Converging outer approximations to global attractors using semidefinite programming

Abstract

This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.

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

“…In [10] attractor sets approximated by using SOS to search for Lyapunov functions outside some handpicked set D ⊂ R n that is known to contain the attractor set. In [11], [12] an alternative SOS based method was proposed for attractor set approximation. Impressively, the method proposed in [11] was shown to provide an arbitrarily close approximation of an attractor set with respect to the Lebesgue measure.…”

“…In [11], [12] an alternative SOS based method was proposed for attractor set approximation. Impressively, the method proposed in [11] was shown to provide an arbitrarily close approximation of an attractor set with respect to the Lebesgue measure. However, the methods in [11], [12] do not yield Lyapunov functions and hence any approximation found cannot be shown to also be an attractor set.…”

Sparse moment-sum-of-squares relaxations for nonlinear dynamical systems with guaranteed convergence

“…Those decompositions allow us to decouple the computations of those sets as well. We apply this to the linear programs for those sets from [10], [12] and [20], which describe the desired sets up to a discrepancy of Lebesgue measure zero.…”

“…We follow the approach from [10], [12] and [20] where outer approximations of the ROA, MPI set and GA are based on infinite dimensional linear programs on continuous functions approximated via the moment-sum-of-squares hierarchy (see [16] for a general introduction and [9] for recent applications).…”

“…We prove general results allowing for a decomposition of these sets provided that the vector field and constraint set posses certain structure. We combine these decompositions with the methods from [10], [12] and [20] based on infinite-dimensional linear programming. For polynomial dynamics, we show that these problems admit a sparse sum-of-squares (SOS) approximation with guaranteed convergence such that the number of variables in the largest SOS multiplier is given by the dimension of the largest subsystem appearing in the decomposition.…”