Minimization with differential inequality and equality constraints applied to complete Lyapunov functions

Figure 1.  We use the algorithm for the system $ \dot{x} = 1 $ with only two points $ \Gamma = \{-1, 1\} $ with equality condition $ { \dot{v}}(\pm 1) = -1 $ and support radius $ 1/c = 1/2 $ of the Radial Basis function. Top left: $ v(x) $, top right $ { \dot{v}}(x) $. The function $ v $ is constant apart from small neighborhoods around $ \pm 1 $. Bottom left: the values $ \beta_i $ at each collocation point, bottom right: the values $ { \dot{v}}(x_i) $ at each collocation point. The coefficients $ \beta_i $ are only zero around $ \pm 1 $ and at the boundary of the interval; in the areas where they are strictly negative, the function is constant

Figure 2.  We use the algorithm for the system $ \dot{x} = 1 $ with only two points $ \Gamma = \{-1, 1\} $ with equality condition $ { \dot{v}}(\pm 1) = -1 $ and support radius $ 1/c = 1/0.3 $ of the Radial Basis function. Top left: $ v(x) $, top right $ { \dot{v}}(x) $. The function $ v $ is strictly decreasing apart from $ 0 $ and the boundary. Bottom left: the values $ \beta_i $ at each collocation point, bottom right: the values $ { \dot{v}}(x_i) $ at each collocation point. The coefficients $ \beta_i $ are mostly zero apart from at $ 0 $ and at the boundary of the interval; this is a consequence of the proof of Proposition 3.3

Figure 3.  The CLF $ v(x, y) $ (left) as well as $ { \dot{v}}(x, y) $ (right). We have used $ { \dot{v}}(x) = -1 $ for $ x\in I $ and $ v(x) \le 0 $ for $ x\in X\setminus I $

Figure 4.  The CLF $ v(x, y) $ (left) as well as $ { \dot{v}}(x, y) $ (right) when using the conditions $ { \dot{v}}(x) = -1 $ for $ x\in M $ and $ { \dot{v}}(x) \le 0 $ for $ x\in X\setminus M $

Figure 5.  The CLF $ v(x, y) $ (left) as well as $ { \dot{v}}(x, y) $ (right) when using the conditions $ { \dot{v}}(x) = -1 $ for $ x\in O $ and $ { \dot{v}}(x) \le 0 $ for $ x\in X\setminus O $

Figure 6.  The CLF $ v(x, y) $ (left) as well as $ { \dot{v}}(x, y) $ (right) when using the conditions $ { \dot{v}}(x) = -1 $ for $ x\in I\cup M \cup O $ and $ { \dot{v}}(x) \le 0 $ for $ x\in X\setminus (I\cup M \cup O) $

Figure 7.  The area where the orbital derivative of the CPA interpolation of the computed CLF candidate fails to have a negative orbital derivative, when using the condition $ { \dot{v}}(x) = -1 $ for $ x\in I $ and $ v(x) \le 0 $ for $ x\in X\setminus I $ (upper left), $ { \dot{v}}(x) = -1 $ for $ x\in M $ and $ v(x) \le 0 $ for $ x\in X\setminus M $ (upper right), $ { \dot{v}}(x) = -1 $ for $ x\in O $ and $ v(x) \le 0 $ for $ x\in X\setminus O $ (lower left), and $ { \dot{v}}(x) = -1 $ for $ x\in I\cup M \cup O $ and $ v(x) \le 0 $ for $ x\in X\setminus (I\cup M \cup O) $ (lower right)

Figure 8.  The area where $ \beta_i $ corresponding to the collocation point $ x_i $ fulfills $ \beta_i \le -10^{-5} $. $ {\dot{v}}(x) = -1 $ for $ x\in I $ and $ v(x) \le 0 $ for $ x\in X\setminus I $ (upper left), $ {\dot{v}}(x) = -1 $ for $ x\in M $ and $ v(x) \le 0 $ for $ x\in X\setminus M $ (upper right), $ {\dot{v}}(x) = -1 $ for $ x\in O $ and $ v(x) \le 0 $ for $ x\in X\setminus O $ (lower left), and $ {\dot{v}}(x) = -1 $ for $ x\in I\cup M \cup O $ and $ v(x) \le 0 $ for $ x\in X\setminus (I\cup M \cup O) $ (lower right)

Figure 9.  Ordering the coefficients $ |\beta_i| $ in decreasing order and plotting $ \log_{10}(|\beta_i|) $ gives the following result for points 320 to 335; this shows that there is a sharp decline between points close to zero and negative points

Figure 10.  The CLF candidate $ v(x, y) $ (left) as well as $ {\dot{v}}(x, y) $ (right) in the second step. From top to bottom we started using the conditions $ {\dot{v}}(x) = -1 $ for $ x\in I $, $ x\in M $, $ x\in O $, and $ x\in I\cup M \cup O $ and $ {\dot{v}}(x) \le 0 $ for the rest

Figure 11.  Second step. The area where the orbital derivative of the CPA interpolation of the computed CLF fails to have a negative orbital derivative, when using the condition $ { \dot{v}}(x) = -1 $ for $ x\in I $ and $ v(x) \le 0 $ for $ x\in X\setminus I $ (upper left), $ { \dot{v}}(x) = -1 $ for $ x\in M $ and $ v(x) \le 0 $ for $ x\in X\setminus M $ (upper right), $ { \dot{v}}(x) = -1 $ for $ x\in O $ and $ v(x) \le 0 $ for $ x\in X\setminus O $ (lower left), and $ { \dot{v}}(x) = -1 $ for $ x\in I\cup M \cup O $ and $ v(x) \le 0 $ for $ x\in X\setminus (I\cup M \cup O) $ (lower right)

Figure 12.  Second step with $ \alpha = 0.18 $. Top: The area in $ \Omega $ where the orbital derivative of the CPA interpolation of the computed CLF fails to have a negative orbital derivative. Bottom: The collocation points $ x_i $, such that the corresponding coefficients satisfy $ \beta_i \le -10^{-5} $. Both sets indicate the chain-recurrent set, in this case a periodic orbit. The approximation is poor due to only $ 668 $ collocation points

Figure 13.  Second step with $ \alpha = 0.12 $. Top: The area in $ \Omega $ where the orbital derivative of the CPA interpolation of the computed CLF fails to have a negative orbital derivative. Bottom: The collocation points $ x_i $, such that the corresponding coefficients satisfy $ \beta_i \le -10^{-5} $. Both sets indicate the chain-recurrent set, in this case a periodic orbit. The approximation uses $ 2532 $ collocation points and is much better than with $ \alpha = 0.18 $ and $ 668 $ collocation points, in particular when using the orbital derivative

Figure 14.  Second step with $ \alpha = 0.06 $. Top: The area in $ \Omega $ where the orbital derivative of the CPA interpolation of the computed CLF fails to have a negative orbital derivative. Bottom: The collocation points $ x_i $, such that the corresponding coefficients satisfy $ \beta_i \le -10^{-5} $. Both sets indicate the chain-recurrent set, in this case a periodic orbit. The approximation uses $ 18133 $ collocation points and is far better than with $ \alpha = 0.12 $ and $ 668 $ collocation points, both when using the orbital derivative and the $ \beta_i $