On the estimation and control of the domain of attraction through rational Lyapunov functions

This paper addresses the estimation and control of the domain of attraction (DA) of equilibrium points through rational Lyapunov functions (LFs). Specifically, continuous-time nonlinear systems with polynomial nonlinearities are considered. The estimation problem consists of computing the largest estimate of the DA (LEDA) provided by a given rational LF. The control problem consists of computing a polynomial static output controller of given degree for maximizing such a LEDA. It is shown that lower bounds of the LEDA in the estimation problem, or the maximum achievable LEDA in the control problem, can be obtained by solving either an eigenvalue problem or a generalized eigenvalue problem with smaller dimension. The conservatism of these lower bounds can be reduced by increasing the degree of some multipliers introduced in the construction of the optimization problems. Moreover, a necessary and sufficient condition for establishing tightness of the found lower bounds is provided. Some numerical examples illustrate the use of the proposed results.