## Riccati equation of stochastic control and stochastic uniform observability in infinite dimensions.(English)Zbl 1071.93014

Barbu, Viorel (ed.) et al., 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 (ISBN 1-4020-7439-5/hbk). 421-432 (2003).
Let $$(w_1,\dots, w_m)$$ be a standard Wiener process. Consider the control problem with quadratic optimality criterion for $dx(t)= [A(t) x(t)+ B(t) u(t)]\,dt+ \sum^m_{i=1} G_i(t) x(t)\,dw_i(t),$ $$u$$ being the the control process. Under suitable stabilizability and observability conditions the corresponding RIM equation has a unique bounded, uniformly positive solution. It is shown by a counterexample that in contrast to the deterministic case uniform controllability does not imply stabilizability.
For the entire collection see [Zbl 1012.00037].

### MSC:

 93B28 Operator-theoretic methods 93B05 Controllability 93B07 Observability