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Discrete and Continuous Dynamical Systems, 2012

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 (subm.) 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.

Journal of Mathematical Analysis and Applications, 1974

Journal of Mathematical Analysis and Applications, 2013

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Discrete and Continuous Dynamical Systems, 2004

Closed physical systems eventually come to rest, the reason being that due to friction of some kind they continuously lose energy. The mathematical extension of this principle is the concept of a Lyapunov function. A Lyapunov function for a dynamical system, of which the dynamics are modelled by an ordinary differential equation (ODE), is a function that is decreasing along any trajectory of the system and with exactly one local minimum. This implies that the system must eventually come to rest at this minimum. Although it has been known for over 50 years that the asymptotic stability of an ODE's equilibrium is equivalent to the existence of a Lyapunov function for the ODE, there has been no constructive method for non-local Lyapunov functions, except in special cases. Recently, a novel method to construct Lyapunov functions for ODEs via linear programming was presented [5], , which includes an algorithmic description of how to derive a linear program for a continuous autonomous ODE, such that a Lyapunov function can be constructed from any feasible solution of this linear program. We will show how to choose the free parameters of this linear program, dependent on the ODE in question, so that it will have a feasible solution if the equilibrium at the origin is exponentially stable. This leads to the first constructive converse Lyapunov theorem in the theory of dynamical systems/ODEs. 1991 Mathematics Subject Classification. 93D05, 93D20, 93D30, 34D05, 34D20.

Journal of Mathematical Analysis and Applications, 2010

In [10] a method to compute Lyapunov functions for systems with asymptotically stable equilibria was presented. The method uses finite differences on finite elements to generate a linear programming problem for the system in question, of which every feasible solution parameterises a piecewise affine Lyapunov function. In it was proved that the method always succeeds in generating a Lyapunov function for systems with an exponentially stable equilibrium. However, the proof could not guarantee that the generated function has negative orbital derivative locally in a small neighborhood of the equilibrium. In this article we give an example of a system, where no piecewise affine Lyapunov function with the proposed triangulation scheme exists. This failure is due to the triangulation of the method being too coarse at the equilibrium, and we suggest a fan-like triangulation around the equilibrium. We show that for any two-dimensional system with an exponentially stable equilibrium there is a local triangulation scheme such that the system possesses a piecewise affine Lyapunov function. Hence, the method might eventually be equipped with an improved triangulation scheme that does not have deficits locally at the equilibrium.

Lecture Notes in Control and Information Sciences, 2005

2003

This paper starts by presenting local stability conditions for limit cycles of piecewise linear systems (PLS), based on analyzing the linear part of Poincaré maps. Local stability guarantees the existence of an asymptotically stable neighborhood around the limit cycle. However, tools to characterize such neighborhood do not exist. This work gives conditions in the form of LMIs that guarantee asymptotic stability of PLS in a reasonably large region around a limit cycle, based on recent results on impact maps and surface Lyapunov functions (SuLF). These are exemplified with a biological application: a 4 th -order neural oscillator, also used in many robotics applications like, for example, juggling and locomotion.

Journal of Mathematical Analysis and Applications, 1969

IEEE Transactions on Circuits and Systems, 1979