Modeling, analysis and optimal control of systems described by hemivariational inequalities

In the paper we present a survey on the mathematical modeling of nonconvex and nonsmooth problems arising in the mathematical theory of contact mechanics which is a growing field in engineering and scientific computing. The approach to such problems is based on the notion of hemivariational inequality and our presentation focuses on three aspects. First we present the ideas leading to inequality problems encountered in mechanics and we formulate different forms of hemivariational inequalities. Then we give results on the existence and uniqueness of solutions to hemivariational inequalities of elliptic, parabolic and hyperbolic types. For these classes of hemivariational inequalities we formulate optimal control problems and provide conditions under which they admit optimal solutions. Finally we indicate the mechanical problems in viscoelasticity, thermoviscoelasticity, heat conduction and the fluid flow problems to which our results can be applied.