Mathematical Sciences: Stochastic Adaptive Control

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A collection of problems in stochastic adaptive control will be investigated in this project. Stochastic adaptive control is the optimal ergodic control of an unknown system and includes identification and control. For finite dimensional, continuous time, unknown linear stochastic systems the research will improve existing results for completely observed systems and result in a better conceptual understanding of partially observed systems. For unknown stochastic linear distributed parameter systems the initial work on adaptive control of systems described by analytic semigroups with boundary or point control will be expanded to stochastic hyperbolic systems. For unknown, finite dimensional nonlinear stochastic systems the results for almost self- optimality and self-optimality will be improved. The adaptive control of unknown stochastic distributed parameter systems will be initiated for families of semilinear equations with cylindrical noise. Most physical systems require the intervention of control mechanisms for their optimal performance. Typically the system is not completely known and often contains random and unknown perturbations. The control of an unknown system subject to unknown random perturbations is called stochastic adaptive control. Most controlled systems require stochastic adaptive control in order to meet system performance criteria. This project will investigate problems in stochastic adaptive control that are motivated by potential applications to physical systems. It is anticipated that significant improvements in the conceptual understanding of adaptive stochastic control processes and their applications will be obtained .