Subdifferentials and Their Applications to Control Theory

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9704203 Zhu The main goals of this project are to investigate several important areas of mathematical control theory in which nonsmoothness is intrinsic to the problem and the usual assumptions of Lipschitz continuity or continuity must be replaced by assumptions of lower semicontinuity even if the original problem is defined by continuous or even smooth data. As an important tool for such investigations, it is also proposed to further develop a basic theory and a calculus of subdifferentials. The control problems that are intended to be investigated by these methods include: (1) Controllability properties of unbounded control and differential inclusion systems; (2) Bilevel optimization problems; (3) Infinite horizon problems with both initial-value and natural system perturbations; (4) Necessary optimality conditions for distributed control problems starting with those defined by quasilinear elliptic partial differential equations; (5) Value function and sensitivity analysis for delayed control problems; and (6) Dynamical programming approach to control problems with discrete stochastic variables. The control problems proposed above naturally arise in resource management, flexible manufacturing system scheduling, transportation scheduling and other industry and management problems. These control problems have been studied with simplified models in which the system performance is assumed to be free of abrupt changes. However, in real life problems such abrupt changes are not only unavoidable but rather often among the most important features of the problem. The discontinuous aspects of these problems will be studied with recent methods of the subdifferential theory. New methods will also be developed as needed. Research results in this project are likely to shed light on other applications that involve similar discontinuous behaviors.