Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability
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 , for finite dimensional stochastic equations or , for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see , ). 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 ℝ + and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known  that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see ).
KeywordsRiccati equation stochastic uniform observability stabilizability quadratic control tracking problem
MSC 200093E20 49K45
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