Construction of a local and global Lyapunov function using radial basis functions

Giesl, Peter

doi:10.1093/imamat/hxn018

Cited by 37 publications

(41 citation statements)

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“…(Johansson, 1999;Johansen, 2000;Marinósson, 2002;Giesl, 2007;Hafstein, 2007;Björnsson et al, 2014b;Kamyar and Peet, 2015;Anderson and Papachristodoulou, 2015;Doban, 2016;Doban and Lazar, 2016) see also the recent review . Our algorithm will be based on the RBF (Radial Basis Function) method, a special case of mesh-free collocation, which approximates the solution of a linear PDE, specifying the orbital deriva-tive.…”

Section: Mesh-free Collocationmentioning

confidence: 99%

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Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Argáez

Hafstein

Giesl

2017

Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

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Abstract:Ordinary differential equations arise in a variety of applications, including e.g. climate systems, and can exhibit complicated dynamical behaviour. Complete Lyapunov functions can capture this behaviour by dividing the phase space into the chain-recurrent set, determining the long-time behaviour, and the transient part, where solutions pass through. In this paper, we present an algorithm to construct complete Lyapunov functions. It is based on mesh-free numerical approximation and uses the failure of convergence in certain areas to determine the chain-recurrent set. The algorithm is applied to three examples and is able to determine attractors and repellers, including periodic orbits and homoclinic orbits.

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Section: Mesh-free Collocationmentioning

confidence: 99%

“…has a negative orbital derivative. Such error estimates were derived, for example in (Giesl, 2007;Giesl and Wendland, 2007), see also (Narcowich et al, 2005;Wendland, 2005).…”

Section: Mesh-free Collocationmentioning

confidence: 99%

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Argáez

Hafstein

Giesl

2017

Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

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“…In [33], Rezaiee-Pajand and Moghaddasie proposed a different collocation method using two classes of basis functions. Giesl [8] proposed a method to construct Lyapunov functions for autonomous systems with an exponentially stable equilibrium by numerically solving a generalized Zubov equation, cf. [36].…”

Section: Introductionmentioning

confidence: 99%

Revised CPA method to compute Lyapunov functions for nonlinear systems

Giesl

Hafstein

2014

Journal of Mathematical Analysis and Applications

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The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov function for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium's basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C 2 and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems.

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“…In [28], the authors proposed a new method for computing Lyapunov functions for PDSs by solving semi-algebraic constraints using the tool DISCOVERER [36]. Approaches for constructing Lyapunov functions beyond polynomials using radial basis functions were proposed in [10].…”

Section: Related Workmentioning

confidence: 99%

Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems

2012

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The notion of Lyapunov function plays a key role in the design and verification of dynamical systems, as well as hybrid and cyber-physical systems. In this paper, to analyze the asymptotic stability of a dynamical system, we generalize standard Lyapunov functions to relaxed Lyapunov functions (RLFs), by considering higher order Lie derivatives. Furthermore, we present a method for automatically discovering polynomial RLFs for polynomial dynamical systems (PDSs). Our method is relatively complete in the sense that it is able to discover all polynomial RLFs with a given predefined template for any PDS. Therefore it can also generate all polynomial RLFs for the PDS by enumerating all polynomial templates.

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Construction of a local and global Lyapunov function using radial basis functions

Cited by 37 publications

References 0 publications

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Revised CPA method to compute Lyapunov functions for nonlinear systems

Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems

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