Various sufficient conditions for approximate controllability of linear evolution systems in abstract spaces have been obtained, but approximate controllability of semilinear control systems usually requires some complicated and limited assumptions. In this paper, we show the approximate controllability of the abstract semilinear deterministic and stochastic control systems under the natural assumption that the associated linear control system is approximately controllable. The results are obtained using new properties of symmetric operators (which are proved in this paper), compact semigroups, the Schauder fixed point theorem, and/or the contraction mapping principle.

We discuss several concepts of controllability for partially observable stochastic systems: complete controllability, approximate controllability, controllability and S-controllability, and show that complete and approximate controllability notions are equivalent, and in turn are equivalent to the controllability for linear stochastic systems controlled by gaussian processes. We derive necessary and sufficient conditions for these concepts of controllability. These criteria reduce to the well-known rank condition.

The new necessary and sufficient conditions, which are formulated in terms of convergence of a certain sequence of operators involving the resolvent of the negative of the controllability operator, are found for deterministic linear stationary control systems to be completely and approximately controllable, respectively. These conditions are applied to study the S-controllability (a property of attaining an arbitrarily small neighborhood of each point in the state space with a probability arbitrarily close to one) and C--controllability (the S--controllability fortified with some uniformity) of stochastic systems. It is shown that the S--controllability (the C--controllability) of a partially observable linear stationary control system with an additive Gaussian white noise disturbance on all the intervals [0,T] for T>0 is equivalent to the approximate (complete) controllability of its deterministic part on all the intervals [0,T] for T>0.