Abstract
Lyapunov functions are an important tool to determine the basin of attraction of exponentially stable equilibria in dynamical systems. In Marinosson (2002), a method to construct Lyapunov functions was presented, using finite differences on finite elements and thus transforming the construction problem into a linear programming problem. In Hafstein (2004), it was shown that this method always succeeds in constructing a Lyapunov function, except for a small, given neighbourhood of the equilibrium. For two-dimensional systems, this local problem was overcome by choosing a fan-like triangulation around the equilibrium. In Giesl/Hafstein (2010) the existence of a piecewise linear Lyapunov function was shown, and in Giesl/Hafstein (2012) it was shown that the above method with a fan-like triangulation always succeeds in constructing a Lyapunov function, without any local exception. However, the previous papers only considered two-dimensional systems. This paper generalises the existence of piecewise linear Lyapunov functions to arbitrary dimensions.
Highlights
In this paper we study the autonomous system of differential equations x = f (x), f ∈ C1(Rn, Rn), and assume that the origin is an exponentially stable equilibrium with basin of attraction denoted by A
In [6], we have shown that the triangulation scheme used in [17, 8, 9, 10] in general does not allow for piecewise affine Lyapunov functions near the equilibrium
We prove the existence of a piecewise linear Lyapunov function w : [−b, b]n → R for any C1 system with an exponentially stable equilibrium at the origin
Summary
In this paper we study the autonomous system of differential equations x = f (x), f ∈ C1(Rn, Rn), and assume that the origin is an exponentially stable equilibrium with basin of attraction denoted by A. The size of this neighborhood is a priori not known and is, except for linear f , in general a poor estimate of A (see, for example, [8] for more details) This method to compute local Lyapunov functions is constructive because there is an algorithm to solve the Lyapunov equation that succeeds whenever it possesses a solution, cf Bartels and Stewart [2]. Giesl proposed in [5] a method to construct Lyapunov functions for autonomous systems with an exponentially stable equilibrium by solving numerically a generalised Zubov equation, cf [21],. A vector x ∈ Rn is assumed to be a column vector and xT is the corresponding row vector
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