On the stability of globally projected dynamical systems. (English) Zbl 0971.37013

The problem of stability of dynamical systems is very hard. Even to practically solve this problem for equilibrium points is not possible in general. So new techniques to attack this problem have been proposed. One of them [P. Dupuis and A. Nagurney [Ann. Oper. Res. 44, 9-42 (1993; Zbl 0785.93044)] constructs a globally projected dynamical system. One can show there is then a close connection of this problem to the variational inequality method. Even more the authors show that there is a relationship to the associated dynamical system with the optimal solution of a mathematical quadratic programming problem. By the above mentioned approach the authors are able to give weaker sufficient conditions for the stability which are less restrictive than those of the paper cited above. The global asymptotic stability conditions given in the paper are the symmetry and monotonicity of the treated functions. Another novelty of the paper is the result on global exponential stability. The projected dynamical systems have important applications in economics, physical equilibrium analysis, and mathematical programming.


37C75 Stability theory for smooth dynamical systems
37N40 Dynamical systems in optimization and economics


Zbl 0785.93044
Full Text: DOI


[1] Dupuis, P., and Nagurney, A., Dynamical Systems and Variational Inequalities, Annals of Operations Research, Vol. 44, pp. 19-42, 1993. · Zbl 0785.93044
[2] Friesz, T. L., Bernstein, D. H., Mehta, N. J., Tobin, R. L., and Ganjlizadeh, S., Day-to-Day Dynamic Network Disequilibria and Idealized Traveler Information Systems, Operations Research, Vol. 42, pp. 1120-1136, 1994. · Zbl 0823.90037
[3] Tanabe, K., A Geometric Method in Nonlinear Programming, Journal of Optimization Theory and Applications, Vol. 30, pp. 181-210, 1980. · Zbl 0396.90078
[4] Flam, S. D., and Ben-israel, A., A Continuous Approach to Oligopolistic Market Equilibrium, Operations Research, Vol. 38, pp. 1045-1051, 1990. · Zbl 0723.90009
[5] Zhang, D., and Nagurney, A., On the Stability of Projected Dynamical Systems, Journal of Optimization Theory and Applications, Vol. 85, pp. 97-124, 1995. · Zbl 0837.93063
[6] Hopfield, J. J., and Tank, D. W., Neural Computation of Decisions in Optimization Problems, Biological Cybernetics, Vol. 52, pp. 141-152, 1985. · Zbl 0572.68041
[7] Kennedy, M. P., and Chua, L. O., Neural Networks for Nonlinear Programming, IEEE Transactions on Circuits and Systems, Vol. 35, pp. 554-562, 1988.
[8] Xia, Y. S., A New Neural Network for Solving Linear Programming Problems and Its Applications, IEEE Transactions on Neural Networks, Vol. 7, pp. 525-529, 1996.
[9] Xia, Y. S., A New Neural Network for Solving Linear and Quadratic Programming Problems, IEEE Transactions on Neural Networks, Vol. 7, pp. 1544-1547, 1996.
[10] Friesz, T. L., Bernstein, D. H., and Stough, R., Dynamic Systems, Variational Inequalities, and Control Theoretic Models for Predicting Time-Varying Urban Network Flows, Transportation Science, Vol. 30, pp. 14-31, 1996. · Zbl 0849.90061
[11] Khalil, H. K., Nonlinear Systems, Prentice-Hall, Upper Saddle River, New Jersey, 1996.
[12] Kinderlehrer, D., and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, NY, 1980. · Zbl 0457.35001
[13] Ortega, J. M., and Rheinboldt, W. G., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, 1970. · Zbl 0241.65046
[14] Smith, T. E., Friesz, T. L., Bernstein, D. H., and Suo, Z., A Comparative Analysis of Two Minimum-Norm Projective Dynamics and Their Relationship to Variational Inequalities, Complementary and Variational Problems, Edited by J. S. Peng and M. C. Ferris, SIAM Press, Philadelphia, Pennsylvania, pp. 405-424, 1995. · Zbl 0879.65042
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