Mathematical Sciences: Nonsmooth Analysis and Control Theory

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ABSTRACT Proposal: DMS-9623406 PI: Wolenski The main emphasis of this proposal is aimed at studying a so-called differential inclusion by using ideas and recent results from nonsmooth analysis. A differential inclusion is a generalization of an ordinary differential equation, and can be viewed as a multidirectional flow. The practical uses of the formulation are motivated by issues in control theory, where it has been known for some time that a control system can be reformulated as a differential inclusion without losing information regarding its trajectories. But as with any reformulated model, one must take care in analysing the precise transfer of assumptions from one to the other. Part of the research in this proposal is aimed at bridging this gap between what is known in differential inclusions and control systems under different sets of assumptions. It is the contention of the Wolenski that many issues in control theory will not be properly understood until the fundamental traits implying the desired behavior are brought out in the associated differential inclusion. Specific topics Wolenski plans to address include a Hamilton-Jacobi type analysis in the minimal time problem, a detailed analysis of the boundary to a reachable set, an extension of the Hopf representation formula, a survey of the uses of semiconvexity in analysis and optimization, and a broadening of the theory of hereditary control systems. Applications of control theory are prevalent in applied science, and its influence on our society will undoubtedly continue to expand. From manufacturing to medicine to engineering to economics, control systems are increasingly being introduced as models for describing complicated system behavior. The many recent mathematical advances in control theory are finding their way into practice, and this exposes the need for further mathematical understanding of control phenomena. Nonsmooth analysis is a mathematical discipline that has developed rapidly in the last twen ty years, mainly motivated by its many applications to optimization and optimal control theory. It now contains a surprisingly complete body of results that both in method and substance extend most of the classical differential calculus. The present proposal is directed toward applying some of the more recent advances to topics in control theory. Since the influence of control theory in applied science can hardly be overestimated, and as more of the recent mathematical advances are finding applications throughout society, a need has arisen for a greater understanding of the properties of control systems. The successful completion of this research will lead to a greater mathematical understanding of the evolution of control systems, and could have significant consequences in the modelling process by allowing the designer to focus on crucial elements of the design that would lead to an improved performance.