Construction of a contraction metric by meshless collocation

Abstract

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.

Review on contraction analysis and computation of contraction metrics

“…The computation of a contraction metric for an equilibrium using meshfree collocation was studied in Giesl & Wendland 2019 [74]. The construction was achieved by approximating the contraction metric M satisfying (59).…”

Computation and verification of contraction metrics for exponentially stable equilibria

“…2.9 Theorem (Perturbation effect on contraction metrics) [14,Theorem 2.4] Let f ∈ C s (R n ; R n ), s ≥ 2. Let x 0 be an exponentially stable equilibrium of ẋ = f (x) with basin of attraction A(x 0 ).…”

“…Then, we calculated the CPA verification over the rectangle [−2.5, 2.5] × [−5.5, 5.5] with 2200 2 vertices, see Figure 1. This example was already used in [13] and [14] to illustrate the RBF approximation of the contraction metric and one can compare this result with them. Here we are able to rigorously verify the conditions of a contraction metric for the CPA interpolation, while in previous work it has been checked for the optimal recovery at finitely many points.…”

“…Much less literature is available on numerical methods to compute contraction metrics, but the methods are often similar to the ones used to compute Lyapunov functions, e.g. in [3] SOS polynomials are parameterized, in [13,14] RBF is used, and in [10] a CPA contraction metric is parameterized using semidefinite programming. While the RBF method is computationally more efficient than the CPA method, the RBF method does not guarantee that the computed metric is indeed a contraction metric.…”

“…x ) δ iµ δ jν . (B.1) for 1 ≤ k, ≤ N , and 1 ≤ i, j, µ, ν ≤ n (see[14, Subsection 3.2] for more details). Calculate the coefficients c k, ,i,j,µ,ν defined asc k, ,i,i,µ,µ = b k, ,i,i,µ,µ c k, ,i,i,µ,ν = 1 2 (b k, ,i,i,µ,ν + b k, ,i,i,ν,µ ) c k, ,i,j,µ,µ = b k, ,i,j,µ,µ = 1 2 (b k, ,i,j,µ,µ + b k, ,j,i,µ,µ ) c k, ,i,j,µ,ν = 1 2 (b k, ,i,j,µ,ν + b k, ,i,j,ν,µ ) = 1 4 (b k, ,i,j,µ,ν + b k, ,j,i,ν,µ + b k, ,i,j,ν,µ + b k, ,j,i,µ,ν ) (B.2)…”