System Control and Rough Paths
This book describes a novel mathematical development with potential for applications to engineering. Intended for mathematicians and engineers with a good mathematical background, the book describes the evolution of complex non-linear systems influenced by rough or rapidly fluctuating stimuli. It focuses on an analysis of the relationship between the control and the response of the system. An essential problem is to identify the point at which two different stimuli produce the same (or almost the same) response from the class of receivers. Here we deal with an essentially non-linear problem that requires new mathematics. The book focuses on systems responding to such rough external stimuli, and demonstrates that the natural reduction approximates the stimuli as a sequence of nilpotent elements. The main result of the book is represented by a theorem on continuity: “the response of the system depends continuously on these nilpotent elements“. An interesting mathematical object is the notion of a rough path, which is based on a combination of the notion of a p-variation of Wiener with the iterated integral expansions of paths introduced by K. T. Chen. The continuity theorem for rough paths gives a new way to construct solutions of stochastic differential equations.