On the stability of projected dynamical systems. (English) Zbl 0837.93063

Summary: A class of projected dynamical systems (PDS), whose stationary points solve the corresponding variational inequality problem (VIP), was recently studied by P. Dupuis and and the second author [Ann. Oper. Res. 44, 9-42 (1993; Zbl 0785.93044)]. 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. In a more direct way, we then establish a series of local and global stability results of the PDS, under various conditions of monotonicity.


93D20 Asymptotic stability in control theory
93B11 System structure simplification


Zbl 0785.93044
Full Text: DOI


[1] Dupuis, P., andNagurney, A.,Dynamical Systems and Variational Inequalities, Annals of Operations Research, Vol. 44, pp. 9–42, 1993. · Zbl 0785.93044
[2] Nagurney, A.,Network Economics: A Variational Inequality Approach, Kluwer Academic Publishers, Boston, Massachusetts, 1993. · Zbl 0873.90015
[3] Nagurney, A., Takayama, T., andZhang, D.,Massively Parallel Computation of Spatial Price Equilibrium Problems as Dynamical Systems, Journal of Economic Dynamics and Control, Vol. 19, pp. 3–37, 1995. · Zbl 0875.90112
[4] Dafermos, S.,Sensitivity Analysis in Variational Inequalities, Mathematics of Operations Research, Vol. 13, pp. 421–434, 1988. · Zbl 0674.49007
[5] Hirsch, M. W., andSmale, S.,Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, New York, 1974. · Zbl 0309.34001
[6] Perko, L.,Differential Equations and Dynamical Systems, Springer Verlag, New York, New York, 1991. · Zbl 0717.34001
[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.
[8] Dupuis, P.,Large Deviation Analysis of Reflected Diffusions and Constrained Stochastic Approximation Algorithms in Convex Sets, Stochastic, Vol. 21, pp. 63–96, 1987. · Zbl 0614.60023
[9] Dafermos, S., andNagurney, A.,Sensitivity Analysis for the Asymmetric Network Equilibrium Problem, Mathematical Programming, Vol. 28, pp. 174–184, 1984. · Zbl 0535.90038
[10] Dafermos, S., andNagurney, A.,Sensitivity Analysis for the General Spatial Economic Equilibrium Problem, Operations Research, Vol. 32, pp. 1069–1088, 1984. · Zbl 0562.90009
[11] Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1972. · Zbl 0224.49003
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