Combinatorics-Based Approaches to Controllability Characterization for Bilinear Systems

TL;DR

This paper introduces combinatorial methods using permutations and graph theory to analyze the controllability of bilinear systems, offering an alternative to traditional Lie algebra techniques and enabling new applications in control theory.

Contribution

It develops novel combinatorial approaches for controllability analysis of bilinear systems, integrating symmetric group and graph theory with Lie algebra decompositions.

Findings

Combinatorial characterization of controllability for bilinear systems.

Compatibility of methods with Lie algebra decompositions.

Application to systems governed by semisimple and reductive Lie algebras.

Abstract

The control of bilinear systems has attracted considerable attention in the field of systems and control for decades, owing to their prevalence in diverse applications across science and engineering disciplines. Although much work has been conducted on analyzing controllability properties, the mostly used tool remains the Lie algebra rank condition. In this paper, we develop alternative approaches based on theory and techniques in combinatorics to study controllability of bilinear systems. The core idea of our methodology is to represent vector fields of a bilinear system by permutations or graphs, so that Lie brackets are represented by permutation multiplications or graph operations, respectively. Following these representations, we derive combinatorial characterization of controllability for bilinear systems, which consequently provides novel applications of symmetric group and graph theory to control theory. Moreover, the developed combinatorial approaches are compatible with Lie algebra decompositions, including the Cartan and non-intertwining decomposition. This compatibility enables the exploitation of representation theory for analyzing controllability, which allows us to characterize controllability properties of bilinear systems governed by semisimple and reductive Lie algebras.