Time-Varying Halanay Inequalities With Application to Stability and Control of Delayed Stochastic Systems

In this article, the Halanay inequality is generalized into the time-varying version under simple and natural conditions, without constant to bound the time-varying coefficients. The solution of the underlying inequality is estimated by virtue of the feasible function pair associated with the coefficients and delays. Corollaries are given for typical special cases. Some approaches are illustrated for the searching of the initial feasible pairs, and an iterative procedure is proposed for the optimization of the feasible pairs. The usage and superiority of the obtained result are exhibited by numerical examples with simulations. As a base, the comparison principle for the functional differential inequalities and equations with arbitrary time-varying delays is studied first. As an application of the main result, the stability and control of the stochastic systems with time-varying coefficients and delays are investigated, with the main concern to the time variance. An asymptotic stability theorem and a criterion are established, and an interesting control strategy with unbounded time delay, namely the pantograph delay sampled data based feedback, is proposed. The stability criterion and the control strategy are illustrated and verified with numerical experiments.