Combinatorial Algebra in Controllability and Optimal Control

Control theory models, analyzes, and designs purposeful interactions with dynamical systems so that these behave in a desired way. Fundamental to the ability to synthesize complex responses by strategically combining simple control building blocks is a desired lack of commutativity of the flows associated to control actions. In its most simple terms this means that the dynamics are not constrained to evolve on lower dimensional integral manifolds. Mathematically this lack of integrability is manifested in terms of (nonabelian) Lie algebras. In recent years it has been rec-ognized that Zinbiel algebras and combinatorial Hopf algebras play an important unifying role for a deeper understanding of the underlying structures. This book chapter presents fundamental concepts in control theory that are inher-ently linked to combinatorial and algebraic structures. It demonstrates how modern combinatorial algebraic tools provide both deeper insight and facilitate analysis, computations, and design. The emphasis is on exhibiting the algebraic structure