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**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.

Reviewer: Ladislav Andrey (Praha)

### MSC:

37C75 | Stability theory for smooth dynamical systems |

37N40 | Dynamical systems in optimization and economics |

### Keywords:

globally projected dynamical systems; stability analysis; variational inequality; neural networks### Citations:

Zbl 0785.93044
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\textit{Y. S. Xia} and \textit{J. Wang}, J. Optim. Theory Appl. 106, No. 1, 129--150 (2000; Zbl 0971.37013)

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### References:

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