J.S. McGough, A.W. Christianson, and R.C. Hoover (USA)
Lyapunov Functions, Grammatical Evolution, Evolutionary Computing, Symbolic Computation, Dynamical Systems.
This paper is concerned with the question of stability
in dynamical systems, speciﬁcally the issue of computing
symbolic forms of Lyapunov functions for given dynami
cal systems. Due to the non-constructive form of the the
Lyapunov constraints, we employ a type of evolutionary
algorithm to construct candidate Lyapunov functions. Evo
lutionary 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 Pro
gramming known as Grammatical Evolution (GE). GE dis
tinguishes itself from more traditional forms of genetic pro
grams 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 be
ing generated by performing a mapping using a problem
speciﬁc grammar. Traditional approaches using Genetic
Programming have been plagued by unrestrained expres
sion growth, stagnation and lack of convergence. These are
addressed by the more biologically realistic gene represen
tation and variations in the genetic operators. Illustrative
examples are presented to validate the proposed technique.