1. Introduction
Stochastic differential equations (SDEs) have been universally modeled to describe a great deal of real phenomena along with stochastic factor (e.g., [
1,
2,
3,
4,
5]). Time delays are a significant and unavoidable factor for the reality. Thus, the stochastic delay differential equations (SDDEs) are well established with serial emphases on the asymptotic behavior; please refer to related Refs. [
5,
6,
7,
8,
9,
10] and the references therein. Generally, an SDDE
can be regarded as the stochastically perturbed system of deterministic delay differential equation (DDE)
, where
maps from
to
maps from
to
,
are time-varying delays,
is a Brownian motion. Then, one must face the significant scheme of stochastic robustness: when this deterministic DDE is bounded/stable, how much stochastic perturbation
can be tolerated without the lost of boundedness/stability? One is referred to as stochastic robust boundedness, and the other as stochastic robust stability.
So far, there are serial assertions on asymptotic boundedness of SDDEs (e.g., [
5,
11,
12,
13,
14]), but few assertions on stochastic robust boundedness are obtained. The pioneer work on stochastic robust boundedness was [
15], which was further developed to functional case by [
16]. Different from stochastic robust boundedness, many more assertions have been poured into the scheme of stochastic robust stability. On this topic, one can refer to Refs. [
15,
17,
18,
19,
20,
21,
22] and the references therein.
It should be noted that, in almost all of the above assertions on stochastic robustness, stochastic perturbations were based on continuous states observations. Stimulated by Ref. [
23], Feng and his coauthors [
24] firstly proposed and studied the scheme of stochastic robustness of discrete stochastic perturbations. Speaking in detail, for a bounded/stable deterministic differential equation
discrete stochastic noise
can be tolerated to keep the properties of boundedness/stability, where
maps from
to
maps from
to
,
stands for the time interval of two consecutive observations, and
stands for the biggest integer not greater than
a. However, two aspects need to be further studied for Ref. [
24], which are listed as follows:
(1) The underlying system considered by Ref. [
24] is just the deterministic differential equations. In reality, the time delays phenomenon is widespread, which is always the source of instability and other dreadful behavior. Thus, it is necessary to consider deterministic differential equations with delays, particularly with time-varying delays [
25]. This is the first goal of our paper.
(2) Stochastic perturbation in Ref. [
24] was mainly based on discrete states observations at serial times
. Nevertheless, when observing the system states to make some estimations, time delays are unavoidable (e.g., data transmission). That is to say, making the observations not only needs to consider the discrete issue but also the delay issue. Hence, stochastic perturbations dependent on delayed discrete states observations (e.g.,
) are more reasonable and practical than that in Ref. [
24], where
is a designed delay. Furthermore, studying the stochastic robustness issue of delayed discrete noises is necessary and significant, which is the second goal of our paper.
Motivated by the aforesaid aspects, this paper is devoted to studying the issue of stochastic robustness (including robust boundedness, robust stability, robust asymptotic stability, and robust exponential stability) of delayed discrete noises for deterministic delay differential equations. So far, this paper is the first study on the stochastic robustness of delayed discrete noises, which makes up for the gap in this field.
Notations:
: positive numbers.
: the set of continuous functions with the norm .
: positive time-varying delay with the boundedness and derivative .
: the trace norm of vector or matrix A.
: a complete probability space satisfying usual conditions.
: an m-dimensional Brownian motion defined on .
: the set of functions with the continuity, twice differentiability about u, and once differentiability about t.
2. Problem Statement
An
n-dimensional DDE
is concerned with initial value
, where
maps from
to
with the local Lipschitz condition (LLC) and
,
. The following condition is assumed for
to ensure the boundedness and stability of DDE (
1).
Assumption 1. There are constants , , , satisfying and Remark 1. Under Assumption A1 and some other conditions, DDE (1) is bounded and stable. For example, choosing Lyapunov function , we can prove the stability and asymptotic stability of DDE (1) similarly to the proof of Theorem 1 with ; Choosing Lyapunov function , we can prove the exponential stability of DDE (1) similarly to the proof of Theorem 2 with and the asymptotic boundedness similarly to the proof of Theorem 3 with . The detailed proofs are left to readers. As mentioned above, we will study the stochastic robustness of delayed discrete noise
satisfying the LLC and following Assumption A2 for DDE (
1) in this section, where
stands for the time interval of two consecutive observations,
stands for the time interval between the system arrival instant and the observed instant.
Assumption 2. There are constants , satisfying, and In other words, the goal is to give the criteria on
, and
such that stochastically perturbed system
retains the properties of boundedness or stability.
Remark 2. Under Assumption 2, is the equilibrium solution of system (4). In addition, system (4) can be rewritten aswhere . Hence, essentially speaking, three types of delays exist for system (4): time-varying delay , with the differentiability, is self-contained for (4); time-varying delay , without the differentiability, is natural interval between current instant t and observation instant ; constant delay is intentionally designed. For the three types of delays, the property analyses are complex. It is noted that
from Remark 2. Hence, system (
4) is essentially an SDDE with bounded delays, and the global solution
can be ensured for system (
4) with
(
) by Theorem 3.2 of Ref. [
5], Assumptions 1 and 2.
For SDDE (
4), the initial values
are required. When
, it is okay. Otherwise, we need to give the values on
. Hence, it is assumed that
and
.
To further deduce the main assertions, for SDDE (
4), the Itô’s operator [
5] of
is cited as
where
.
Lemma 1 (Barbalat Lemma). Assume that is uniformly continuous on with , then .
3. Robust Stability
In this part, we will study the issue of stochastic robust stability of delayed discrete noises, which is one of the main goals.
Theorem 1. Under Assumptions 1 and 2 with , if there exist constants and satisfyingandthen, solution of SDDE (4) is moment stable and moment asymptotic stable, i.e.,where with represent the two solutions of Please see the detailed proof of Theorem 1 in
Appendix A.
Remark 3. Essentially, condition (7) is implied by condition (9). Condition (7) here is especially given as a simplified bound for parameter . Theorem 1 has studied the
stability and asymptotic stability of SDDE (
4), but it does not consider the convergent rate. In the following, it will be studied.
Theorem 2. Under the conditions of Theorem 1, the solution of SDDE (4) is moment exponentially stable and almost surely exponentially stable, i.e.,where , and represents the positive solution of Please see the detailed proof of Theorem 2 in
Appendix B.
Remark 4. It is noted that is monotonically increasing. Hence, from the definition of and , we have for . Furthermore, from the definition of , we have for .
Furthermore, the following assertion, as the applications of obtained assertions (i.e., Theorems 1 and 2), can be proposed.
Corollary 1. Under conditions (7)–(9), stable/asymptotically stable/ exponentially stable DDE (1) with Assumption A1 and can tolerate the delayed discrete noise with Assumption A2 and . 4. Robust Boundedness
In the above section, under Assumptions 1 and 2 with , the scheme of robust stability of delayed discrete noises, in means of stability, asymptotic stability and exponential stability, has been studied. Then, without the constraint of , what assertions can be obtained? That is the goal of this part, i.e., the scheme of robust boundedness of delayed discrete noises.
Theorem 3. Under Assumptions 1 and 2, if there exist constants , and satisfyingandthen, solution of SDDE (4) is asymptotically bounded, i.e.,where the expression of is given in the proof, , , represents the positive solution ofand with represent the two solutions of Please see the detailed proof of Theorem 3 in
Appendix C.
Remark 5. Similarly to Remark 3, condition (16) is implied by condition (18) and given as a simplified bound for parameter . Remark 6. Similarly to Remark 4, from the monotonically increasing property of , the definitions of , and , we have for and even for .
Similarly to Corollary 1, the assertion on the robust stability can be yielded.
Corollary 2. Under conditions (16)–(18), asymptotically bounded DDE (1) with Assumption 1 can tolerate the delayed discrete noise with Assumption 2. Remark 7. Compared with the obtained assertions on stochastic robustness in Refs. [15,16,17,18,19,20,21,22], the novelty and significance of our theory are as follows. (1) Stochastic perturbations here are based on delayed discrete states observations rather than continuous states observations in Refs. [15,16,17,18,19,20,21,22]; (2) It is noted that the methods in Refs. [15,16,17,18,19,20,21,22] can not be applied to deal with discrete term . Hence, the Lyapunov functionals of (A2), of (A9) and of (A13) are especially designed to deal with the discrete term.