Grid refinement in the construction of Lyapunov functions using radial basis functions

2023

JCD

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<p style='text-indent:20px;'>Lyapunov functions are functions with negative derivative along solutions of a given ordinary differential equation. Moreover, sublevel sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. One of the numerical construction methods for Lyapunov functions uses meshless collocation with radial basis functions.</p><p style='text-indent:20px;'>Recently, this method was combined with a grid refinement algorithm (GRA) to reduce the number of collocation points needed to construct Lyapunov functions. However, depending on the choice of the initial set of collocation point, the algorithm can terminate, failing to compute a Lyapunov function. In this paper, we propose a modified grid refinement algorithm (MGRA), which overcomes these shortcomings by adding appropriate collocation points using a clustering algorithm. The modified algorithm is applied to two- and three-dimensional examples.</p>

“…To estimate the second derivative of the orbital derivative in (21) we use the following Theorem 10 from [9], which makes use of the special form of the approximant v.…”

Section: (A) 1b: 1st Extension Processmentioning

confidence: 99%

mentioning

confidence: 99%

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Modified refinement algorithm to construct Lyapunov functions using meshless collocation

Mohammed

2023

JCD

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“…The advantages of using RBF include that the method is computationally efficient and that scattered collocation points can be used. For example, one can employ refinement strategies, using more collocation points in areas where the approximation S is not sufficiently good; these areas can easily be found by checking for which points x the function F (S)(x) is not negative definite or S(x) is not positive definite, see [29] for a similar refinement algorithm for the computation of Lyapunov functions using RBF.…”

Section: Introductionmentioning

confidence: 99%

Computation and verification of contraction metrics for exponentially stable equilibria

Journal of Computational and Applied Mathematics

Hafstein

Mehrabinezhad

2021

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“…However, as the third run in Example 2 suggests, this might be the case and we are currently examining this. For further studies on refinement for the RBF method, see also [27].…”

Section: ⊃Cmentioning

confidence: 99%

Computation and Verification of Lyapunov Functions

SIAM J. Appl. Dyn. Syst.

Hafstein²

2015

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Abstract. Lyapunov functions are an important tool to determine the basin of attraction of equilibria in Dynamical Systems through their sublevel sets. Recently, several numerical construction methods for Lyapunov functions have been proposed, among them the RBF (Radial Basis Function) and CPA (Continuous Piecewise Affine) methods. While the first method lacks a verification that the constructed function is a valid Lyapunov function, the second method is rigorous, but computationally much more demanding. In this paper, we propose a combination of these two methods, using their respective strengths: we use the RBF method to compute a potential Lyapunov function. Then we interpolate this function by a CPA function. Checking a finite number of inequalities, we are able to verify that this interpolation is a Lyapunov function. Moreover, sublevel sets are arbitrarily close to the basin of attraction. We show that this combined method always succeeds in computing and verifying a Lyapunov function, as well as in determining arbitrary compact subsets of the basin of attraction. The method is applied to two examples.Key words. Lyapunov function, basin of attraction, mesh-free collocation, radial basis function, continuous piecewise affine interpolation, computation, verification AMS subject classifications. 37B25, 65N35, 93D30, 65N15, 37M99, 34D20DOI. 10.1137/1409888021. Introduction. Given an autonomous ordinary differential equation (ODE), we are interested in the determination of the basin of attraction of an exponentially stable equilibrium.The basin of attraction can be computed using a variety of methods: Invariant manifolds form the boundaries of basins of attraction, and their computation can thus be used to find a basin of attraction [24, 23]. The cell mapping approach [20] or set oriented methods [4] divide the phase space into cells and compute the dynamics between these cells; they have also been used to construct Lyapunov functions [16,22,14].Lyapunov functions [26] are a natural way of analyzing the basin of attraction. A Lyapunov function is a function which is decreasing along solutions of the ODE; sublevel sets of the Lyapunov function are subsets of the basin of attraction. The explicit construction of Lyapunov functions for a given system, however, is a difficult problem.In the last decades, several numerical methods to construct Lyapunov functions have been developed; for a review, see [12]. These methods include the SOS (sums of squares) method, which is usually applied to polynomial vector fields, introduced in [29] and available as a MATLAB tool box [28]. It can also be applied to more general systems as shown in [30,31] and it constructs a polynomial Lyapunov function by semidefinite optimization.