Applications of Hybrid Dynamical Systems to Dynamics of Equilibrium Problems

Many mathematical models generally consist of either a continuous system like that of a system of differential equations, or a discrete system such as a discrete game theoretic model; however, there exist phenomena in which neither modeling approach alone is sufficient for capturing the behaviour of the intended real world system. This leads to the need to explore the use of combinations of such discrete and continuous processes, namely the use of mathematical modeling with what are known as hybrid dynamical systems. In what follows, we provide a blueprint for one approach to study several classes of equilibrium problems in non-equilibrium states through the direct use of hybrid dynamical systems. The motivation of our work stems from the fact that the real world is rarely, if ever, in a state of perfect equilibrium and that the behaviour of equilibrium problems in non-equilibrium states is just as complex and interesting (if not more so) than standard equilibrium solutions. Our approach consists of an association of classes of traffic equilibrium problems, noncooperative games, minimization problems, and complementarity problems to a class of hybrid dynamical system called projected dynamical systems. The purposed connection between equilibrium problems and projected dynamical system is made possible through mutual connections to the robust framework of variational inequalities. The results of our work include theoretical contributions such as showing how evolution solutions (non-equilibrium solutions) can be analyzed from a theoretical point of view and how they relate to equilibrium solutions; computational methods for tracking and visualizing evolution solutions and the development of numerical algorithms for simulation; and applications such as the effect of population vaccination decisions in the spread of infectious disease, dynamic traffic networks, dynamic vaccination games, and nonsmooth electrical circuits. iii Acknowledgements Over the course of my educational journey a number of people have inspired and aided me along the way. To mention everyone would require a very long and extensive list, however I feel that I must mention the following people. Firstly, this thesis would not have been possible without the invaluable advice, encouragement and patience of my supervisor, Monica Cojocaru; you have shown me the ropes of academia and helped me to become the mathematician I am today and for that I will always be grateful. In addition, I also wish to thank my internship supervisor Bernard Brogliato, as the work presented in Section 4, would not have been possible without the opportunity that he presented to me for joining him and his research team in France. As I have been in school for nearly 23 years there are several teachers whom I wish to acknowledge for having a significant impact on the direction of my life. To mathematics teachers Mike Harris and Harry Houser, thank-you for introducing me to the humor and joy of the mathematical world. I also owe a special debt of thanks to teachers, Lynda Ferguson, and Ron Netwon for helping to set me on the path towards educational excellence. My family has also played an integral role in my academic achievements. Thus, I am in need of thanking Melissa Barnard, for her love, support and structural & stylistic editorial skills; and my brother Chris, as his perseverance, dedication and determination in life inspired me to continue on when things seemed bleakest (and for that I will always be grateful). Above all, I would like to thank my parents, John & Sue, for the heights I have reached are due to their endless love, support, motivation and belief in my abilities.