Abstract and Applied Analysis
Volume 2013, Issue 1 103894
Research Article
Open Access

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

Veysel Fuat Hatipoğlu, 

Corresponding Author

Veysel Fuat Hatipoğlu

Department of Mathematics, Faculty of Science, Muğla University, Kötekli Campus, 48000 Muğla, Turkey mu.edu.tr

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Deniz Uçar, 

Deniz Uçar

Department of Mathematics, Faculty of Sciences and Arts, Usak University, 1 Eylul Campus, 64200 Usak, Turkey usak.edu.tr

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Zeynep Fidan Koçak, 

Zeynep Fidan Koçak

Department of Mathematics, Faculty of Science, Muğla University, Kötekli Campus, 48000 Muğla, Turkey mu.edu.tr

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First published: 10 April 2013
https://doi.org/10.1155/2013/103894
Citations: 3
Academic Editor: Stefan Siegmund

Abstract

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

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 the ψ-exponential stability of the zero solution of the first-order nonlinear impulsive dynamic system
()
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 . 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 scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. For t ∈ 𝕋 we define the forward jump operator σ : 𝕋 → 𝕋 by
()
while the backward jump operator ρ : 𝕋 → 𝕋 is defined by
()
If σ(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 function μ : 𝕋 → [0, ∞) is defined by
()

We introduce the set 𝕋κ 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 function f 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
()

The function p : 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.

Let p ∈ ℜ and μ(t) ≠ 0 for all t ∈ 𝕋. The exponential function on 𝕋, defined by
()
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,
()
and Δ derivative of type I Lyapunov function as follows:
()

We start introducing notations that will be used in the following sections. In the Euclidean n-space, norm of a vector 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 of ψ-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,

()
where d is a positive constant and 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

()

Now, we shall present sufficient conditions for the ψ-exponential stability, ψ uniformly exponential stability, and ψ globally exponentially stability of(1).

Theorem 3. Assume that D ⊂ ℝn contains the origin and there exists a type I Lyapunov function such that, for all and t ∈ [tk−1, tk), k = 1,2, …, n,

()
()
()
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 . Then the trivial solution to (1) is ψ exponentially stable on [0, ∞).

Proof. Let x be a solution to (1) that stays in D for all t ≥ t0. As ,  eM(t, t0) is well defined and positive. Thus . Consider

()
Integrating both sides of above inequality from t0 to t with x0 = x(t0), we obtain, for t ∈ [tk−1, tk),
()
From condition
()
Letting
()
we get,
()
By condition (11), we have
()
And by the fact that λ1(t) ≥ λ1(t0), we obtain
()
From (18) and (20) we obtain the result for all, t ∈ [tk−1, tk), k = 1,2, …, n,
()
By Definition 1 system (1) is ψ exponentially stable.

If we consider ψ as scaler function independent of t, then we get a sufficient condition for ψ uniformly exponential stability as stated below.

Theorem 4. In Theorem 3 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 Theorem 3 by taking and , hence omitted.

Theorem 5. Assume that D ⊂ ℝn contains the origin and there exists a type I Lyapunov function such that, for all and t ∈ [tk−1, tk), k = 1,2, …, n,

()
()
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. Let x 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

()
Integrating both sides of the above inequality from t0 to t, we obtain, for t ∈ [tk−1, tk),
()
This implies that
()
From (26) and by invoking condition (22) we obtain, for all t ∈ [tk−1, tk), k = 1,2, …, n,
()
By Definition 1 system (1) is ψ uniformly exponentially stable.

Theorem 6. Assume that D ⊂ ℝn contains the origin and there exists a type I Lyapunov function such that, for all and t ∈ [tk−1, tk), k = 1,2, …, n,

()
()
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. Let x 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

()
Integrating both sides of the above inequality from t0 to t, t ≠ tk, with x0 = x(t0), we obtain,
()
This implies that
()
From (32), and by invoking condition (28), we obtain, for all t ∈ [tk−1, tk), k = 1,2, …, n,
()
If we set , then (33) can be written as
()
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,

()
()
where δ > 0 is a constant x0 ∈ ℝ. If there is a constant 0 < M < δ such that
()
()
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 Theorem 4 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,

()
we have the following comparison:
()
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
()
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. If 𝕋 = 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. If 𝕋 = (1/2)ℤ, then μ(t) = 1/2. In this case rewriting (37) we have

()
then (37) holds for 2/3 > M > 0. Also for , condition (38) is satisfied. Therefore for δ > 2/3, then the trivial solution to (35) is ψ uniformly exponentially stable.

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