On the stability domain estimation via a quadratic Lyapunov function: convexity and optimality properties for polynomial systems

The problem of estimating the stability domain of the origin of an n-order polynomial system is considered. Exploiting the structure of this class of systems it is shown that, for a given quadratic Lyapunov function, an estimate of the stability domain can be obtained by solving a suitable convex optimization problem. This estimate is shown to be optimal for an important subclass including both quadratic and cubic systems, and its accuracy in the general polynomial case is discussed via several examples.