Optimal ellipsoidal stability domain estimates for odd polynomial systems

The algorithms for computing estimates of the domain of attraction of an equilibrium point essentially consists of two distinct steps: 1) a Lyapunov function is selected according to some rules; 2) an estimate of the domain of attraction is computed for the chosen Lyapunov function. While step (1) strongly depends on the algorithm used, step (2) is common to all the algorithms and it can be cast as a non-convex minimization problem that is in general difficult to solve for the presence of local extreme. Tesi et al. (1996) have proposed a convex optimization method for obtaining optimal ellipsoidal estimates of polynomial systems having a single homogeneous nonlinear term other than the linear one. Motivated by these results, we show how optimal ellipsoidal estimates can be obtained for general odd polynomial systems by solving a sequence of convex optimization problems.