Stochastic suppression and almost surely stabilization of non-autonomous hybrid system with a new general one-sided polynomial growth condition

This paper investigates the stochastic suppression and almost surely stabilization of non-autonomous hybrid system with a new general one-sided polynomial growth condition. Given an unstable non-autonomous hybrid system dy(t)=f(y(t),r(t),t)dt with a new general one-sided polynomial growth condition, we introduce two independent Brownian noise and perturb this system into stochastic hybrid system dx(t)=f(x(t),r(t),t)dt+q(r(t))δ(t)x(t)dB1(t)+σ(r(t))δ(t)|x(t)|βx(t)dB2(t). It shows that the polynomial form of Brownian noise may suppress the potential explosion of hybrid system. Under a stronger condition, another linear Brownian noise will make the perturbed stochastic hybrid system is almost surely stable with general decay rate.