Control Lyapunov Functions and Zubov's Method

Fabio Camilli,Fabian Wirth,Lars Grüne

doi:10.1137/06065129x

Abstract

For finite-dimensional nonlinear control systems we study the relation between asymptotic null-controllability and control Lyapunov functions. It is shown that control Lyapunov functions (CLFs) may be constructed on the domain of asymptotic null-controllability as viscosity solutions of a first order PDE that generalizes Zubov's equation. The solution is also given as the value function of an optimal control problem from which several regularity results may be obtained.

Highlights

A fundamental question in the qualitative theory of dynamical systems concerns the stability of fixed points or more generally attractors. In his seminal thesis Lyapunov showed that a sufficient stability condition can be obtained in terms of a positive definite function that decreases along the trajectories of the system, or as we say today the existence of a Lyapunov function implies asymptotic stability
While for linear systems a constructive procedure to find Lyapunov functions has already been given by Lyapunov, the first general constructive procedure to find Lyapunov functions was obtained by Zubov [37]
A Lyapunov function on the domain of attraction of an asymptotically stable fixed point x∗ ∈ Rn of the system x (t) = f (x(t)), t ∈ R, x ∈ Rn may be found by solving the 1st order PDE, called Zubov’s equation, Dv(x)f (x) = −h(x)(1 − v(x)) 1 + f 2 x ∈ Rn, under the condition that v(0) = 0

Summary

Introduction

A fundamental question in the qualitative theory of dynamical systems concerns the stability of fixed points or more generally attractors. In contrast to the case of ordinary differential equations, where smooth Lyapunov functions always exist for asymptotically stable systems, it is not reasonable to require too many regularity properties of Lyapunov functions for controllability questions For this reason it is standard to formulate the concept of a control Lyapunov function in nondifferential terms. In general asymptotic nullcontrollability does not imply the existence of continuous stabilizing feedback as there may be topological obstructions to this which even carry over to the case of upper semicontinuous set-valued feedbacks, [7, 12, 27] For this reason discontinuous feedbacks and associated solution concepts have been one of the focal points of the research on CLF’s in recent times starting with [10]. It is shown that for the classical linear quadratic control problem the general equations of this paper reduce to the standard algebraic Riccati equation

The domain of null controllability

Characterization of D0 using Optimal Control

Characterizations of V and v by Zubov’s Method

Approximation with bounded control values

Examples