Computation and Verification of Lyapunov Functions

Abstract

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.

Computation of a Contraction Metric for a Periodic Orbit Using Meshfree Collocation

“…In particular, one could start with a coarse set of collocation points and refine, where the conditions of a contraction metric are not fulfilled. A further improvement could include a posteriori estimates, that can be obtained by using Taylor-type estimates or by interpolating with a CPA function, similar to [10] and [9] for Lyapunov functions. To determine a positively invariant set, one could first seek to compute a Lyapunov function.…”

Kernel-Based Discretization for Solving Matrix-Valued PDEs

“…Contraction analysis can be used to study the distance between trajectories, without reference to an attractor, establishing (exponential) attraction of adjacent trajectories, see [28,24], see also [20,Section 2.10]; it can be generalised to the study of a Finsler-Lyapunov function [14].…”

Computation and verification of contraction metrics for exponentially stable equilibria

“…Computing a Lyapunov function analytically is usually not feasible for a nonlinear system, therefore a plethora of numerical methods has been developed. To name a few, a sum of squared (SOS) polynomials Lyapunov function can be parameterized by using semidefinite programming [2,6,30,31] or with different methods [23,32,33], an approximate solution to the Zubov equation [38] can be obtained using series expansion [33,38] or by using radial basis functions (RBF) [8], or linear programming can be used to parameterize a continuous and piecewise affine (CPA) Lyapunov function [11,15,21,22,28] or to verify a Lyapunov function candidate computed by other methods [5,12,18].…”