Computation of Lyapunov functions for systems with multiple local attractors

Jóhann Björnsson; Peter Giesl; Sigurdur F. Hafstein; Christopher M. Kellett

doi:10.3934/dcds.2015.35.4019

Article Contents

2015, Volume 35, Issue 9: 4019-4039. Doi: 10.3934/dcds.2015.35.4019

This issue Previous Article State constrained $L^\infty$ optimal control problems interpreted as differential games Next Article Value iteration convergence of $\epsilon$-monotone schemes for stationary Hamilton-Jacobi equations

Computation of Lyapunov functions for systems with multiple local attractors

1.
School of Science and Engineering, Reykjavik University, Menntavegi 1, Reykjavik, IS-101
2.
Department of Mathematics, University of Sussex, Falmer BN1 9QH
3.
School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan, New South Wales 2308

Received: May 31, 2014

Revised: September 30, 2014

Published: May 2017

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Abstract

We present a novel method to compute Lyapunov functions for continuous-time systems with multiple local attractors. In the proposed method one first computes an outer approximation of the local attractors using a graph-theoretic approach. Then a candidate Lyapunov function is computed using a Massera-like construction adapted to multiple local attractors. In the final step this candidate Lyapunov function is interpolated over the simplices of a simplicial complex and, by checking certain inequalities at the vertices of the complex, we can identify the region in which the Lyapunov function is decreasing along system trajectories. The resulting Lyapunov function gives information on the qualitative behavior of the dynamics, including lower bounds on the basins of attraction of the individual local attractors. We develop the theory in detail and present numerical examples demonstrating the applicability of our method.

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Mathematics Subject Classification: Primary: 37B25, 37M99; Secondary: 37C10.

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References

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