Stochastic Boundedness of State Trajectories of Stable LTI Systems in the Presence of Nonvanishing Stochastic Perturbation

This paper studies stochastic boundedness of trajectories of a nonvanishing stochastically perturbed stable LTI system. First, two definitions on stochastic boundedness of stochastic processes are presented, then the boundedness is analyzed via Lyapunov theory. In this proposed theorem, it is shown that under a condition on the Lipchitz constant of the perturbation kernel, the trajectories remain stochastically bounded in the sense of the proposed definitions and the bounds are calculated. Also, the limiting behavior of the trajectories have been studied. At the end an illustrative example is presented, which shows the effectiveness of the proposed theory.