Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability.(English)Zbl 1224.93135

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 S. Chen and X. Y. Zhou [SIAM J. Control Optimization 39, No. 4, 1065–1081 (2000; Zbl 1023.93072)]), for finite dimensional stochastic equations or V. M. Ungureanu [Stud. Univ. Babeş-Bolyai, Math. 50, No. 4, 73–81 (2005; Zbl 1113.60061)], for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see G. Da Prato and A. Ichikawa [SIAM J. Control Optimization 28, No. 2, 359–381 (1990; Zbl 0692.49006)] and V. M. Ungureanu [Analysis and optimization of differential systems. IFIP TC7/WG 7.2 international working conference, Constanta, Romania, September 10–14, 2002. Boston, MA: Kluwer Academic Publishers. 421–432 (2003; Zbl 1071.93014)]). 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 $$\mathbb {R}_{+}$$ and stabilizing solution. Using this result we find an optimal control and the optimal cost. It is known (see V. M. Ungureanu [Analysis and optimization of differential systems. IFIP TC7/WG 7.2 international working conference, Constanta, Romania, September 10–14, 2002. Boston, MA: Kluwer Academic Publishers. 421–432 (2003; Zbl 1071.93014)]) that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see G. Da Prato and A. Ichikawa [SIAM J. Control Optimization 28, No. 2, 359–381 (1990; Zbl 0692.49006)]).

MSC:

 93E20 Optimal stochastic control 49K45 Optimality conditions for problems involving randomness 93B07 Observability 49N10 Linear-quadratic optimal control problems
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References:

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