the other linear, densely defined and maximal monotone operator. Then, we give an application to a unilateral contact problem of viscoelasticity which is modeled by a dynamic hemivariational inequality. †Dedicated to Professor N.U. Ahmed on the occasion of his 70th birthday.

Weak solvability of a piezoelectric contact problem

We consider a mathematical model which describes the frictional contact between a piezoelectric body and a foundation. The material behaviour is modelled with a non-linear electro-elastic constitutive law, the contact is bilateral, the process is static and the foundation is assumed to be electrically conductive. Both the friction law and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system of two coupled hemi-variational inequalities for the displacement and the electric potential fields, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proof is based on an abstract result on operator inclusions in Banach spaces.

MODELING AND ANALYSIS OF AN ANTIPLANE PIEZOELECTRIC CONTACT PROBLEM

We study a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is static, the material behavior is described with a linearly electro-elastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is in the form of a system of two coupled hemivariational inequalities for the displacement and the electric potential fields, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proofs are based on an abstract result on operator inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our results are valid.

Nonconvex Inequality Models for Contact Problems of Nonsmooth Mechanics

This review paper deals with selected nonsmooth and nonconvex inequality problems for dynamic frictional contact between a viscoelastic body and a foundation. The process is modeled by general nonmonotone possibly multivalued multidimensional Clarke subdifferential contact boundary conditions. The problems of frictional contact with both short and long memory, thermoviscoelastic frictional contact, bilateral frictional contact and bilateral contact for piezoelectric materials with adhesion are considered. The formulations and results on existence, and uniqueness of solutions are presented.

Existence results for evolutionary inclusions and variational–hemivariational inequalities

We consider an abstract first-order evolutionary inclusion in a reflexive Banach space. The inclusion contains the sum of L-pseudomonotone operator and a maximal monotone operator. We provide an existence theorem which is a generalization of former results known in the literature. Next, we apply our result to the case of nonlinear variational–hemivariational inequalities considered in the setting of an evolution triple of spaces. We specify the multivalued operators in the problem and obtain existence results for several classes of variational–hemivariational inequality problems. Finally, we illustrate our existence result and treat a class of quasilinear parabolic problems under nonmonotone and multivalued flux boundary conditions.

Evolutionary Inclusions and Hemivariational Inequalities

In this chapter we study evolutionary inclusions of second order. These are multivalued relations which involve the second-order time derivative of the unknown. We start with a basic existence result for such inclusions. Then we provide results on existence and uniqueness of solutions to evolutionary inclusions of the subdifferential type, i.e., inclusions involving the Clarke subdifferential operator of locally Lipschitz functionals. We also prove an existence and uniqueness result for integro-differential evolutionary inclusions. Next, we consider a class of hyperbolic hemivariational inequalities for which we provide a theorem on existence of solutions and, under stronger hypotheses, their uniqueness. We conclude this chapter with a result on existence and uniqueness of solutions to the evolutionary integro-differential hemivariational inequality with the Volterra integral term. The results provided below represent the dynamic counterparts of theorems presented in Chap. 4 and will be used in the study of the dynamic frictional contact problems in Chap. 8.

Variational-Hemivariational Approach to a Quasistatic Viscoelastic Problem with Normal Compliance, Friction and Material Damage

This work studies a model for quasistatic frictional contact between a viscoelastic body and a reactive foundation. The constitutive law is assumed to be nonlinear and contains damage effects modeled by a parabolic differential inclusion. Contact is described by the normal compliance condition and a subdifferential frictional condition. A variational-hemivariational formulation of the problem is provided and the existence and uniqueness of its solution is proved. The proof is based on a surjectivity result for pseudomonotone coercive operators and a fixed point argument.