Construction of lyapunov functions on the domain of asymptotic nullcontrollability: Numerics

Fabio Camilli,Lars Grüne,Fabian Wirth

doi:10.1016/s1474-6670(17)31309-5

Abstract

We present a numerical method for the computation of control Lyapunov functions on the domain of nullcontrollability of a nonlinear system. We apply an adaptive semi-Lagrangian discretization technique to a generalized version of the Zubov equation whose solutions provide such Lyapunov functions. In particular, we address regularization issues which need to be resolved before the scheme is applicable and discuss an adaptive space discretization technique.

Highlights

In this paper we continue the theoretical work presented in the companion paper (Camilli et al, 2004) on the construction of Lyapunov functions on the domain of nullcontrollability
In (Camilli et al, 2004) it was shown that the desired Lyapunov functions can be characterized as (i) optimal value functions to suitable optimal control problems and (ii) as viscosity solutions to a suitable Hamilton–Jacobi PDE, which is a generalization of Zubov’s equation
In this paper we will use both characterizations as we apply a semi–Lagrangian discretization tech

Summary

INTRODUCTION

In this paper we continue the theoretical work presented in the companion paper (Camilli et al, 2004) on the construction of Lyapunov functions on the domain of nullcontrollability. Zubov’s equation has already been used as the basis for numerical computations, e.g. in (Dubljevic and Kazantsis, 2002), where the solutions to Zubov’s equation for a fixed control value are approximated by truncation of series solutions and the resulting Lyapunov function is used for controller design. In order to achieve uniformity in u we restrict ourselves to a compact subset of the control range U , because in this way the global Lipschitz property follows from the assumptions in (Camilli et al, 2004). Under mild local Lipschitz conditions on f and g and when g : Rn → R is nonnegative, vanishes at 0 and satisfies appropriate growth properties, (2) admits a unique viscosity solution v which approximation by the compact sets Uk = {u ∈ U | u ≤ k} for k ∈ N.

RESTRICTION TO COMPACT U

REGULARIZATION OF THE EQUATION

THE NUMERICAL SCHEME II

A NUMERICAL EXAMPLE