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

Ungureanu, Viorica Mariela

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].

Reviewer: H. Hering (Göttingen)

MSC:

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

Keywords:

stochastic quadratic control; Riccati equation; stochastic uniform observability; stabilizability; detectability; uniform controllability