Combinatorial dynamical system theory: General framework and controllability criteria

Abstract

A combinatorial analogue of the dynamical system theory is developed in a matroid-theoretic framework. The combinatorial dynamical system is described by a combinatorial analogue of the state-space equation xk + 1 =Axk + Buk; the matrices A and B are to be replaced by bimatroids (or linking systems). Related concepts such as controllability are defined and their fundamental properties are investigated. In particular, a sequence of matroids {Rk} determined by a “stationary iteration” Rk + 1 =A *Rk∨N is considered, whereA * Rk is the matroid induced fromRk by a bimatroidA, andN is a matroid.