Stabilization of Systems by Delayed Noisy States

In this paper, stabilization of systems by delayed noisy states is investigated further. Compared with related literature, the time delays in the stabilizing noisy states are no longer restricted to the point and distributed types, which are extended into the general form, including the unbounded ones. To support this novelty, a series of preliminaries on related principles are firstly carried out: 1. the familiar Doob martingale inequality in the continuous version is improved; 2. the equivalence principle, which says that the exponential stability in moment of an anhysteretic stochastic system infers the same property of the corresponding hysteretic system, is extended to the cases with the newly proposed semi-Lipschitz condition and the general time delays; 3. the implication theorem, which indicates that the exponential stability in moment of arbitrary order of a delayed stochastic system implies almost sure exponential stability, is generalized to the stochastic systems with time delays of general forms only under the linear growth condition, to finally confirm the stability of the controlled systems. The possible active contribution of the existing noise in the underlying system is discovered and applied sufficiently. Technically, non-zero property is not needed for the derivations towards to the stability in low order moments, variable substitution is applied to time to map the function spaces for the functional differential equation models with unbounded time delays into those with bounded ones, and the existence of the solutions of the systems is confirmed only under the linear growth condition in general function space. Unified framework for the stability analysis for both bounded and unbounded time delays is built with the mapping for the time and the function space, the decay rates, including those for polynomial stability, for estimates of the solutions of systems with time delays of general form are formulated. A technical mean is proposed to counteract the restriction to the system parameters caused by the equivalence inference. At the end of the paper, typical strategies, including the divided state feedback with unbounded time delays, for the stabilizing noise are presented, a numerical example is proposed to illustrate the method and to show the efficiency of the results of the paper.