Stochastic uniform observability in infinite dimensions
by
Viorica Mariela Ungureanu
Constantin Brancusi University

The purpose of this paper is to discuss the stochastic uniform observability property of a class of general linear stochastic differential equations (LSDEs for short) with multiplicative white noise defined on infinite dimensional Hilbert spaces. It is known that under stochastic uniform observability and stabilizability conditions there exist some global bounded and stabilizing solutions of the differential Riccati equations arising in stochastic control. Similar results were obtained by G. Da Prato and A Ichikawa under detectability and stabilizability conditions. Unlike the deterministic case the concept of stochastic uniform observability considered in this paper doesn't imply detectability and the results obtained under stochastic uniform observability conditions are different to those obtained under detectability hypothesis. However we can ask the question: Why to choose stochastic uniform observability instead of detectability? This talk will show that the answer is complex. Using perturbation theory for evolution operators on spaces of Hilbert Schmidt or nuclear operators we give deterministic characterizations of the stochastic uniform observability property by using Lyapunov equations. These results show that generally stochastic uniform observability property is more easy to verify than detectability. On the other hand, we proved that there exists a large class of LSDEs which are never stochastic uniformly observable. In fact we proved that if the operator in the drift of LSDE generates a compact evolution operator and if the observation operator is strongly continuous, then the LSDE is never stochastically uniformly observable. In this case detectability remains the only alternative.

Date received: June 15, 2008