PHD Math Defence: Nonunique Equilibria of Projected Dynamical Systems and Their Applications

CANDIDATE:  FATIMA ETBAIGHA

ABSTRACT:  

Projected dynamical systems were formulated in the 1990s by Dupuis and Nagurney. In contrast to classical dynamical systems, projected dynamical systems have discontinuous right hand sides that are associated with a projection operator. By the projection mechanism, the whole Hilbert space H is projected onto a non-empty, closed convex set K ⊂ H. Critical points of a projected dynamical system coincide with the solutions of a corresponding variational inequality problem; therefore, the applications of projected dynamical systems have been found in various fi such as economics, operations research and engineering. 

In this thesis, we study applications of equilibria of projected dynamical systems for market equilibrium problems, games and compartmental population models. First, the well-known market disequilibrium model with excess supply and demand is investigated to determine if it exhibits changes in the structure and the number of equilibrium states for specific choices of parameter values. We study the bifurcation problem (i.e., a qualitative change in equilibrium states) as a parametrized variational inequality problem. We conduct our analysis by modeling the markets via a projected dynamical system. Second, we present a combination of theoretical and computational results meant to give insights into the question of the existence of nonunique Nash equilibria for N-player nonlinear games. Our inquiries make use of the theory of variational inequalities and projected systems to classify cases where multiplayer Nash games with parametrized payoffs exhibit changes in the number of Nash equilibria, depending on given parameter values. Finally, we use the compartmental population model, namely, the deterministic Susceptible-Exposed-Infectious-Recovered model, to analyze the dynamics of influenza infection of a farrow-to-finish swine farm, and we explore the reinfection at the farm level. We further examine the effectiveness of two control strategies: vaccination and reduction of indirect contact. In this case, we show that the model is a projected dynamical system but the projection does not add any relevant applied meaning to the results. Therefore, we show that the problem can be studied using classical dynamical systems.