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On controllability of linear stochastic systems. (English) Zbl 1031.93033
Linear, continuous-time, stochastic control systems are considered. The concepts of complete controllability, approximate controllability and $$S$$-controllability are introduced and discussed. It is generally assumed that the stochastic control system is partially observable and that values of controls are not constrained. Using methods taken from functional analysis and the theory of stochastic differential equations, necessary and sufficient conditions for different types of controllability are formulated and proved. It is shown that the notions of complete controllability and approximate controllability are equivalent. Moreover, they are also equivalent to the controllability of linear stochastic systems controlled by Gaussian processes. Several remarks and comments on stochastic controllability problems are also presented. It should be pointed out that similar stochastic controllability problems have been considered in the paper [J. Klamka and L. Socha, IEEE Trans. Autom. Control 22, 880-881 (1977; Zbl 0363.93048)].

##### MSC:
 93B05 Controllability 93E03 Stochastic systems in control theory (general) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 93C05 Linear systems in control theory
Zbl 0363.93048
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