Combinatorial Algebra in Controllability and Optimal Control

This chapter presents fundamental concepts in control theory that are inherently linked to combinatorial and algebraic structures. It demonstrates how modern combinatorial algebraic tools provide both deeper insight and facilitate analyses, computations and designs. Control theory models, analyzes and designs purposeful interactions with dynamical systems, with the objective of making them behave in desired way. A decisive role is played by the lack of commutativity: in general, control actions taken in different orders result in different outcomes. Control theory is a broad discipline that studies a broad range of dynamical systems, ranging from discrete systems to systems governed by stochastic partial differential equations. The chapter reviews the expansion in a form that immediately gives a coordinate representation of a universal free nilpotent system that covers every nilpotent control system. The exponential product expansion is useful for the theoretical analysis of controllability and optimality properties, and for proving that certain feedback schemes indeed stabilize a system.