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

Abstract

This paper develops sparse moment-sum-of-squares approximations for three problems from nonlinear dynamical systems: region of attraction, maximum positively invariant set and global attractor. 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. The dimension of such subsystems depends on the sparse structure of the vector field and the constraint set and can allow for a significant reduction of the size of the semidefinite program (SDP) relaxations, thereby allowing to address far larger problems without compromising convergence guarantees. The method is simple to use and based on convex optimization. Numerical examples demonstrate the approach.

Finding unstable periodic orbits: a hybrid approach with polynomial optimization

“…As a result, the largest polynomial degree that can currently be considered for the nine-dimensional system in subsection 4.3 is approximately 10 on a workstation with 64GB of RAM, and reduces to no more than 4 or 6 for ODEs with a few tens of states. Nevertheless, removing computational bottlenecks in general polynomial optimization and in its applications to dynamical systems are problems that have attracted significant interest in recent years (see, e.g., [28][29][30][31][32][33][34][35][36][37]), so we expect that our UPO search strategy will become practical for ODEs of moderate dimension in the near future. With computational aspects in mind, a particularly attractive aspect of the control strategy proposed in this paper is that its four steps do not depend on the particular algorithms used to carry them out.…”

Spatio-Temporal Decomposition of Sum-of-Squares Programs for the Region of Attraction and Reachability

“…Finally, a natural way to decrease the computational complexity further is to combine the approach with the sparsity-based decomposition method of [24], where the time and state-space splitting would be applied to the subsystems that cannot decomposed using the method of [24].…”

Exploiting Term Sparsity in Moment-SOS hierarchy for Dynamical Systems

“…To this end, different speed-up techniques have been proposed to reduce the size of SDPs via exploiting structure of the dynamical system. Among these are symmetries (see [9] for symmetry exploitation of polynomial optimization, as well as [2] exploiting symmetries in the context of dynamical systems) or correlative sparsity as in [15] for polynomial optimization, or in [11] where a specific sparsity structure was used to decompose the SDP while preserving convergence guarantees. In this paper we present the use of a recent term sparsity approach [19,20] which has been already proven useful for a wide range of polynomial optimization problems, involving complex [17] or noncommutative [18] variables, and fast approximation of joint spectral radius of sparse matrices [16].…”

“…Whereas the approach of [11] is concerned with the sparsity among the state variables themselves, the approach proposed here exploits sparsity in the algebraic description of the dynamics, in particular among the monomial terms appearing in the components of the polynomial vector field. The method proceeds by searching non-negativity certificates comprised of polynomials with specific sets of terms only which in turn are enlarged in an iterative scheme.…”