# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2020378
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## System specific triangulations for the construction of CPA Lyapunov functions

 1 Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom 2 Faculty of Physical Sciences, University of Iceland, 107 Reykjavik, Iceland

Received  September 2020 Early access December 2020

Fund Project: The research in this paper was partly supported by the Icelandic Research Fund (Ranní s) grant number 163074-052, Complete Lyapunov functions: Efficient numerical computation

Recently, a transformation of the vertices of a regular triangulation of ${\mathbb {R}}^n$ with vertices in the lattice $\mathbb{Z}^n$ was introduced, which distributes the vertices with approximate rotational symmetry properties around the origin. We prove that the simplices of the transformed triangulation are $(h, d)$-bounded, a type of non-degeneracy particularly useful in the numerical computation of Lyapunov functions for nonlinear systems using the CPA (continuous piecewise affine) method. Additionally, we discuss and give examples of how this transformed triangulation can be used together with a Lyapunov function for a linearization to compute a Lyapunov function for a nonlinear system with the CPA method using considerably fewer simplices than when using a regular triangulation.

Citation: Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020378
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Left: The triangulation ${\mathcal T}^\text{ std}_{K}$, $K = 4$, with a triangle fan at the origin. Right: The transformed approximately rotationally symmetric triangulation ${\mathcal T}_{\Phi, K}$
, right, are mapped by the linear transformation ${\bf x} \mapsto P^{-\frac{1}{2}} {\bf x}$, where $P^{-\frac{1}{2}}$ is a symmetric and positive definite matrix. This triangulation is adapted to the structure of the system with a local Lyapunov function $V( {\bf x}) = {\bf x}^\text{T}P {\bf x}$">Figure 2.  The vertices of the triangulations of Figure 1, right, are mapped by the linear transformation ${\bf x} \mapsto P^{-\frac{1}{2}} {\bf x}$, where $P^{-\frac{1}{2}}$ is a symmetric and positive definite matrix. This triangulation is adapted to the structure of the system with a local Lyapunov function $V( {\bf x}) = {\bf x}^\text{T}P {\bf x}$
Left: CPA Lyapunov function for system (3.1) with $\alpha = 0.5$ and $\beta = -0.3$ using a rectangular grid. Right: The rectangular grid, with a triangle fan at the origin, used for the computation. Level-sets of the Lyapunov function are drawn in red on both figures
">Figure 4.  Left: CPA Lyapunov function for system (3.1) with $\alpha = 0.5$ and $\beta = -0.3$ using a transformed grid. Right: The transformed grid, with a triangle fan at the origin, used for the computation. Level-sets of the Lyapunov function are drawn in red on both figures. Note that the triangulation is much better adapted to the shape of the level-sets than when using a rectangular grid as in Figure 3
Left: CPA Lyapunov function for system (3.1) with $\alpha = 0.5$ and $\beta = -0.4$ using a rectangular grid. Right: The rectangular grid, with a triangle fan at the origin, used for the computation. Level-sets of the Lyapunov function are drawn in red on both figures
. Both the area covered by the triangle fan, in both cases with $64$ triangles, as well as the area covered overall are much larger than when using the rectangular grid, see Figure 5">Figure 6.  Left: CPA Lyapunov function for system (3.1) with $\alpha = 0.5$ and $\beta = -0.4$ using a transformed grid. Right: The transformed grid, with a triangle fan at the origin, used for the computation. Level-sets of the Lyapunov function are drawn in red on both figures. Note that the triangulation is much better adapted to the shape of the level-sets than when using a rectangular grid as in Figure 5. Both the area covered by the triangle fan, in both cases with $64$ triangles, as well as the area covered overall are much larger than when using the rectangular grid, see Figure 5
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