Time Dependent Variational Inequalities — Some Recent Trends

Abstract

In this paper we attempt to survey several classes of time dependent variational inequalities that model various constrained evolution problems, in particular with unilateral constraints encountered in applied sciences. Here we are mainly concerned with the challenging evolution problems where the state of the system lives in an infinite dimensional space; the wide field of differential inclusions in finite dimensional space is outside of the scope of the present paper. Also we refrain here from covering the literature on the existence and regularity theory of parabolic and hyperbolic evolution inequalities.At first we address time dependent variational inequalities where time enters as an additional parameter in the variational inequality. This class of time dependent variational inequalities has been recently introduced to study certain time dependent traffic flow problems. In addition we draw the attention to work on constrained evolution problems that include the time history via memory terms.Then we turn to time dependent variational inequalities that generalize classical ordinary differential equations. In particular, we report on recent extensions of the sweeping process introduced by Moreau. Moreover, we deal with projected dynamical systems in a Hilbert space framework. Quite recently, this class of time dependent variational inequalities has been introduced and studied in finite dimensions to treat various time dependent network problems in operations research, particularly in traffic science. It is shown that projected dynamical systems are equivalent to a class of differential inclusions that were already analysed twenty years ago.Finally we deal with a central issue of evolution problems and equilibrium theory, namely the asymptotics of the time processes and their convergence to steady-state solutions.KeywordsVariational InequalityEquilibrium ProblemMonotone OperatorDifferential InclusionQuasivariational InequalityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.