Lagrangian methods for approximating the viability kernel in high-dimensional systems

Abstract

While a number of Lagrangian algorithms to approximate reachability in dozens or even hundreds of dimensions for systems with linear dynamics have recently appeared in the literature, no similarly scalable algorithms for approximating viable sets have been developed. In this paper we describe a connection between reachability and viability that enables us to compute the viability kernel using reach sets. This connection applies to any type of system, such as those with nonlinear dynamics and/or non-convex state constraints; however, here we take advantage of it to construct three viability kernel approximation algorithms for linear systems with convex input and state constraint sets. We compare the performance of the three algorithms and demonstrate that the two based on highly scalable Lagrangian reachability–those using ellipsoidal and support vector set representations–are able to compute the viability kernel for linear systems of larger state dimension than was previously feasible using traditional Eulerian methods. Our results are illustrated on a 6-dimensional pharmacokinetic model and a 20-dimensional model of heat conduction on a lattice.