Non-equilibrium Solutions of Dynamic Networks: A Hybrid System Approach
Many dynamic networks can be analyzed through the framework of equilibrium problems. While traditionally, the study of equilibrium problems is solely concerned with obtaining or approximating equilibrium solutions, the study of equilibrium problems not in equilibrium provides valuable information into dynamic network behavior. One approach to study such non-equilibrium solutions stems from a connection between equilibrium problems and a class of parametrized projected differential equations. However, there is a drawback of this approach: the requirement of observing distributions of demands and costs. To address this problem we develop a hybrid system framework to model non-equilibrium solutions of dynamic networks, which only requires point observations. We demonstrate stability properties of the hybrid system framework and illustrate the novelty of our approach with a dynamic traffic network example.
KeywordsDynamic networks Hybrid systems Variational inequalities Equilibrium problems
Scott Greenhalgh and Monica-Gabriela Cojocaru would like to thank the referees comments which led to a clearer presentation of this work. Monica-Gabriela Cojocaru graciously acknowledges the support received from the Natural Sciences and Engineering Research Council (NSERC) of Canada.
- 1.J.P. Aubin, A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer (1984)Google Scholar
- 2.J.P. Aubin, A. Cellina, Differential inclusions. J. Appl. Math. Mech. 67 (2), 100 (1987)Google Scholar
- 5.M.-G. Cojocaru, Double-Layer Dynamics Theory and Human Migration After Catastrophic Events (Bergamo University Press, Bergamo, 2007)Google Scholar
- 6.M.G. Cojocaru, Piecewise solutions of evolutionary variational inequalities. Consequences for the doublelayer dynamics modelling of equilibrium problems. J. Inequal. Pure Appl. Math. 8 (2), 17 (2007)Google Scholar
- 11.M. Cojocaru, P. Daniele, A. Nagurney, Projected dynamical systems, evolutionary variational inequalities, applications, and a computational procedure, in Pareto Optimality, Game Theory …, Springer (2008), pp. 387–406Google Scholar
- 13.S. Dafermos, Congested transportation networks and variational inequalities, in Flow Control of Cogested Networks (Springer, Berlin, 1987)Google Scholar
- 22.G. Stampacchia, Variational inequalities, in Theory and Applications of Monotone Operators Proceedings of a Nato Advance Study Inst. Vienice, Italy (1969), pp. 101–192Google Scholar