Finding a stable solution of a system of nonlinearequations arising from dynamic systems

Carl. T. Kelley; Liqun  Qi; Xiaojiao Tong; Hongxia Yin

doi:10.3934/jimo.2011.7.497

Article Contents

2011, Volume 7, Issue 2: 497-521. Doi: 10.3934/jimo.2011.7.497

This issue Previous Article Optimality conditions for approximate solutions of vector optimization problems Next Article

Finding a stable solution of a system of nonlinear equations arising from dynamic systems

1.
Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205
2.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon
3.
Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, 421002
4.
Department of Mathematics and Statistics, Minnesota State University Mankato, Mankato, MN 56001

Received: July 31, 2009

Revised: February 28, 2011

Published: March 2011

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Abstract

In this paper, we present a new approach for finding a stable solution of a system of nonlinear equations arising from dynamical systems. We introduce the concept of stability functions and use this idea to construct stability solution models of several typical small signal stability problems in dynamical systems. Each model consists of a system of constrained semismooth equations. The advantage of the new models is twofold. Firstly, the stability requirement of dynamical systems is controlled by nonlinear inequalities. Secondly, the semismoothness property of the stability functions makes the models solvable by efficient numerical methods. We introduce smoothing functions for the stability functions and present a smoothing Newton method for solving the problems. Global and local quadratic convergence of the algorithm is established. Numerical examples from dynamical systems are also given to illustrate the efficiency of the new approach.

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Mathematics Subject Classification: Primary: 65H10, 65K10; Secondary: 90C30.

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