Stochastic suppression and stabilisation of non‐linear hybrid delay systems with general one‐sided polynomial growth condition and decay rate

Abstract

This study investigates the stochastic suppression and stabilisation of non-linear hybrid delay systems with general one-sided polynomial growth condition and decay rate. Given an unstable non-linear hybrid delay system x ˙ ( t ) = f ( x ( t ) , x ( t − τ ( t ) ) , r ( t ) , t ) with general one-sided polynomial growth condition, the authors introduce two independent Brownian noise and perturb this system into stochastic hybrid delay system d x ( t ) = f ( x ( t ) , x ( t − τ ( t ) ) , r ( t ) , t ) d t + h ( x ( t ) , r ( t ) , t ) d B 1 ( t ) + g ( x ( t ) , r ( t ) , t ) d B 2 ( t ) . It is shown that the non-linear diffusion term g ( x ( t ) , r ( t ) , t ) may suppress the potential explosion of hybrid delay system. Under a stronger condition, another linear diffusion term h ( x ( t ) , r ( t ) , t ) will make the perturbed stochastic hybrid delay system is almost surely stable with general decay rate.