On concepts of controllability for deterministic and stochastic systems

Abstract

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.