Linear stochastic control systems defined in infinite-dimensional Hilbert spaces are considered. Three types of stochastic controllability are studied: approximate, complete, and -controllability. The relationships between different types of controllability are explained using methods of stochastic differential equations and the theory of stochastic processes. Moreover, several conditions for these types of controllability are formulated and proved. As an application of the theoretical results, the controllability of the wave equation with a distributed control and disturbances described by a Wiener process is studied. The relationships to deterministic controllability are also discussed.
Finally, it should be stressed that similar stochastic controllability problems have been considered in the papers [J. Klamka and L. Socha, IEEE Trans. Autom. Control. 22, 880-881 (1997; Zbl 0363.93048)] and [N. I. Mahmudov and A. Denker, Int. J. Control 73, 144-151 (2000; Zbl 1031.93033)].