Computation of Lyapunov Functions under State Constraints using Semidefinite Programming Hierarchies *
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.
Cite
@article{arxiv.2009.06885,
title = {Computation of Lyapunov Functions under State Constraints using Semidefinite Programming Hierarchies *},
author = {Marianne Souaiby and Aneel Tanwani and Didier Henrion},
journal= {arXiv preprint arXiv:2009.06885},
year = {2020}
}