Computation of Lyapunov Functions under State Constraints using Semidefinite Programming Hierarchies *

TL;DR

This paper develops algorithms using hierarchies of linear and semidefinite programs to compute Lyapunov functions for systems with state constraints, enabling stability analysis within convex sets.

Contribution

It introduces novel discretization and semidefinite programming hierarchies for efficiently computing Lyapunov functions under convex state constraints.

Findings

Hierarchical linear programs for conic constraints.

Semidefinite programming hierarchy for semi-algebraic sets.

Algorithms facilitate stability verification within constrained domains.

Abstract

We provide algorithms for computing a Lyapunov function for a class of systems where the state trajectories are constrained to evolve within a closed convex set. The dynamical systems that we consider comprise a differential equation which ensures continuous evolution within the domain, and a normal cone inclusion which ensures that the state trajectory remains within a prespecified set at all times. Finding a Lyapunov function for such a system boils down to finding a function which satisfies certain inequalities on the admissible set of state constraints. It is well-known that this problem, despite being convex, is computationally difficult. For conic constraints, we provide a discretization algorithm based on simplicial partitioning of a sim-plex, so that the search of desired function is addressed by constructing a hierarchy (associated with the diameter of the cells in the partition) of linear programs. Our second algorithm is tailored to semi-algebraic sets, where a hierarchy of semidefinite programs is constructed to compute Lyapunov functions as a sum-of-squares polynomial.