Symbolic Computation of Lyapunov Functions using Evolutionary Algorithms

This paper is concerned with the question of stability in dynamical systems, specifically the issue of computing symbolic forms of Lyapunov functions for given dynamical systems. Due to the non-constructive form of the the Lyapunov constraints, we employ a type of evolutionary algorithm to construct candidate Lyapunov functions. Evolutionary Algorithms have demonstrated results in a vast array of optimization problems and are regularly employed in engineering design. We study the application of a variant of Genetic Programming known as Grammatical Evolution (GE). GE distinguishes itself from more traditional forms of genetic programs in that it separates the internal representation of a potential solution from the actual target expression. Strings of integers are evolved, with the candidate expressions being generated by performing a mapping using a problemspecific grammar. Traditional approaches using Genetic Programming have been plagued by unrestrained expression growth, stagnation and lack of convergence. These are addressed by the more biologically realistic gene representation and variations in the genetic operators. Illustrative examples are presented to validate the proposed technique.