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

In this paper, we present a new approach for finding a stablesolution of a system of nonlinear equations arising from dynamicalsystems. We introduce the concept ofstability functions and use this idea to constructstability solution models of severaltypical small signal stability problems in dynamical systems.Each model consists of a system of constrainedsemismooth equations. The advantage of the new models is twofold.Firstly, the stability requirement of dynamical systems iscontrolled by nonlinear inequalities. Secondly, the semismoothnessproperty of the stability functions makes the models solvable byefficient numerical methods. We introduce smoothing functions forthe stability functions and present a smoothing Newton methodfor solving the problems. Global and local quadratic convergence ofthe algorithm is established. Numerical examples from dynamicalsystems are also given to illustrate the efficiency of the newapproach.