Abstract
We consider stochastic suppression and stabilization for nonlinear delay differential system. The system is assumed to satisfy local Lipschitz condition and one-side polynomial growth condition. Since the system may explode in a finite time, we stochastically perturb this system by introducing independent Brownian noises and Lévy noise feedbacks. The contributions of this paper are as follows. (a) We show that Brownian noises or Lévy noise may suppress potential explosion of the solution for some appropriate parameters. (b) Using the exponential martingale inequality with jumps, we discuss the fact that the sample Lyapunov exponent is nonpositive. (c) Considering linear Lévy processes, by the strong law of large number for local martingale, sufficient conditions for a.s. exponentially stability are investigated in Theorem 13.
Highlights
For a given nonlinear system ẋ (t) = f (x (t), t) or delay system ẋ (t) = f (x (t), x (t − δ (t)), t), (1)f satisfies the polynomial growth condition; the solutions may explode in finite time
The key of this proof in Theorem 3 is the boundedness of LV(x, y), under the conditions of Assumptions 1 and 2 and 2β > max{α, 2β}, which imply that the Brownian noise σ(r(t))|x(t)|βx(t)dW1(t) plays the important role to suppress potential explosion of the solution and guarantees the existence of the global solution
This section is devoted to considering the stable effect of noises under some appropriate conditions; we show that jump processes may stabilize the given unstable nonlinear delay system ẋ(t) = f(x(t), x(t − τ), t)
Summary
To suppress explosion and stabilize system (2), the dependent scalar Brownian noises are introduced in [1] and they disturbed system (2) into ẋ (t) = x (t) [1 + x (t)] dt + 2x (t) dW1 (t) + x2 (t) dW2 (t) ; (3). (ii) Discuss the global solution of the stochastic equation under polynomial growth condition; in particular, we compare the effect of suppression solution of different type of noises. To the best of our knowledge, under the assumption of polynomial growth condition, analysis of nonlinear stochastic system with jumps has not been fully investigated and few results have been available so far.
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