On the stability of projected dynamical systems
A class of projected dynamical systems (PDS), whose stationary points solve the corresponding variational inequality problem (VIP), was recently studied by Dupuis and Nagurney (Ref. 1). This paper initiates the study of the stability of such PDS around their stationary points and thus gives rise to the study of the dynamical stability of VIP solutions. Examples are constructed showing that such a study can be quite distinct from the classical stability study for dynamical systems (DS). We give the definition of a regular solution to a VIP and introduce the concept of a minimal face flow induced by a PDS, which is a standard DS of a lower dimension. We then show that, at the regular solutions of the VIP, the local stability of the PDS is essentially the same as that of its minimal face flow. Hence, we reduce the problem, in this case, to one of the classical stability study of DS, a more developed discipline. In a more direct way, we then establish a series of local and global stability results of the PDS, under various conditions of monotonicity.
Key WordsProjected dynamical systems variational inequalities stability theory minimal face flows
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- 2.Nagurney, A.,Network Economics: A Variational Inequality Approach, Kluwer Academic Publishers, Boston, Massachusetts, 1993.Google Scholar
- 4.Dafermos, S.,Sensitivity Analysis in Variational Inequalities, Mathematics of Operations Research, Vol. 13, pp. 421–434, 1988.Google Scholar
- 5.Hirsch, M. W., andSmale, S.,Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, New York, 1974.Google Scholar
- 6.Perko, L.,Differential Equations and Dynamical Systems, Springer Verlag, New York, New York, 1991.Google Scholar
- 7.Smith, M. J.,The Stability of a Dynamic Model of Traffic Assignment: An Application of a Method of Lyapunov, Transportation Science, Vol. 18, pp. 245–252, 1984.Google Scholar
- 8.Dupuis, P.,Large Deviation Analysis of Reflected Diffusions and Constrained Stochastic Approximation Algorithms in Convex Sets, Stochastic, Vol. 21, pp. 63–96, 1987.Google Scholar
- 9.Dafermos, S., andNagurney, A.,Sensitivity Analysis for the Asymmetric Network Equilibrium Problem, Mathematical Programming, Vol. 28, pp. 174–184, 1984.Google Scholar
- 10.Dafermos, S., andNagurney, A.,Sensitivity Analysis for the General Spatial Economic Equilibrium Problem, Operations Research, Vol. 32, pp. 1069–1088, 1984.Google Scholar
- 11.Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1972.Google Scholar