Computation and verification of Lyapunov functions

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

Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming

We study the global asymptotic stability in probability of the zero solution of linear stochastic differential equations with constant coefficients. We develop a sum-of-squares program that verifies whether a parameterized candidate Lyapunov function is in fact a global Lyapunov function for such a system. Our class of candidate Lyapunov functions are naturally adapted to the problem. We consider functions of the form $V(\mathbf{x}) = ||\mathbf{x}||^p_Q = (\mathbf{x}^\top Q \mathbf{x})^{\frac{p}{2}}$, where the parameters are the positive definite matrix $Q$ and the number p > 0. We give several examples of our proposed method and show how it improves previous results

Computation of Lyapunov functions for systems with multiple attractors

We present a novel method to compute Lyapunov functions for continuous-time systems with multiple local attractors. In the proposed method one first computes an outer approximation of the local attractors using a graphtheoretic approach. Then a candidate Lyapunov function is computed using a Massera-like construction adapted to multiple local attractors. In the final step this candidate Lyapunov function is interpolated over the simplices of a simplicial complex and, by checking certain inequalities at the vertices of the complex, we can identify the region in which the Lyapunov function is decreasing along system trajectories. The resulting Lyapunov function gives information on the qualitative behavior of the dynamics, including lower bounds on the basins of attraction of the individual local attractors. We develop the theory in detail and present numerical examples demonstrating the applicability of our method

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

Lyapunov functions are a main tool to determine the domain of attraction of equilibria in dynamical systems. Recently, several methods have been presented to construct a Lyapunov function for a given system. In this paper, we improve the construction method for Lyapunov functions using Radial Basis Functions. We combine this method with a new grid refinement algorithm based on Voronoi diagrams. Starting with a coarse grid and applying the refinement algorithm, we thus manage to reduce the number of data points needed to construct Lyapunov functions. Finally, we give numerical examples to illustrate our algorithms

Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

Heritage, endangerment and participation: alternative futures in the Lake District

Cultural heritage policy in the UK puts a high value on participation, and heritage agencies often encourage that participation through appealing to the endangered status of the landscapes, sites and monuments in their care. Participation takes many forms, and can involve influencing policy, contributing to cultural outputs and enjoying cultural activities. This paper critically examines the literature and discourse underpinning the endangerment/participation axis and presents a case study of heritage participation in the English Lake District. In order to ground critique in empirical investigation, the case study focusses on the practice of a particular fell shepherd, whose participation in heritage is not motivated by endangerment. The paper then explores the implications of this research for wider thinking about heritage and public life, arguing for the importance of moving beyond endangerment narratives for the creation of resilient heritage futures

Cultural Contestation in China: Ethnicity, Identity and the State

This chapter explores how the political-administrative design of the Chinese state, characterized as “multi-level governance”, might be the cause of more subtle forms of resistance. By looking at the formulation of heritage policies of Lancang County, Christina Maags illustrates how the administrative fragmentation resulted in both administrative contestation and cultural contestation, with a threatened local identity at its core

A constructive converse Lyapunov theorem on exponential stability

An ordinary differential equation’s (ODE) equilibrium is asymptotically stable, if and only if the ODE possesses a Lyapunov function, that is, an energy-like function decreasing along any trajectory of the ODE and with exactly one local minimum. Theorems regarding the ‘only if ’ part are called converse theorems. Recently, the author presented a linear programming problem, of which every feasible solution parameterizes a Lyapunov function for the nonlinear autonomous ODE in question. In 2004 the author proved the first general constructive converse theorem by showing that if the equilibrium of the ODE is exponentially stable, then the linear programming problem possesses a feasible solution. In this paper we prove a constructive converse theorem on asymptotic stability for nonlinear autonomous ODEs and so improve the 2004 results. The only restriction on the ODE _x fðxÞ is that f is a class C 2 function. Note, that these results imply that the algorithm presented by the author in 2002 is capable of constructing a Lyapunov function for all nonlinear systems, of which the equilibrium is asymptotically stable. 1

Existence of piecewise linear Lyapunov functions in arbitrary dimensions

Lyapunov functions are an important tool to determine the basin of attraction of exponentially stable equilibria in dynamical systems. In Marinósson (2002), a method to construct Lyapunov functions was presented, using finite differences on finite elements and thus transforming the construction problem into a linear programming problem. In Hafstein (2004), it was shown that this method always succeeds in constructing a Lyapunov function, except for a small, given neighbourhood of the equilibrium. For two-dimensional systems, this local problem was overcome by choosing a fan-like triangulation around the equilibrium. In Giesl/Hafstein (2010) the existence of a piecewise linear Lyapunov function was shown, and in Giesl/Hafstein (2012) it was shown that the above method with a fan-like triangulation always succeeds in constructing a Lyapunov function, without any local exception. However, the previous papers only considered two-dimensional systems. This paper generalises the existence of piecewise linear Lyapunov functions to arbitrary dimensions