Bilinear systems emerge in a wide variety of fields as natural models fordynamical systems ranging from robotics to quantum dots. Analyzingcontrollability of such systems is of fundamental and practical importance, forexample, for the design of optimal control laws, stabilization of unstablesystems, and minimal realization of input-output relations. Tools from Lietheory have been adopted to establish controllability conditions for bilinearsystems, and the most notable development was the Lie algebra rank condition(LARC). However, the application of the LARC may be computationally expensivefor high-dimensional systems. In this paper, we present an alternative andeffective algebraic approach to investigate controllability of bilinearsystems. The central idea is to map Lie bracket operations of the vector fieldsgoverning the system dynamics to permutation multiplications on a symmetricgroup, so that controllability and controllable submanifolds can becharacterized by permutation cycles. The method is further applicable tocharacterize controllability of systems defined on undirected graphs, such asmulti-agent systems with controlled couplings between agents and Markov chainswith tunable transition rates between states, which in turn reveals a graphrepresentation of controllability through the graph connectivity.
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