# Article

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Keywords:
Riccati equation; stochastic uniform observability; stabilizability; quadratic control; tracking problem
Summary:
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}).
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