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Projected dynamical systems: Stability analysis and computation with applications to transportation and economic systems
Abstract
This dissertation studies the theory and applications of a new and nonstandard class of dynamical system, called projected dynamical systems. We focus on three subjects: (1) the proposal of dynamical models, for social and economic systems that are subject to certain constraints, (2) the establishment of the stability analysis for such models, and (3) the development of discrete time algorithms for the computation of equilibria. The applications that this thesis is devoted to include: elastic and fixed demand transportation network problems, spatial price market problems, and oligopoly problems. The theoretical foundation is laid down through rigorous mathematics. (1) Projected dynamical system models. In each application chapter, a projected dynamical system model is proposed to describe the underlying competitive system of the equilibrium problem. These models extend the static variational inequality formulations of the equilibrium problems to an additional dimension of time so as to allow the study of the disequilibrium behavior that leads to the equilibrium states. (2) Stability analysis. An important achievement of this thesis is the establishment of the stability theory of projected dynamical systems, which is then utilized to provide stability analysis for the application models. The local and global stability results presented here address the following questions: Will the competitive system eventually approach an equilibrium and at what rate? If a competitive behavior starts near an equilibrium, will it stay close to it forever? Is a certain equilibrium state stable to some local perturbation and to what extent. In particular applications, these questions represent practical concerns of interest. (3) Discrete time algorithms. Another contribution is the development of efficient algorithms for the computation of equilibria that can be implemented on parallel computer architectures. These algorithms are derived through time discretization of the projected dynamical system models of the application problems, and hence, besides serving for computational purposes, they track the dynamic behavior of the competitive systems. We have established the convergence of these algorithms under reasonable conditions in the context of the specific applications.
Subject Area
Industrial engineering|Operations research
Recommended Citation
Zhang, Ding, "Projected dynamical systems: Stability analysis and computation with applications to transportation and economic systems" (1996). Doctoral Dissertations Available from Proquest. AAI9639057.
https://scholarworks.umass.edu/dissertations/AAI9639057