# American Institute of Mathematical Sciences

September  2019, 24(9): 4955-4981. doi: 10.3934/dcdsb.2019040

## Verification estimates for the construction of Lyapunov functions using meshfree collocation

 1 Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom 2 Department of Mathematical Sciences, Umm Al-qura University, Saudi Arabia

* Corresponding author

The second author acknowledges funding for her PhD studies from the Saudi Government

Received  May 2018 Revised  September 2018 Published  February 2019

Lyapunov functions are functions with negative derivative along solutions of a given ordinary differential equation. Moreover, sub-level sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. One of the numerical construction methods for Lyapunov functions uses meshfree collocation with radial basis functions (RBF). In this paper, we propose two verification estimates combined with this RBF construction method to ensure that the constructed function is a Lyapunov function. We show that this combination of the RBF construction method and the verification estimates always succeeds in constructing and verifying a Lyapunov function for nonlinear ODEs in $\mathbb{R}^d$ with an exponentially stable equilibrium.

Citation: Peter Giesl, Najla Mohammed. Verification estimates for the construction of Lyapunov functions using meshfree collocation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4955-4981. doi: 10.3934/dcdsb.2019040
##### References:

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##### References:
The two point sets $S_{h_1}$ (square grid) and $C_{h_1}$ (centered square grid) in $\mathbb R^2$; $h_1$ is the distance between the square grid points in both directions
The $1$-norm balls with radius $h^* = \frac{1}{2} h_1$. The square $[0, h_1]^2$ is completely covered with closed $1$-norm balls of radius $h^* = \frac{1}{2}h_1$, centered at the vertices and the center of the square, so in $C_{h_1}$
The standard and the centered triangulation in $\mathbb R^2$
Collocation points (blue)
Approximation with $\phi_{6, 4}$. Left: Orbital derivative $v'(x, y)$, which approximates $-\|(x, y)\|^2$ well. Right: Constructed Lyapunov function $v(x, y)$
Approximation with too few points. Left: Orbital derivative $v'(x, y)$, which does not approximate $-\|(x, y)\|^2$ well. Right: Collocation points (blue) and the level set $v'(x, y) = 0$ (red), which indicates an area where $v'(x, y)>0$ near the origin
Approximation with $\phi_{7, 5}$. Left: Orbital derivative $v'(x, y)$, which approximates $-\|(x, y)\|^2$ well. Right: Constructed Lyapunov function $v(x, y)$
Approximation with $\exp(-\epsilon^2r^2)$. Left: Orbital derivative $v'(x, y)$, which approximates $-\|(x, y)\|^2$ well. Right: Constructed Lyapunov function $v(x, y)$
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