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Construction of a contraction metric by meshless collocation

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  • A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of an equilibrium and it is robust to small perturbations of the system, including those varying the position of the equilibrium.

    The contraction metric is described by a matrix-valued function $ M(x) $ such that $ M(x) $ is positive definite and $ F(M)(x) $ is negative definite, where $ F $ denotes a certain first-order differential operator. In this paper, we show existence, uniqueness and continuous dependence on the right-hand side of the matrix-valued partial differential equation $ F(M)(x) = -C(x) $. We then use a construction method based on meshless collocation, developed in the companion paper [12], to approximate the solution of the matrix-valued PDE. In this paper, we justify error estimates showing that the approximate solution itself is a contraction metric. The method is applied to several examples.

    Mathematics Subject Classification: Primary: 37B25, 65N35; Secondary: 37M99, 65N15.

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  • Figure 1.  System (45) with $ \epsilon = 0 $. The collocation points used for the approximation together with the boundaries of the areas where $ \mathop{\mathrm{sign}}\limits ( \mathop{\mathrm{tr}}\limits F(S)(x,y))- \mathop{\mathrm{sign}}\limits (\det F(S)(x,y)) = -2 $ (red) and $ \mathop{\mathrm{sign}}\limits ( \mathop{\mathrm{tr}}\limits S(x,y))+ \mathop{\mathrm{sign}}\limits (\det S(x,y)) = 2 $ (blue). Blue and red lines are lines where one of the requirements of a contraction metric is violated. The constructed metric is thus a valid contraction metric where the collocation points are placed, but not beyond the first red or blue line

    Figure 2.  $ \mathop{\mathrm{sign}}\limits ( \mathop{\mathrm{tr}}\limits F_\epsilon(S)(x,y))- \mathop{\mathrm{sign}}\limits (\det F_\epsilon(S)(x,y)) $. If this function is $ -2 $, then $ F_\epsilon(S)(x,y) $ is negative definite, which is one of the requirements for $ S $ to be a contraction metric for the system with $ \epsilon = 0.1 $

    Figure 3.  The collocation points used for the approximation with $ f_0 $ together with the boundaries of the areas where $ \mathop{\mathrm{sign}}\limits ( \mathop{\mathrm{tr}}\limits F_\epsilon(S)(x,y))- \mathop{\mathrm{sign}}\limits (\det F_\epsilon(S)(x,y)) = -2 $ (red) and $ \mathop{\mathrm{sign}}\limits ( \mathop{\mathrm{tr}}\limits S(x,y))+ \mathop{\mathrm{sign}}\limits (\det S(x,y)) = 2 $ (blue). Blue and red lines are lines where one of the requirements of a contraction metric is violated. Hence, there are collocation points, where the constructed metric is not a contraction metric, since it was computed using a different system, namely with $ \epsilon = 0 $

    Figure 4.  The collocation points used for the approximation together with the boundary of the area where $ F(S) $ is not negative definite (green). Note that $ S $ is positive definite in the whole area displayed. Hence, the constructed metric is a contraction metric inside the cube bounded by the green areas

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