Classical converse theorems in Lyapunov's second method
Christopher M. Kellett
Discrete & Continuous Dynamical Systems - B 2015, 20(8): 2333-2360 doi: 10.3934/dcdsb.2015.20.2333
Lyapunov's second or direct method is one of the most widely used techniques for investigating stability properties of dynamical systems. This technique makes use of an auxiliary function, called a Lyapunov function, to ascertain stability properties for a specific system without the need to generate system solutions. An important question is the converse or reversability of Lyapunov's second method; i.e., given a specific stability property does there exist an appropriate Lyapunov function? We survey some of the available answers to this question.
keywords: Lyapunov functions difference and differential inclusions. stability theory
Computation of Lyapunov functions for systems with multiple local attractors
Jóhann Björnsson Peter Giesl Sigurdur F. Hafstein Christopher M. Kellett
Discrete & Continuous Dynamical Systems - A 2015, 35(9): 4019-4039 doi: 10.3934/dcds.2015.35.4019
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 graph-theoretic 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.
keywords: numerical method. asymptotic stability multiple local attractors Lyapunov function dynamical system
Computing continuous and piecewise affine lyapunov functions for nonlinear systems
Sigurdur F. Hafstein Christopher M. Kellett Huijuan Li
Journal of Computational Dynamics 2015, 2(2): 227-246 doi: 10.3934/jcd.2015004
We present a numerical technique for the computation of a Lyapunov function for nonlinear systems with an asymptotically stable equilibrium point. The proposed approach constructs a partition of the state space, called a triangulation, and then computes values at the vertices of the triangulation using a Lyapunov function from a classical converse Lyapunov theorem due to Yoshizawa. A simple interpolation of the vertex values then yields a Continuous and Piecewise Affine (CPA) function. Verification that the obtained CPA function is a Lyapunov function is shown to be equivalent to verification of several simple inequalities. Numerical examples are presented demonstrating different aspects of the proposed method.
keywords: Lyapunov functions continuous and piecewise affine functions computational techniques stability theory ordinary differential equations.

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