# ψ-Exponential Stability of Nonlinear Impulsive Dynamic Equations on Time Scales

Authors: Veysel Fuat Hatipoğlu; Deniz Uçar; Zeynep Fidan Koçak
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The purpose of this paper is to present the sufficientψ-exponential, uniform exponential, and global exponential stability conditions for nonlinear impulsive dynamic systems on time scales.


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## Body

## 1. Introduction

In recent years, a significant progress has been made in the stability theory of impulsive systems [1, 2], and in [3] authors studied the ψ-exponential stability for nonlinear impulsive differential equations. There are various types of stability of dynamic systems on time scales such as asymptotic stability [4, 5], exponential and uniform exponential stability [6–8], and h-stability [9]. In the past decade, many authors studied impulsive dynamic systems on time scales [10–14]. There are some papers on the theory of the stability of impulsive dynamic systems on time scales. In [15], stability criteria for impulsive systems are given and in [16], authors studied ψ-uniform stability of linear impulsive dynamic systems.In this paper, we consider theψ-exponential stability of the zero solution of the first-order nonlinear impulsive dynamic system
(1)xΔ(t)=f(t,x(t)),t∈𝕋t0+,t≠tk,x(tk+)-x(tk-)=Ik(x(tk-)),t=tk,k=1,2,…,n,x(t0+)=x0,
where 𝕋 is a time scale which has at least finitely many right-dense points of impulsive tk, f:[0,∞)×ℝn→ℝn is a nonlinear function and rd continuous in (tk-1,tk]×ℝn, Ik∈Crd[ℝn,ℝn], and 0≤t0<t1<t2<⋯<tn<t are fixed moments of impulsive effect. Let ψi:𝕋→(0,∞), i=1,2,…,n, be rd continuous functions and let ψ=diag[ψ1,ψ2,…,ψn]. Throughout the paper, we assume that f(t,0)=0, for all t in the time scale interval [0,∞), and call the zero function the trivial solution of (1) and we consider 𝕋t0+={t∈𝕋:t≥t0}. Existence and uniqueness of solutions of (1) have been studied in [10].In the following part we present some basic concepts about time scale calculus and we refer the reader to resource [17] for more detailed information on dynamic equations on time scales.


## 2. Preliminaries

A time scale𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. For t∈𝕋 we define the forward jump operatorσ:𝕋→𝕋 by
(2)σ(t)∶=inf{s∈𝕋:s>t}
while the backward jump operatorρ:𝕋→𝕋 is defined by
(3)ρ(t)∶=sup{s∈𝕋:s<t}.Ifσ(t)>t, we say that t is right scattered, while if ρ(t)<t, we say that t is left scattered. Also, if σ(t)=t, then t is called right dense, and if ρ(t)=t, then t is called left dense. The graininess functionμ:𝕋→[0,∞) is defined by
(4)μ(t)∶=σ(t)-t.We introduce the set𝕋κ which is derived from the time scale 𝕋 as follows. If 𝕋 has a left-scattered maximum m, then 𝕋κ=𝕋-{m}; otherwise 𝕋κ=𝕋.A functionf on 𝕋 is said to be delta differentiable at some point t∈𝕋 if there is a number fΔ(t) such that for every ε>0 there is a neighborhood U⊂𝕋 of t such that
(5)|f(σ(t))-f(s)-fΔ(t)(σ(t)-s)|≤ε|σ(t)-s|,;s∈U.The functionp:𝕋→ℝ is said to be regressive provided 1+μ(t)p(t)≠0 for all t∈𝕋κ. The set of all regressive rd-continuous functions f:𝕋→ℝ is denoted by ℜ.Letp∈ℜ and μ(t)≠0 for all t∈𝕋. The exponential function on 𝕋, defined by
(6)ep(t,s)=exp(∫st1μ(z)log(1+μ(z)p(z))Δz),
is the solution to the initial value problem yΔ=p(t)y, y(s)=1. Properties of the exponential function on 𝕋 are given in [6].In [6] authors defined the Lyapunov function on time scales, type I Lyapunov function V as,
(7)V(x)=∑i=1nVi(xi)=V1(x1)+⋯+Vn(xn),
and Δ derivative of type I Lyapunov function as follows:
(8)[V(x(t))]Δ={∑i=1n[Vi(xi+μ(t)fi(t,x))-Vi(xi)]μ(t)forμ(t)≠0,∇V(x)·f(t,x)forμ(t)=0.We start introducing notations that will be used in the following sections. In the Euclideann-space, norm of a vector x={x1,x2,…,xn}T is given by ∥x∥=max{|x1|,|x2|,…,|xn|}. The induced norm of an n×n matrix A is defined to be ∥A∥=sup∥x∥≤1∥Ax∥.Now, we give definition ofψ-exponential, ψ-uniform exponential, ψ-global exponential stability, and stability conditions for the solution of nonlinear impulsive dynamic system (1).


## 3.ψ-Exponential Stability

Definition 1.
The trivial solution to (1) is ψ exponentially stable on [0,∞) if any solution x(t,t0,x0) of the system (1) satisfies for all t∈[tk-1,tk), k=1,2,…,n,
(9)∥ψ(t)x(t,t0,x0)∥≤C(∥x0∥,t0)(e⊖M(t,t0))d,
where d is a positive constant and C(h,t)∈ℝ+×𝕋t0+→ℝ+ is a nonnegative increasing function, M>0. If the function C is independent of t0, then the trivial solution to system (1) is said to be ψ uniformly exponentially stable on [0,∞).Definition 2.
The trivial solution to (1) is ψ globally exponentially stable on [0,∞) if there exist some constants δ>0 and M≥1 such that any solution x(t,t0,x0) of (1), for all t∈[tk-1,tk), k=1,2,…,n, we have
(10)∥ψ(t)x(t,t0,x0)∥≤Me⊖δ(t,t0).Now, we shall present sufficient conditions for theψ-exponential stability, ψ uniformly exponential stability, and ψ globally exponentially stability of(1).Theorem 3.
Assume thatD⊂ℝn contains the origin and there exists a type I Lyapunov function V:𝕋t0+×D→[0,∞) such that, for all (t,x)∈𝕋t0+×D and t∈[tk-1,tk), k=1,2,…,n,
(11)λ1(t)∥ψ(t)x(t)∥p≤V(t,x)≤λ2(t)∥ψ(t)x(t)∥q,(12)VΔ(t,x)≤-λ3(t)∥ψ(t)x(t)∥r-L(M⊖δ)e⊖δ(t,t0)1+Mμ(t),(13)V(t,x)-Vr╱q(t,x)≤γe⊖δ(t,t0),
where λ1(t), λ2(t), and λ3(t) are positive functions, where λ1(t) is nondecreasing; p,q,r, and γ are positive constants; L is a nonnegative constant, and δ>M∶=inft≥0λ3(t)╱[λ2(t)]r╱q>0. Then the trivial solution to (1) is ψ exponentially stable on [0,∞).Proof.
Letx be a solution to (1) that stays in D for all t≥t0. As M∶=inft≥0λ3(t)╱[λ2(t)]r╱q>0,  eM(t,t0) is well defined and positive. Thus λ3(t)╱[λ2(t)]r╱q≥M. Consider
(14)[V(t,x(t))eM(t,t0)]Δ=VΔ(t,x(t))eMσ(t,t0)+V(t,x(t))eMΔ(t,t0),≤(-λ3(t)∥ψ(t)x(t)∥r-L(M⊖δ)e⊝δ(t,t0))eM(t,t0).=+MV(t,x(t))eM(t,t0)=(-λ3(t)∥ψ(t)x(t)∥r+MV(t,x(t))-L(M⊖δ)e⊖δ(t,t0)).=×eM(t,t0)≤(-λ3(t)[λ2(t)]r╱qVr╱q(t,x(t))+MV(t,x(t))=Zz-L(M⊖δ)e⊖δ(t,t0)-λ3(t)[λ2(t)]r/q)eM(t,t0)≤(M(V(t,x(t))-Vr╱q(t,x(t)))-L(M⊖δ)e⊖δ(t,t0)).=×eM(t,t0)≤(Mγ-L(M⊖δ))eM⊖δ(t,t0).
Integrating both sides of above inequality from t0 to t with x0=x(t0), we obtain, for t∈[tk-1,tk),
(15)V(t,x)eM(t,t0)≤V(t0,x0)+∫t0t(Mγ-L(M⊖δ))eM⊖δ(τ,t0)Δτ=V(t0,x0)+(MγM⊖δ-L)eM⊖δ(t,t0)+Mγδ⊖M+L≤V(t0,x0)+Mγδ⊖M+L.
From condition V(t0,x0)≤λ2(t0)∥ψ(t0)x0∥q(16)V(t,x)eM(t,t0)≤λ2(t0)∥ψ(t0)x0∥q+Mγδ⊖M+L.
Letting
(17)λ2(t0)∥ψ(t0)x0∥q+Mγδ⊖M+L=C(∥x0∥,t0)>0
we get,
(18)V(t,x)eM(t,t0)≤C(∥x0∥,t0).
By condition (11), we have
(19)∥ψ(t)x(t)∥≤λ1-1╱p(t)(V(t,x))1╱p
And by the fact that λ1(t)≥λ1(t0), we obtain
(20)∥ψ(t)x(t)∥≤λ1-1╱p(t0)(V(t,x))1╱p.
From (18) and (20) we obtain the result for all, t∈[tk-1,tk), k=1,2,…,n,
(21)∥ψ(t)x(t)∥≤λ1-1╱p(t0)(C(∥x0∥,t0))1╱pe⊖M(t,t0)1╱p.
By Definition 1 system (1) is ψ exponentially stable.If we considerψ as scaler function independent of t, then we get a sufficient condition for ψ uniformly exponential stability as stated below.Theorem 4.
In Theorem3 if ψ is a constant function independent of t and λi(t)=λi, i=1,2,3, are positive constants, then the trivial solution to system (1) is ψ uniformly exponentially stable on [0,∞).Proof.
The proof is similar to proof of Theorem3 by taking δ>λ3╱[λ2]r╱q and M=λ3╱[λ2]r╱q, hence omitted.Theorem 5.
Assume thatD⊂ℝn contains the origin and there exists a type I Lyapunov function V:𝕋t0+×D→[0,∞) such that, for all (t,x)∈𝕋t0+×D and t∈[tk-1,tk), k=1,2,…,n,
(22)λ1∥ψx(t)∥p≤V(x),(23)VΔ(t,x)≤-λ2V(x)-L(M⊖δ)e⊖δ(t,0)1+Mμ(t),
where ψ is a constant function independent of t. λ1,λ2,p,δ>0,  L≥0 are constants and 0<M<min{λ2,δ}. Then the trivial solution to (1) is ψ uniformly exponentially stable on [0,∞).Proof.
Letx be a solution to (1) that stays in D for all t≥t0. Since M∈ℜ+, eM(t,0) is well defined and positive. Now consider
(24)[V(x(t))eM(t,0)]Δ=VΔ(t,x(t))eMσ(t,0)+MV(x(t))eM(t,0),≤(-λ2V(x(t))-L(M⊖δ)e⊖δ(t,0))eM(t,0)=.+MV(x(t))eM(t,0)=(-λ2V(x(t))+MV(x(t))-L(M⊖δ)e⊖δ(t,0))eM(t,0)≤((M-λ2)V(x(t))-L(M⊖δ)e⊖δ(t,0))eM(t,0)≤-L(M⊖δ)e⊖δ(t,0)eM(t,0)=-L(M⊖δ)eM⊖δ(t,0).
Integrating both sides of the above inequality from t0 to t, we obtain, for t∈[tk-1,tk),
(25)V(x(t))eM(t,0)≤V(x0)eM(t0,0)-LeM⊖δ(t,0)+LeM⊖δ(t0,0)≤V(x0)eM(t0,0)+LeM⊖δ(t0,0)≤(V(x0)+L)eM(t0,0).
This implies that
(26)V(x(t))≤((V(x0)+L)eM(t0,0))e⊖M(t,0)=(V(x0)+L)e⊖M(t,t0).
From (26) and by invoking condition (22) we obtain, for all t∈[tk-1,tk), k=1,2,…,n,
(27)∥ψx(t)∥≤λ1-1╱p((V(x0)+L)e⊖M(t,t0))1╱p.
By Definition 1 system (1) is ψ uniformly exponentially stable.Theorem 6.
Assume thatD⊂ℝn contains the origin and there exists a type I Lyapunov function V:𝕋t0+×D→[0,∞) such that, for all (t,x)∈𝕋t0+×D and t∈[tk-1,tk), k=1,2,…,n,
(28)λ1∥ψ(t)x(t)∥p≤V(x)≤λ2∥ψ(t)x(t)∥p,(29)VΔ(t,x)≤-λ3∥ψ(t)x(t)∥p-L(K⊖δ)e⊖δ(t,0)1+Kμ(t),
where λ1,λ2,λ3, and p are positive constants, K=λ3/λ2, L≥λ1 is a nonnegative constant, and δ>λ3/λ2. Then the trivial solution to (1) is ψ globally exponentially stable on [0,∞).Proof.
Letx be a solution to (1) that stays in D for all t≥t0. Since K=λ3/λ2, eK(t,0) is well defined and positive. For all t∈[tk-1,tk), k=1,2,…,n, consider
(30)[V(x(t))eK(t,0)]Δ=VΔ(t,x(t))eKσ(t,0)+V(x(t))eKΔ(t,0),≤(-λ3∥ψ(t)x(t)∥p-L(K⊖δ)e⊖δ(t,0))eK(t,0)=.+KV(x(t))eK(t,0)=(-λ3∥ψ(t)x(t)∥p+KV(x(t))-L(K⊖δ)e⊖δ(t,0))=.×eK(t,0)≤(-λ3λ2V(x(t))+KV(x(t))-L(K⊖δ)e⊖δ(t,0))eK(t,0)=(-L(K⊖δ)e⊖δ(t,0))eK(t,0)=-L(K⊖δ)eK⊖δ(t,0).
Integrating both sides of the above inequality from t0 to t,t≠tk, with x0=x(t0), we obtain,
(31)V(x(t))eK(t,0)≤V(x0)eK(t0,0)+L(eK⊖δ(t0,0)-eK⊖δ(t,0))≤V(x0)eK(t0,0)+LeK⊖δ(t0,0)≤(V(x0)+L)eK(t0,0).
This implies that
(32)V(x(t))≤((V(x0)+L)eK(t0,0))e⊖K(t,0)=(V(x0)+L)e⊖K(t,t0).
From (32), and by invoking condition (28), we obtain, for all t∈[tk-1,tk), k=1,2,…,n,
(33)∥ψ(t)x(t)∥≤λ1-1╱p((V(x0)+L)e⊖K(t,t0))1╱p≤λ1-1╱p((V(x0)+L)e⊖K(t,t0))1╱p.
If we set M∶=((V(x0)+L)/λ1)1╱p, then (33) can be written as
(34)∥ψ(t)x(t)∥≤M(e⊖K(t,t0))1╱p.
Since M≥1, by Definition 2 system (1) is ψ globally exponentially stable.


## 4. Examples

Example 7.
We consider Example (35) in [7] and extend the example by using impulse condition,
(35)xΔ=-x+15x1/3e⊖δ(t,0),t≠tk,t∈𝕋,(36)x(tk+)=-13,t=k,k=1,2,…,n,
where δ>0 is a constant x0∈ℝ. If there is a constant 0<M<δ such that
(37)(μ(t)-1)(1+Mμ(t))≤-M,(38)(23(125μ(t))3/2+|(2/5)-(2/5)μ(t)|33)(1+Mμ(t))≤-L(M⊖δ)(t),
for some constant L≥0 and all t≠k, (35) is ψ uniformly exponentially stable.
Under above assumptions, we will show that the conditions of Theorem4 are satisfied. Let ψ(t)=1/2, choose D=ℝ and V(x)=x2, t≠k, then (11) holds with p=q=2, λ1=λ2=4. If we calculate VΔ, for all t≠k,
(39)VΔ=2x(-x+15x1/3e⊖δ(t,0))+μ(t)(-x+15x1/3e⊖δ(t,0))2,
we have the following comparison:
(40)VΔ=2x(-x+15x1/3e⊖δ(t,0))+μ(t)(-x+15x1/3e⊖δ(t,0))2≤(μ(t)-1)x2+[23(125μ(t))3/2+|(2/5)-(2/5)μ(t)|33]e⊖δ(t,0).
Dividing and multiplying the right-hand side by (1+Mμ(t)), we see that (12) holds under the above assumptions with r=2 and λ3=4M. Also, since p=q=2, we have
(41)V(x)-Vr╱q(x)=x2-(x2)2/2=0≤γe⊖δ(t,t0),
for all t≠k. Therefore (13) is satisfied. Hence, all hypotheses of Theorem 4 are satisfied and we conclude that the trivial solution to (35) is ψ uniformly exponentially stable. We consider following two special cases of (35).Case 1.
If𝕋=ℝ, then μ(t)=0. It is easy to see that (37) holds for M=1. Also for L=8/[375(δ-M)], condition (38) is satisfied. Hence, we conclude that if δ>1, then the trivial solution to (35) is ψ uniformly exponentially stable.Case 2.
If𝕋=(1/2)ℤ, then μ(t)=1/2. In this case rewriting (37) we have
(42)(-12)(1+M2)≤-M,
then (37) holds for 2/3>M>0. Also for L=((6+2)/2250(δ-M))(1-(M/2))(1-(δ/2)), condition (38) is satisfied. Therefore for δ>2/3, then the trivial solution to (35) is ψ uniformly exponentially stable.



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