Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability

Ungureanu, Viorica Mariela

Czechoslovak Mathematical Journal, Tome 59 (2009) no. 2, pp. 317-342

Abstract (VO)

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}).

Détail

Classification : 49K45, 93E20
Keywords: Riccati equation; stochastic uniform observability; stabilizability; quadratic control; tracking problem

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