# American Institute of Mathematical Sciences

April  2011, 7(2): 497-521. doi: 10.3934/jimo.2011.7.497

## 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, United States 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong 3 Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, 421002, China 4 Department of Mathematics and Statistics, Minnesota State University Mankato, Mankato, MN 56001, United States

Received  August 2009 Revised  March 2011 Published  April 2011

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.
Citation: Carl. T. Kelley, Liqun Qi, Xiaojiao Tong, Hongxia Yin. Finding a stable solution of a system of nonlinear equations arising from dynamic systems. Journal of Industrial and Management Optimization, 2011, 7 (2) : 497-521. doi: 10.3934/jimo.2011.7.497
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