Stochastic suppression and stabilization of functional differential equations

Without the linear growth condition or the one-sided linear growth condition, this paper discusses whether or not stochastic noise feedback can stabilize a given unstable nonlinear functional system ẋ(t)=f(xt,t). Since ff may defy the linear growth condition or the one-sided linear growth condition, this system may explode in a finite time. To stabilize this system, this paper stochastically perturbs this system into the stochastic functional differential system dx(t)=f(xt,t)dt+qx(t)dw1(t)+σ|x(t)|βx(t)dw2(t) by two independent Brownian motions w1(t)w1(t) and w2(t)w2(t). This paper shows that the Brownian motion w2(t)w2(t) may suppress the potential explosion of the solution for appropriate ββ. Moreover, for sufficiently large qq, this stochastic functional system is exponentially stable. These results can be used to examine stochastic stabilization.

Research highlights
► Consider noise stabilization for high nonlinear functional systems.
► Introduce the polynomial noise to suppress the potential explosion.
► Then introduce the linear noise to stabilize the suppressed system.