A Converse Sum of Squares Lyapunov Function for Outer Approximation of Minimal Attractor Sets of Nonlinear Systems

Abstract

Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose 1-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.

Converging outer approximations to global attractors using semidefinite programming

“…We begin with presenting the infinite dimensional optimization problems for global attractors from [21] and [11].…”

“…Some approximate the attractor by following trajectories for long but finite time T ∈ [0, ∞) or using set oriented methods as in [5]. Another approach motivated by a relaxation of the notion of Lyapunov functions was given in [11]. There the authors showed that the weaker concept of Lyapunov function they use is still sufficient to envelope the attractor by positively invariant sets while at the same time SOS polynomials can be used.…”

“…In this paper we marry both approaches by combining their techniques and get converging outer approximations of the global attractor consisting of positively invariant sets. Because both the methods from [21] and [11] are based on convex optimization via sum-of-squares techniques the same is true for our proposed method. The method is easy to use and numerical examples illustrate the procedure.…”

“…

In this paper we combine the approaches from [21] and [11] for approximating global attractors. In [21] the global attractors is arbitrarily well approximated by sets that are not necessarily positively invariant.

…”

“…In [21] the global attractors is arbitrarily well approximated by sets that are not necessarily positively invariant. On the contrary, the method from [11] provides supersets of the global attractor which are positively invariant but not necessarily converging. In this paper we marry both approaches by combining their techniques and get converging outer approximations of the global attractor consisting of positively invariant sets.…”