Verification estimates for the construction of Lyapunov functions using meshfree collocation

Figure 1.  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

Figure 2.  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} $

Figure 3.  The standard and the centered triangulation in $ \mathbb R^2 $

Figure 4.  Collocation points (blue)

Figure 5.  Approximation with $ \phi_{6, 4} $. Left: Orbital derivative $ v'(x, y) $, which approximates $ -\|(x, y)\|^2 $ well. Right: Constructed Lyapunov function $ v(x, y) $

Figure 6.  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

Figure 7.  Approximation with $ \phi_{7, 5} $. Left: Orbital derivative $ v'(x, y) $, which approximates $ -\|(x, y)\|^2 $ well. Right: Constructed Lyapunov function $ v(x, y) $

Figure 8.  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) $