Viorica Mariela Ungureanu, Department of Mathematics, "Constantin Brancusi" University, Tg. Jiu, B-dul Republicii, nr. 1, jud. Gorj, Romania, e-mail: vio@utgjiu.ro
Abstract: In this paper we study the existence of the optimal (minimizing) control for a tracking problem, as well as a quadratic cost problem subject to linear stochastic evolution equations with unbounded coefficients in the drift. The backward differential Riccati equation (BDRE) associated with these problems (see \cite{chen}, for finite dimensional stochastic equations or \cite{UC}, for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see \cite{1990}, \cite{ukl}). Under stabilizability and uniform observability conditions and assuming that the control weight-costs are uniformly positive, we establish that BDRE has a unique, uniformly positive, bounded on ${\mathbf{R}}_+$ and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known \cite{ukl} that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see \cite{1990}).
Keywords: Riccati equation, stochastic uniform observability, stabilizability, quadratic control, tracking problem
Classification (MSC 2000): 93E20, 49K45
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