Stochastic suppression and stabilization of delay differential systems

Abstract

SUMMARYIn this paper, we investigate stochastic suppression and stabilization for nonlinear delay differential systeṁ, where (t) is the variable delay and f satisfies the one-sided polynomial growth condition. Since f 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 by Brownian noises, we stochastically perturb this system into the nonlinear stochastic differential system dx(t) = f (x(t), x(t − (t)), t)dt +qx(t)dw 1 (t)+ |x(t)| x(t)dw 2 (t) by introducing two independent Brownian motions w 1 (t) and w 2 (t). This paper shows that the Brownian motion w 2 (t) may suppress the potential explosion of the solution of this stochastic system for appropriate choice of under the condition = 0. Moreover, for sufficiently large q, the Brownian motion w 1 (t) may exponentially stabilize this system.

The Almost Sure Asymptotic Stability and Boundedness of Stochastic Functional Differential Equations with Polynomial Growth Condition

“…Through Theorem 16, we claim that, for any given initial data ∈ , the solution of system (49) is almost surely asymptotically stable; that is, (This technique has been used by many researchers, e.g., [10,14].) From Lemma 5, we have ∫ Since is arbitrary, we must have that ∞ = ∞ a.s. and this completes the proof.…”

“…Stability and boundedness are two of the most important topics in the study of SDDEs or SFDEs in modern control theory. Many researchers have done a lot of works for these two topics (see, e.g., [6][7][8][9][10][11][12][13][14][15][16][17][18]). …”

New criteria for stochastic suppression and stabilization of hybrid functional differential systems

“…Details on delay‐dependent stability analysis and stabilization for stochastic time‐delay systems can be found in the work of Song et al The results of stabilization by linear form Brownian noise are extended to hybrid systems with continuous‐time state observation feedback control and discrete‐time observation feedback control . Wu and Hu investigated stochastic suppression and stabilization for nonlinear delay differential systems.…”

A Class of Stochastic Nonlinear Delay System with Jumps

“…It is well know that noises can stabilize the given unstable system or make this system even more stable if it is already stable. To stabilize the system, the authors in [1] introduced a polynomial Brownian noise to suppress potential explosion and then considered another linear Brownian noise to stabilize the suppressed equation. For more details, refer to [1][2][3].…”

“…We simulate the discrete equations for delay equation (77).Then the strong law of large numbers yields lim ∈ Ω 0 and ( − 1) ≤ ≤ , → 0, it follows from (69) and (− < 0, system (60) is a.s. exponentially stable; the proof is completed.Remark 14. Compared with results in[1], authors discussed the SDE: that polynomial Brownian noise | ( )| ( ) 2 ( ) can suppress this potential explosion and another linear Brownian noise ( ) 1 ( ) has an effect to stabilize the suppressed equation.Theorem 13 shows that under Assumptions 1, 2, and 12, choosing appropriate function , together with 2 > , ̸ = 0, Brownian noises may suppress the given deterministic equation( ) = ( ( ), ( − ), ). Linear jump process has stable effect on system (60); it has the same role as linear Brownian noise.…”