# 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  September 2019 Early access  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 and Continuous Dynamical Systems - B, 2019, 24 (9) : 4955-4981. doi: 10.3934/dcdsb.2019040
##### References:
 [1] R. Baier, L. Grüne and S. Hafstein, Linear programming based Lyapunov function computation for Differential Inclusions, Discrete Contin. Dyn. Syst. Ser. B., 17 (2012), 33-56.  doi: 10.3934/dcdsb.2012.17.33. [2] M. Buhmann, Radial Basis Functions: Theory and Implementations, volume 12 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511543241. [3] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math. 1904, Springer, 2007. [4] P. Giesl, Construction of a local and global Lyapunov function using Radial Basis Functions, IMA J. Appl. Math., 73 (2008), 782-802.  doi: 10.1093/imamat/hxn018. [5] P. Giesl and S. Hafstein, Computation and verification of Lyapunov functions, SIAM J. Appl. Dyn. Syst., 14 (2015), 1663-1698.  doi: 10.1137/140988802. [6] P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331.  doi: 10.3934/dcdsb.2015.20.2291. [7] P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to Dynamical Systems, SIAM J. Numer. Anal., 45 (2007), 1723-1741.  doi: 10.1137/060658813. [8] S. Hafstein, C. Kellett and H. Li, Computing continuous and piecewise affine Lyapunov functions for nonlinear systems, J. Comp. Dyn., 2 (2015), 227-246.  doi: 10.3934/jcd.2015004. [9] C. M. Kellett, Classical converse theorems in Lyapunov's second method, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 2333-2360.  doi: 10.3934/dcdsb.2015.20.2333. [10] J. Massera, On Liapounoff's conditions of stability, Ann. of Math., 50 (1949), 705-721.  doi: 10.2307/1969558. [11] N. Mohammed, Grid Refinement and Verification Estimates for the RBF Construction Method of Lyapunov Functions, Doctoral thesis (PhD), University of Sussex, 2016. [12] N. Mohammed and P. Giesl, Grid refinement in the construction of Lyapunov functions using radial basis functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2453-2476.  doi: 10.3934/dcdsb.2015.20.2453. [13] M. J. D. Powell, The theory of radial basis function approximation in 1990, In Advances in Numerical Analysis, Vol. II (Lancaster, 1990), Oxford Sci. Publ., pages 105-210. Oxford Univ. Press, New York, 1992. [14] R. Schaback and H. Wendland, Kernel techniques: From machine learning to meshless methods, Acta Numer., 15 (2006), 543-639.  doi: 10.1017/S0962492906270016. [15] H. Wendland, Error estimates for interpolation by compactly supported Radial Basis Functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272.  doi: 10.1006/jath.1997.3137. [16] H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.

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##### References:
 [1] R. Baier, L. Grüne and S. Hafstein, Linear programming based Lyapunov function computation for Differential Inclusions, Discrete Contin. Dyn. Syst. Ser. B., 17 (2012), 33-56.  doi: 10.3934/dcdsb.2012.17.33. [2] M. Buhmann, Radial Basis Functions: Theory and Implementations, volume 12 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511543241. [3] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Math. 1904, Springer, 2007. [4] P. Giesl, Construction of a local and global Lyapunov function using Radial Basis Functions, IMA J. Appl. Math., 73 (2008), 782-802.  doi: 10.1093/imamat/hxn018. [5] P. Giesl and S. Hafstein, Computation and verification of Lyapunov functions, SIAM J. Appl. Dyn. Syst., 14 (2015), 1663-1698.  doi: 10.1137/140988802. [6] P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331.  doi: 10.3934/dcdsb.2015.20.2291. [7] P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to Dynamical Systems, SIAM J. Numer. Anal., 45 (2007), 1723-1741.  doi: 10.1137/060658813. [8] S. Hafstein, C. Kellett and H. Li, Computing continuous and piecewise affine Lyapunov functions for nonlinear systems, J. Comp. Dyn., 2 (2015), 227-246.  doi: 10.3934/jcd.2015004. [9] C. M. Kellett, Classical converse theorems in Lyapunov's second method, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 2333-2360.  doi: 10.3934/dcdsb.2015.20.2333. [10] J. Massera, On Liapounoff's conditions of stability, Ann. of Math., 50 (1949), 705-721.  doi: 10.2307/1969558. [11] N. Mohammed, Grid Refinement and Verification Estimates for the RBF Construction Method of Lyapunov Functions, Doctoral thesis (PhD), University of Sussex, 2016. [12] N. Mohammed and P. Giesl, Grid refinement in the construction of Lyapunov functions using radial basis functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2453-2476.  doi: 10.3934/dcdsb.2015.20.2453. [13] M. J. D. Powell, The theory of radial basis function approximation in 1990, In Advances in Numerical Analysis, Vol. II (Lancaster, 1990), Oxford Sci. Publ., pages 105-210. Oxford Univ. Press, New York, 1992. [14] R. Schaback and H. Wendland, Kernel techniques: From machine learning to meshless methods, Acta Numer., 15 (2006), 543-639.  doi: 10.1017/S0962492906270016. [15] H. Wendland, Error estimates for interpolation by compactly supported Radial Basis Functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272.  doi: 10.1006/jath.1997.3137. [16] H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.
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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