Exponential Stability Results on Random and Fixed Time Impulsive Differential Systems with Infinite Delay

In this paper, we investigated the stability criteria like an exponential and weakly exponential stable for random impulsive infinite delay differential systems (RIIDDS). Furthermore, we proved some extended exponential and weakly exponential stability results for RIIDDS by using the Lyapunov function and Razumikhin technique. Unlike other studies, we show that the stability behavior of the random time impulses is faster than the fixed time impulses. Finally, two examples were studied for comparative results of fixed and random time impulses it shows by simulation.


Introduction
Impulses occur in a short duration of time which makes a sudden change in the nature or behavior of the differential system; we call this system an impulsive differential system. Most of the impulsive differential equation models deal with the fixed time of occurrence of impulse action. Many authors contributed to analyzing the fixed time impulsive differential systems (IDS) with the finite or infinite delay because this system arises in many fields like science, engineering, biotechnology, neural networks, and control systems-see the monographs [1,2]. The study of qualitative behavior like the stability of impulsive differential systems is also important. Generally, stability behavior for IDS with delays can have two types of results: (i) impulsive perturbation and (ii) impulsive stabilization. For the past several decades, many authors have studied the stability behavior of various types of impulsive systems by using the Lyapunov functions and Razumikhin technique. Moreover, the Lyapunov functional method plays an important role in the stability theory of functional differential systems it used to obtain the minimal class of functional from the corresponding derivative of the Lyapunov functions; for example, in [3,4], the authors proved the exponential stability by using the Lyapunov and Razumikhin technique and the authors in [5][6][7] investigated the Razumikhin-type theorems for weakly exponentially stable and exponentially stable. Recently, the authors in [8] established some new Razumikhin-technique for studying the uniform stability behavior of the systems. However, impulses used to control for the unstable differential systems can be stabilized to the equilibrium point; this is shown in [9,10]. Furthermore, several interesting results have been established in [11][12][13][14][15][16] and the references therein. However, the impulses happen not only in fixed time on the system states, but it is also possible to happen randomly; we know that the real world system states often change randomly. From this point of view, we develop random impulses in differential systems.
Very few attempts are made in the study of the random time occurrence of impulses. This changing nature from a deterministic system to a stochastic system differs from the stochastic differential equation-for example, in [17], the authors investigated the existence, uniqueness and stability results for random IDS. In [18], the author studied the p th moment exponential stability results and the authors [19] discussed the distribution nature for random IDS and proved the exponential stability. For further study, refer to [20][21][22][23][24][25][26][27][28][29] and references therein. Still, now there was no paper reported on the exponential stability for RIIDDS based on the Lyapunov and Razumikhin approach. Therefore, it is necessary to identify the exponential stability results for RIIDDS.
Inspired by the above discussions in this paper, we construct some new sufficient conditions for exponential stability by employing the random impulses. Furthermore, we discuss the random time impulses are faster than fixed time impulses. Finally, we show the stability behavior of random time impulses and the fixed time impulses. The rest of this paper is as follows: there are some definitions and lemmas in the preliminaries in Section 2. In Section 3, we prove the exponential stability and weakly exponential stability results for RIIDDS by using the Lyapunov and Razumikhin technique. Then, in Section 4, two numerical examples and their simulations are discussed and, finally, in Section 5, conclusions are given.
Notations: Let denote the set of all real numbers, + the set of all non-negative real numbers and Z + the set of all positive integers.
Let n be the Euclidean space equipped with norm · , and (Ω, F , P) be a probability space. We use Γ = P C ((−∞, 0] , n ) to denote the set of all piecewise right continuous real valued random variables ϕ : denotes a set of all bounded piecewise right continuous real valued random variables ϕ.

Preliminaries
Let {τ m } ∞ m=1 be a sequence of independent exponentially distributed random variables with parameter γ defined on sample space Ω and {ξ m } ∞ m=0 be the increasing sequence of random variables. Note that ξ 0 = t 0 , where t 0 ≥ 0 is a fixed point and ξ m = ξ m−1 + τ m for m = 1, 2, · · · , where τ m defines the waiting time between two consecutive impulses and provides ∞ ∑ m=1 τ m = ∞ with probability 1.
We assume the existence and uniqueness solution for the initial value problem (1), and denoted as y(t, t 0 , φ). Since g(t, 0) = 0, I m (ξ m , 0) = 0, m = 1, 2, ..., then y(t) = 0 is the trivial solution of system (1). Remark 1. Define {ξ m } ∞ m=0 be the increasing sequence of points, where ξ m is a value of the corresponding random variable ξ m , ∀ m = 1, 2, · · · , and {τ m } ∞ m=1 is a sequence of points, where τ m are arbitrary values of the random variable τ m , ∀ m = 1, 2, · · · . For convenience, we define ξ 0 = t 0 and ξ m = ξ m−1 + τ m , ∀ m = 1, 2, · · · , where τ m denotes the value of the waiting time. Then, system (1) becomes The solution of system (2) depends not only on the initial condition; it also depends on the moments of impulses ξ m , m = 1, 2, · · · . That is, the solution depends on the chosen arbitrary values τ m of the random variable τ m , ∀ m = 1, 2, · · · . We denote the solution of (2) by y(t; t 0 , φ, {τ m }) and will assume y(ξ m ) = lim Moreover, the collection of all solutions of system (2) is called a sample path solution of system (1). Thus, the sample path solution generates a stochastic process. We will say that it is a solution of system (1), and it is denoted by y(t; t 0 , φ, τ m ) .

Lemma 1.
From [19,28], when there will be exactly m impulses until the time t, t ≥ t 0 , and the waiting time between two consecutive impulses follow exponential distribution with parameter γ, then the probability Remark 2. From [19,28], if y(t) is the solution of the random impulsive differential equations, then where ξ m is the impulse moments. (ii) W(t, y) is locally Lipschitzian with respect to y and W(t, 0) ≡ 0.
Let α 1 and α 2 be strictly non-decreasing continuous functions satisfying For any ρ 1 > 0 and > 0, we may choose Let y(t), t ≥ t 0 be a solution of system (1) through (t 0 , φ), and it follows a stochastic process. For any φ ∈ PCB δ (t 0 ), we shall prove that We will prove (3) with the aid of the sample path solution of (1). Thus, first, it is enough to prove that there are m impulses moments until time t, t ≥ t 0 , We shall prove that there are m = k impulses moments until time t, t ≥ t 0 , First, it is clear that, for t ∈ (−∞, t 0 ), Thus, y(t) p < < ρ 1 , t ∈ (−∞, t 0 ]. Assuming k = 0, i.e., no impulse moments, then we prove that Supposing not, then there exists t ∈ [t 0 , Thus, y(t) p < < ρ 1 , t ∈ [t 0 , ξ 1 ), which gives that y(ξ − 1 ) ∈ S(ρ 1 ), y(ξ 1 ) ∈ S(ρ). Considering the condition (iii), we get Furthermore, we claim that there are m = k impulses moments until time t, t ≥ t 0 First, we prove that Supposing not, then which gives that Thus, we obtain where τ is the value of the random variable τ . By (ii), we have Then, we get however noting that where µ is the value of the random variable µ . This is a contradiction.
Apply a similar process as in (8) . However, this is in contradiction to the fact that Thus, we have proven (10). Next, we need to show that there are m = k impulses moments until time t, t ≥ t 0 Supposing not, then there exists some t ∈ [ξ k , ξ k+1 ) such that Meanwhile, we obtain in view of the fact that On the other hand, we note Therefore, we can define Thus, considering (11), we have Hence, by (ii) and (v), a similar process as in (8), we can obtain D + W(t)e λ(t−t 0 ) < 0, which gives that W(t)e λ(t−t 0 ) is non-increasing in t for t ∈ [t * ,t]. In particular, W(t * )e λ(t * −t 0 ) ≥ W(t)e λ(t−t 0 ) . This contradicts the fact that (9) holds. Thus, we have, for t ∈ [ξ k , ξ k+1 ), . Thus, by induction principle, there are m impulses moments until time t, t ≥ t 0 Thus, (4) holds. Using assumption (i), we derive at Thus, solutions generate a stochastic process that is defined by Taking expectations on both sides, by using Lemma 1 and Remark 2, then we get
Proof. Notice that µ q > InM + λτ gives that conditions (iv) and (v) in Theorem 1 hold. Finally, there are m impulses moments until time t, t ≥ t 0 , then we get Ma 2 , then Thus, solutions generate a stochastic process that is defined by Taking expectations on both sides, by using Lemma 1 and Remark 2, then we get

Remark 4.
If the condition h < s holds, the derivative of V is non-negative; then, we get the next exponential stability result.

Corollary 2.
Assume that there exists a function W(t, y) ∈ ω 0 and constants w m > 0, κ > 0, such that E [w m ] ≤ κ, m ∈ Z + , and the following conditions hold: Then, (1) is p th moment exponentially stable.
Proof. For any > 0, we may choose δ = δ( ) > 0, such that β 2 (δ) ≤ M −2 min {β 1 ( ), }. Let'y(t), t ≥ t 0 be a solution of system (1) and it follows a stochastic nature. Then, we shall prove that where where τ is the value of the random variable τ . We will prove (14) with the aid of a sample path solution of system (1). Thus, first, we have enough to prove that there are m impulses moments until t, t ≥ t 0 , For convenience, we take W(t) = W(t, y(t)), and V(t 0 ) = MW(t 0 ). Then, we shall prove that there are m = k impulses moments until time t, t ≥ t 0 , It is obvious that then t ∈ (−∞, t 0 ) Thus, y(t) p < < ρ 1 . Assuming that k = 0 i.e., no impulse moments. First, we prove for t ∈ [ξ 0 , ξ 1 ) that Then,t > t 0 , W(t)e η(t−t 0 ) = V(t 0 ) and W(t)e η(t−t 0 ) ≤ V(t 0 ), t ∈ [t 0 ,t) since Furthermore, we note that From (ii), D + W(t) ≤ q(t)c(W(t)) holds for t ∈ [t * ,t]. Hence, we have where However, note that This is a contradiction. Hence, W(t)e η(t−t 0 ) ≤ V(t 0 ), t ∈ [t 0 , ξ 1 ). Meanwhile, we take for t ∈ [t 0 , ξ 1 ) which gives y(t) p < < ρ 1 and y(ξ − 1 ) ∈ S(ρ 1 ), y(ξ 1 ) ∈ S(ρ). We assume that it is true for m = k − 1 impulses moments until time t, t ≥ t 0 , which implies Next, we shall prove that m = k impulses moments until time t, t ≥ t 0 , Supposing not, then there exists some t ∈ [ξ k , ξ k+1 ) such that In addition, from (19), we know that noting that Furthermore, we define We can deduce that which gives that Hence, we can deduce that where l(t) = q(t) · sup s>0 c(s) s + η. We have However, we note that which is contradiction. Thus, Equation (20) holds. Using the induction method, there are m impulses moments until time t, t ≥ t 0 , Using assumption (i), we derive at Thus, solutions generate a stochastic process that is defined by taking expectations on both side, by using Lemma 1 and Remark 2, then we get

Remark 5.
The above all theorems and corollaries work in fixed time impulses.

Example
In this part, we shall verify examples to analyze our theorems by using random impulses.
Therefore, system (23) is mean square exponentially stable at the origin by Corollary 1; see the comparative results Figures 1 and 2.

Conclusions
In this paper, we obtained several sufficient conditions for exponential stability and weakly exponential stability of RIIDDS by using the Lyapunov function and Razumikhin technique. Furthermore, we showed that random impulses are fast convergence compared with the fixed time impulses. Thus, we conclude that the random impulses are a better way to stabilize the various unstable differential systems in the future.