"Nonsmooth dynamics associated to variational inequalities, generalized Nash games and applications"

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This proposal investigates the relationship between three important mathematical constructs: nonsmooth dynamical systems, quasivariational inequalities, and generalized Nash games, and their appropriate applications. Variational inequalities (VI) were introduced in the early 60s within the framework of partial differential equations. Later it was shown they equivalently reformulate large classes of equilibrium problems (such as Nash, Wardrop, Walras, Cournot, traffic equilibrium etc.), thus becoming prominent in operations research. Due to this equivalence, solutions of VI are often called equilibria. Of specific interest in my work is a particular class of VI, called "quasivariational" (QVI). Essentially a QVI models equilibrium problems with constraint sets dependent on the equilibrium solution. At their core VI/QVI problems are static. To describe the dynamics of such problems, approaches range from inequalities of evolution to recent concepts of evolutionary and differential variational inequalities. My approach relies on associating VI/QVI to dynamical systems given by solutions to a differential equation or inclusion so that two conditions are met: the solutions of the VI/QVI coincide with the equation's or inclusion's stationary points; the solutions of the equation/inclusion remain inside the constraint set at all times. These systems belong to the area of nonsmooth dynamics. Results from QVI theory imply existence of stationary solutions for the associated dynamics. However, existence, uniqueness and stability analysis of non-stationary solutions is largely an open problem. Investigating these questions is my first objective. This objective is a natural extension of some of my previous work relating VI with nonsmooth systems. My second objective is to use results above to provide novel ways to solve generalized Nash (GN) games which can be equivalently reformulated as QVI problems. Solutions for some GN games are known, but in general this is an open question. My third objective concerns applications of results above in dynamic population models of decision making, in time-dependent traffic network problems with changes in network topology, and in epidemiology.