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Revised CPA method to compute Lyapunov functions for nonlinear systems

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posted on 2023-06-08, 21:10 authored by Peter GieslPeter Giesl, Sigurdur F Hafstein
The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov function for nonlinear systems with asymptotically stable equilibria. In 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. 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 C² 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 we have shown these results for planar systems, in this paper we cover general n-dimensional systems.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Journal of Mathematical Analysis and Applications

ISSN

0022-247X

Publisher

Elsevier

Issue

1

Volume

410

Page range

292-306

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2015-06-16

First Open Access (FOA) Date

2017-01-17

First Compliant Deposit (FCD) Date

2017-01-17

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