The Stochastic Linear Quadratic Control Problem with Singular Estimates

We study an infinite dimensional finite horizon stochastic linear quadratic control problem in an abstract setting. We assume that the dynamics of the problem are generated by a strongly continuous semigroup, while the control operator is unbounded and the multiplicative noise operators for the state and the control are bounded. We prove an optimal feedback synthesis along with well posedness of the Riccati equation for the finite horizon case. Our results extend the ones proposed in [C. Hafizoglu, Ph.D. Thesis, University of Virginia, Charlottesville, VA, 2006.] to the case in which disturbance in the control is considered and a final time penalization term is included in the quadratic cost functional.

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