The current issue of DCDS-B contains a special section on computational methods for Lyapunov functions. Quoting from the editorial:
Lyapunov functions, introduced by Lyapunov more than 100 years ago, are to this day one of the most important tools in the stability analysis of dynamical systems. They are functions which decrease along solution trajectories of systems, and they can be used to show stability of an invariant set, such as an equilibrium, as well as to determine its basin of attraction. Lyapunov functions have been considered for a variety of dynamical systems, such as continuous-times, discrete-time, linear, non-linear, non-smooth, switched, etc. Lyapunov functions are used and studied in different communities, such as Mathematics, Informatics and Engineering, often using different notations and methods.
Since Lyapunov functions provide insight into the dynamics, it is an important question how to find them. A first answer is given by so-called converse theorems, which ensure the existence of a certain type of Lyapunov function given a certain type of stability. These theorems, however, are not constructive in nature as they usually use the solution trajectories to construct the Lyapunov function. This means, that they cannot be used directly to find an explicit Lyapunov function for most concrete examples. Therefore, computational methods have been derived to construct Lyapunov functions, using as diverse methods as optimization, Linear Matrix Inequalities, numerical solutions to Partial Differential Equations using collocation or other methods, graph theoretic methods, algebraic methods, and others.
This special issue brings together all these aspects: starting with two surveys on computational methods and converse theorems for Lyapunov functions and then continuing with six research articles which cover a wide range of current research for the construction of Lyapunov functions. The contributions also bring experts from different disciplines together in one volume
The publisher told me that this newly published issue’s full texts remain accessible to the public for a couple of weeks. Among other contributions, this special section contains some very nice review articles that could serve as a good starting point for anyone interested in the subject. More can be found here.
Update 04/10/2015 Free full text access is no longer available, unfortunately.