# h-Stability for Differential Systems Relative to Initial Time Difference

Authors: Peiguang Wang; Xiaowei Liu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper investigates the relationship between an unperturbed differential system and a perturbed differential system that have initial time difference. Notions ofh-stability for differential systems with initial time difference are introduced, and stability criteria are formulated by using variation of parameter techniques.


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## Body

## 1. Introduction

It is well known that, in applications, asymptotic stability is more important than stability, because the desirable feature is to know the size of the region of asymptotic stability. However, when we study the asymptotic stability, it is not easy to deal with nonexponential types of stability. Pinto [1] introduced the notion of h-stability with the intention of obtaining results about stability for a weakly stable system (at least, weaker than those given exponential stability and uniform Lipschitz stability) under some perturbations and developed the study of exponential stability to a variety of reasonable systems called h-systems. Since then, Choi and Ryu [2], Choi et al. [3], and Choi and Koo [4] investigated h-stability problem for the nonlinear differential systems respectively, and Choi et al. [5, 6] characterized the h-stability in variation for nonlinear difference systems via n∞-similarity and Lyapunov functions and obtained some relative results. For the detailed results of h-stability of impulsive dynamic systems on time scale and others systems can be found in [7–10].At present, the investigation of differential systems with initial time difference has attracted a lot of attention. This is mainly because of the fact that when considering initial value problems, it is impossible not to make errors in the starting time in dealing with real world phenomena, that is, the solutions of the unperturbed differential system may start at some initial time and the solutions of the perturbed systems may start at a different initial time. When we consider such a change of initial time for each solution, we need to deal with the problem of comparing between any two solutions which start at different times. At present, there are two methods to discuss the stability problem with initial time difference: one is the differential inequalities and comparison principle, and the other is the method of variation of parameters. For the pioneering works in this area we can refer to the papers [11, 12]. After that, there are many stability results for various of differential and difference systems; see [13–20]. However, the above results were obtained by using comparison principle and differential inequalities; there are few stability criteria by using the method of variation of parameters; see [21–24].In this paper, we attempt to extend the notion ofh-stability to differential systems with initial time difference, namely, initial time difference h-stability (ITDhS) and then establish some stability criteria for such differential systems by using the method of variation of parameters. The remainder of this paper is organized in the following manner. Some preliminaries are presented in Section 2. The notions of h-stability for differential systems with initial time difference are given in this section. In Section 3, several stability criteria are established. Finally, an example is added to illustrate the result obtained.


## 2. Preliminaries

LetR+=[0,+∞) and Rn denotes the n-dimensional Euclidean space with appropriate norm ∥·∥.Consider the differential systems:(1)x′=f(t,x),x(t0)=x0,t≥t0,t0∈R+,(2)x′=f(t,x),x(τ0)=y0,t≥τ0,τ0∈R+
and the perturbed differential system of (2):
(3)y′=F(t,y),y(τ0)=y0,t≥τ0,τ0∈R+,
where f,F∈C[R+×Rn,Rn] are locally Lipschitzian and f has continuous partial derivatives ∂f/∂x on R+×Rn. The above assumptions imply the existence and uniqueness of solutions through (t0,x0) and (τ0,y0). A special case of (3) is where F(t,y)=f(t,y)+R(t,y), R(t,y) is the perturbation term. Let η=τ0-t0>0. Furthermore, suppose that x(t,t0,x0) is the given solution with respect to which we shall study stability criteria.Let us begin by defining the following notions.Definition 1.
The solutionx(t,τ0,y0) of the system (2) through (τ0,y0) is said to be initial time difference h-stability (ITDhS) with respect to the solution x(t-η,t0,x0), where x(t,t0,x0) is any solution of the system (1), if and only if there exist c≥1 and a positive bounded continuous function h defined on R+ such that
(4)∥x(t,τ0,y0)-x(t-η,t0,x0)∥≤c[∥y0-x0∥+|τ0-t0|]h(t)h-1(τ0)
for t≥τ0 and h-1(t)=1/h(t).Similarly, we can define initial time differenceh-stability (ITDhS) with respect to the solution y(t,τ0,y0) of the system (3) through (τ0,y0).We are now in a position to give the Alekseev’s formula, which is an important tool in the subsequent discussion.Lemma 2 (see [25]).
Ifx(t,τ0,y0) is the solution of (2) and exists for t≥τ0, any solution y(t,τ0,y0) of (3), with y(τ0)=y0, satisfies the integral equation:
(5)y(t,τ0,y0)=x(t,τ0,y0)+∫τ0t‍Φ(t,s,y(s,τ0,y0))R(s,y(s,τ0,y0))ds
for t≥τ0, where Φ(t,τ0,y0)=∂x(t,τ0,y0)/∂y0.The following lemma will also be needed in our investigations.Lemma 3 (see [25]).
Assume thatx(t,t0,x0) is the solution of (1) through (t0,x0), which exists for t≥t0, and then
(6)x(t,t0,x0)=[∫01‍Φ(t,t0,sx0)ds]x0,
where Φ(t,t0,x0)=∂x(t,t0,x0)/∂x0.


## 3. Stability Criteria

We shall present, in this section, the stability criteria for differential systems with initial time difference.Theorem 4.
Letx(t,τ0,y0) and x(t-η,t0,x0) be the solutions of (2) and (1) through (τ0,y0) and (t0,x0), respectively, t≥τ0. Assume that (i)
v(t,τ0,v0)=x(t,τ0,y0)-x(t-η,t0,x0), in which v0=y0-x0;(ii)
there exists a positive bounded continuously differentiable functionh(t) on R+ such that
(7)liminfδ→0-∥v(t,τ0,v0)+(f~(t,v,η))δ∥-∥v(t,τ0,v0)∥δ≤h′(t)h-1(t)∥v(t,τ0,v0)∥,wheref~(t,v,η)=f(t,x(t-η,t0,x0)+v(t,τ0,v0))-f(t,x(t-η,t0,x0));(iii)
f is locally Lipschitzian in time such that
(8)∥f(t,x(t-η,t0,x0))-f(t-η,x(t-η,t0,x0))∥≤L1(t)|η|L2(τ0),whereL2(τ0)=∫τ0+∞‍h-1(s)h(τ0)L1(s)ds,L1(s)∈C[R+,R+].Then the solutionx(t,τ0,y0) of the system (2) is ITDhS with respect to the solution x(t-η,t0,x0).Proof.
Definez(t)=∥v(t,τ0,v0)∥ for t≥τ0, and then z(τ0)=∥y0-x0∥. Also,
(9)v′(t,τ0,v0)=x′(t,τ0,y0)-x′(t-η,t0,x0)=f(t,v(t,τ0,v0)+x(t-η,t0,x0))-f(t-η,x(t-η,t0,x0)).
Using a Taylor approximation for v(t,τ0,v0) and the conditions (i) and (ii), we arrive at
(10)D-z(t)=limδ→0inf∥v(t,τ0,v0)+v′(t,τ0,v0)δ∥-∥v(t,τ0,v0)∥δ≤h′(t)h-1(t)z(t)+L1(t)|η|L2(τ0).
And then, from (10), we have
(11)z(t)≤h(t)h-1(τ0)×(z(τ0)+|η|L2(τ0)∫τ0+∞‍h-1(s)h(τ0)L1(s)ds).
Moreover, using the condition (iii), we obtain
(12)z(t)≤h(t)h-1(τ0)(∥y0-x0∥+|η|).
Then from (12), we get
(13)∥x(t,τ0,y0)-x(t-η,t0,x0)∥≤[∥y0-x0∥+|τ0-t0|]h(t)h-1(τ0).
So by Definition 1 with c=1, the solution x(t,τ0,y0) of (2) is ITDhS with respect to the solution x(t-η,t0,x0). This completes the proof.Remark 5.
Seth(t)=e-β0t, and then we can obtain Theorem 3.4 in [8].Theorem 6.
Lety(t,τ0,y0) be the solution of (3) through (τ0,y0). Assume that (i)
the solutionx(t,τ0,y0) of (2) is ITDhS with respect to the solution x(t-η,t0,x0) for t≥τ0, where x(t,t0,x0) is any solution of (1);(ii)
there existc≥1, α>0 and a positive bounded continuous function h defined on R+ such that
(14)∥Φ(t,s,y(s))∥≤ch(t)h-1(s),∥R(s,y(s))∥≤r(s)∥y(s)∥,provided thaty(s,τ0,y0)≤α,r(s)∈C(R+,R+) and ∫τ0+∞‍r(s)ds<+∞.Then the solutiony(t,τ0,y0) of (3) is ITDhS with respect to the solution x(t-η,t0,x0).Proof.
Definev(t,τ0,v0)=x(t,τ0,y0)-x(t-η,t0,x0) and z(t)=∥v(t,τ0,v0)∥, and then z(τ0)=∥y0-x0∥. The condition (i) yields
(15)∥x(t,τ0,y0)-x(t-η,t0,x0)∥≤c[∥y0-x0∥+|τ0-t0|]h(t)h-1(τ0).
By Lemma 2, it follows that
(16)y(t,τ0,y0)-x(t-η,t0,x0)=v(t,τ0,v0)+∫τ0t‍Φ(t,s,y(s))R(s,y(s))ds.
Now taking the norms of both sides and using the triangle inequality, we have
(17)∥y(t,τ0,y0)-x(t-η,t0,x0)∥≤z(t)+∫τ0t‍∥Φ(t,s,y(s))∥∥R(s,y(s))∥ds.
From (15), we obtain
(18)∥y(t,τ0,y0)-x(t-η,t0,x0)∥≤c[∥y0-x0∥+|τ0-t0|]h(t)h-1(τ0)+∫τ0t‍∥Φ(t,s,y(s))∥∥R(s,y(s))∥ds.
Setting M*(t)=∥y(t,τ0,y0)-x(t-η,t0,x0)∥ and using the triangle inequality, we have
(19)M*(t)≤c[∥y0-x0∥+|τ0-t0|]h(t)h-1(τ0)+∫τ0t‍ch(t)h-1(s)r(s)M*(s)ds+∫τ0t‍ch(t)h-1(s)r(s)∥x(s-η,t0,x0)∥ds.
By using Lemma 3 and the condition (ii), we obtain
(20)∥x(t-η,t0,x0)∥≤αch(t)h-1(τ0),for∥x0∥≤α.
Hence,
(21)M*(t)≤c{∥y0-x0∥+|τ0-t0|}h(t)h-1(τ0)+∫τ0t‍ch(t)h-1(s)r(s)M*(s)ds+c2αh(t)h-1(τ0)∫τ0t‍r(s)ds.
Then we have
(22)N*(t)≤c{∥y0-x0∥+|τ0-t0|}+∫τ0t‍r*(s)N*(s)ds+c2αN1(τ0),
where r*(t)=cr(t),N*(t)=h-1(t)h(τ0)M*(t), and ∫τ0+∞‍r(s)ds=N1(τ0).
By Gronwall's inequality, one gets(23)M*(t)≤{c[∥y0-x0∥+|τ0-t0|]+c2αN1(τ0)}×h(t)h-1(τ0)ecN1(τ0).
Moreover, set
(24)c1{∥y0-x0∥+|τ0-t0|}={c[∥y0-x0∥+|τ0-t0|]+c2αN1(τ0)}ecN1(τ0)
and c1≥1, we get
(25)∥y(t,τ0,y0)-x(t-η,t0,x0)∥≤c1{∥y0-x0∥+|τ0-t0|}h(t)h-1(τ0).
From Definition 1, it follows that the solution of (3) is ITDhS with respect to the solution x(t-η,t0,x0). This completes the proof.


## 4. Example

Now, we shall illustrate Theorem6 by a simple example. Consider the differential systems
(26)x′=-x,x(t0)=x0,t≥t0,t0∈R+,(27)x′=-x,x(τ0)=y0,t≥τ0,τ0∈R+,
and the perturbed differential system of (27):
(28)y′=-y+1t2y,y(τ0)=y0,t≥τ0,τ0∈R+.
Define z(t)=x(t,τ0,y0)-x(t-η,t0,x0); by direct calculation, we have the solution of (27) given by x(t,τ0,y0)=y0e-t+τ0, which exists for all t≥τ0, and Φ(t,τ0,y0)=∂x(t,τ0,y0)/∂y0=e-t+τ0,  Φ(τ0,τ0,y0)=I. ∥z(t)∥≤(∥y0-x0∥+|τ0-t0|)e-t+τ0, and then the solution of system (27) is ITDhS with respect to x(t-η,τ0,x0).Now, let us begin to consider the perturbation termF(t,y)=(1/t2)y of (28), and we have ∥(1/t2)y∥≤(1/t2)∥y∥, where ∫τ0+∞‍(1/t2)dt<+∞. Then by Theorem 6, we can conclude that the solution y(t,τ0,y0) of (28) is ITDhS with respect to the solution x(t-η,t0,x0).



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