Robust exponential stability of uncertain impulsive delays differential systems

Author’s Accepted Manuscript Robust exponential stability of uncertain impulsive delays differential systems Dengwang Li, Xiaodi Li

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S0925-2312(16)00058-8 http://dx.doi.org/10.1016/j.neucom.2016.01.011 NEUCOM16634

To appear in: Neurocomputing Received date: 13 May 2015 Revised date: 8 October 2015 Accepted date: 18 January 2016 Cite this article as: Dengwang Li and Xiaodi Li, Robust exponential stability of uncertain impulsive delays differential systems, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.01.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Robust exponential stability of uncertain impulsive delays differential systems Dengwang Lia , Xiaodi Lia,b∗ a.Shandong Province Key Laboratory of Medical Physics and Image Processing Technology, School of Physics and Electronics, Shandong Normal University,Ji’nan, 250014, P.R. China b.School of Mathematical Sciences, Shandong Normal University, Ji’nan, 250014, P.R. China

Abstract This paper deals with the robust stability of a class of uncertain impulsive control systems with infinite delays. By employing the formula for the variation of parameters and estimating the Cauchy matrix, several criteria on robust exponential stability of the systems are derived, these criteria are less restrictive than those in the earlier publications. Moreover, the criteria can be applied to stabilize the unstable continuous systems with infinite delays and uncertainties by utilizing impulsive control. Finally, two numerical examples are given to illustrate the effectiveness and advantages of the proposed method. Key words: Exponential stability; Uncertainty; Impulsive control systems; Infinite delays; Robustness.

1. Introduction

the stability of the corresponding continuous systems and so it can be more widely applied to stabilize the unstable continuous systems with infinite delays and uncertainties by using impulsive control. Finally, two examples are given to show the effectiveness and advantages of the obtained results.

Recently, impulsive control has attracted great interest of many researchers[1–6]. Such control systems arise naturally in a wide variety of applications, such as dosage supply in pharmacokinetics [1], orbital transfer of satellite [2, 3], ecosystems management [4, 5] and control of saving rates in a financial market [6]. Moreover, time delays and uncertainties [7–11] occur frequently in engineering, biological and economical systems, and sometimes they depend on the histories heavily and result in oscillation and instability of systems [8, 12]. [14–22] are the cases of finite delays. Yang and Xu presented several interesting criteria on robust stability for uncertain impulsive control systems with time-varying delays [16]. Liu established several criteria on asymptotic stability for impulsive control systems with time delays[18]. [23–25] are the cases of infinite delays. However, the corresponding theory for impulsive control systems with infinite delays has been relatively less developed. In fact, an infinite delays deserve study intensively because they are not only an extension of finite delays but also describing the adequate mathematical models in many fields [17]. Therefore, it is necessary to further investigate the stability of uncertain impulsive control systems with infinite delays. Meanwhile, it is challenging to address the issue since we must utilize impulsive effects to handle the instability which may be caused by the infinite delays and uncertainties. Hence, techniques and methods for uncertain impulsive control systems with infinite delays should be further developed and explored. This paper is inspired by [16]. In this paper, we present some criteria for the robust exponential stability of uncertain impulsive control systems with infinite delays by using the formula for the variation of parameters and estimating the Cauchy matrix. More importantly, the robust stability criteria don’t require ∗ Corresponding

2. Preliminaries Let N = 1, 2, · · ·, I be the identity matrix, λmin (·) and λmax (·) be the smallest and the largest eigenvalues of a symmetrical matrix, respectively. For φ : R → Rn , denote φ(t+ ) = lim+ φ(t + s→0

s), φ(t− ) = lim− φ(t + s). For x ∈ Rn and A ∈ Rn×n , let || x || be s→0

any vector norm, ||φ||α = sup s≤0 ||φ(s)|| and denote the induced matrix norm and the matrix measure, respectively, by || Ax || || I + hA || −1 , µ(A) = lim+ . h→0 || x || h

|| A ||= sup x,0

The usual norms and measures of vectors and matrices are: || x ||1 =

n X

| x j |, || A ||1 = max

1≤ j≤n

j=1

µ1 (A) = max {a j j + 1≤ j≤n

|| x ||2 =

v tX n

n X

n X

| ai j |,

i=1

| ai j |};

i, j

x2j , || A ||2 =

p

λmax(AT A) ,

j=1

µ2 (A) = 12 λmax (A + AT ); || x ||∞ = max | xi |, || A ||∞ = max 1≤i≤n

Author. Email: [email protected]

1

1≤i≤n

n X j=1

| ai j |,

µ∞ (A) = max {aii + 1≤i≤n

n X

We always assume that the solution x(t) of (3) are right continuous at t = tk , i.e., x(tk ) = x(tk+ ). That is, the solutions are the piecewise continuous functions with discontinuities of the first kind only at t = tk , k ∈ N. For more details on the impulsive delay systems, one can refer to [1, 13, 17] and references therein. In order to prove our main results, we need the following definitions and lemmas:

| ai j |}.

j,i

Consider the uncertain infinite delays differential system:   x˙(t) = [A + ∆A]x(t) + [B + ∆B]x(t − r(t))   Z +∞      +[C + ∆C] h(s)x(t − s)ds, t , tk , (1)    0      y(t) = Ex(t), t ≥ 0, n

Definition 2.1. ([1]) Assume x(t) = x(t, t0 , φ) to be the solution of (3) through (t0 , φ). Then the zero solution of (3) is said to be globally exponentially stable, if for any initial data xt0 = φ, there exist two positive numbers λ > 0, M ≥ 1 such that || x(t) ||≤ M || φ ||α e−λ(t−t0 ) , t ≥ t0 .

m

in which x ∈ R is the state variable, y ∈ R is the output variable, r(t) is the time-varying delay function with 0 ≤ r(t) ≤ τ, τ is a given positive constant, A, B, C ∈ Rn×n and E ∈ Rm×n are known constant matrices, ∆A, ∆B and ∆C are the uncertain matrices, which vary within the range of || ∆A ||≤ a, || ∆B ||≤ b, || ∆C ||≤ c,

Definition 2.2. ([16]) The uncertain impulsive dynamical system (3) is called robustly exponentially stable, if the zero solution x = 0 of the system is globally exponentially stable for any ∆A, ∆B, ∆C satisfying (2).

(2)

where a, b,Zc are known non-negative constant, h(s) ∈ C(R+ , R) +∞

satisfying

The following lemma introduces a norm of vector, a matrix norm and a matrix measure (we call them P-norm and Pmeasure). Lemma 2.1. ([16]) Let√P ∈ Rn×n be symmetrical and positive definite. Then || x ||P = xT Px is a norm of a vector x ∈ Rn , the induced norm and measure of matrix A ∈ Rn×n are, respectively,

| h(s) | eηs ds < ∞, where η > 0 is a given

0

constant. An impulsive control law of (1) can be presented in form of the following control sequence {tk , U(k, x(tk− ))} (see [6, 18]): 0 ≤ t0 < t1 < · · · < tk < · · · , lim tk = +∞, k→+∞

∆x(tk ) =

x(tk+ )

−

x(tk− )

= U(k,

x(tk− )), k

|| A ||P =|| DAD−1 ||2 , µP (A) = µ2 (DAD−1 ), ∈ N.

where DT D = P. Lemma 2.2. ([16]) Let W(t, t0 ) be the Cauchy matrix of the linear systems:

Let U(k, x) = Bk y, Bk ∈ Rn×m and Ck = Bk E. Then we obtain the uncertain impulsive control system with infinite delays as follows:  x˙(t) = [A + ∆A]x(t) + [B + ∆B]x(t − r(t))   Z +∞       +[C + ∆C] h(s)x(t − s)ds, t , tk ,    0 (3)    + − −   ∆x(t ) = x(t ) − x(t ) = C x(t ), k ∈ N,  k k  k k k     x(t) = φ(t), t ≤ 0,

    x˙(t) = Ax(t), t , tk ,    ∆x(tk ) = x(t+ ) − x(t− ) = Ck x(t− ), k ∈ N. k k k

Given a constant || I + Ck ||≤ γ for all k ∈ N, we have the following: Case (1) if 0 < γ < 1 and ρ = supk∈N {tk − tk−1 } < ∞, then

where φ : (−∞, 0] → Rn is continuous. In particular, if h(s) = 0, then system (3) becomes the impulsive control system with finite delays given in [16]:  x˙(t) = [A + ∆A]x(t) + [B + ∆B]x(t − r(t)), t , tk ,       ∆x(tk ) = x(tk+ ) − x(tk− ) = Ck x(tk− ), k ∈ N,       x(t) = φ(t), −τ ≤ t ≤ 0,

|| W(t, t0 ) ||≤ γ1 e(µ(A)+

ln γ ρ )(t−t0 )

, t ≥ t0 ;

Case (2) if γ ≥ 1 and θ = inf k∈N {tk − tk−1 } > 0, then || W(t, t0 ) ||≤ γe(µ(A)+

ln γ c )(t−t0 )

, t ≥ t0 .

(4) 3. Main results

where φ : [−τ, 0] → Rn is continuous. Also, if ∆A = ∆B = ∆C = 0, then system (3) becomes the deterministic impulsive control systems with infinite delays:  x˙(t) = Ax(t) + Bx(t − r(t))   Z +∞       +C h(s)x(t − s)ds, t , tk ,    0      ∆x(tk ) = x(tk+ ) − x(tk− ) = Ck x(tk− ), k ∈ N,       x(t) = φ(t), α ≤ t ≤ 0.

(6)

Theorem 3.1. Let ρ = sup{tk − tk−1 } < ∞. Suppose that there k∈N

exists a constant 0 < γ < 1 satisfy || I + Ck ||≤ γ and a + b+ || B || +(c+ || C ||)M ln γ + µ(A) + < 0, γ ρ

(5)

where M =

Z 0

+∞

| h(s) | eηs ds. Then the zero solution of the

system (3) is robustly exponentially stable. 2

Proof. Since a+b+||B||+(c+||C||)M + µ(A) + lnργ < 0, then we choose γ small enough λ ∈ (0, η) such that

|| φ ||α (µ(A)+ lnργ )t∗ || x(t∗ ) ||≤ e Z t∗ γ 1 (µ(A)+ lnργ )(t∗ −s) e + [a || x(s) || +(b+ || B ||) 0 γ

λτ

a + (b+ || B ||)e + (c+ || C ||)M ln γ + µ(A) + + λ < 0. γ ρ Furthermore, for any ε ∈ (0, λ), we have 0≤

(λ−ε)τ

a+(b+||B||)e

γ

+(c+||C||)M

×Z || x(s − r(s)) || +(c+ || C ||)

≤ −µ(A) −

ln γ ρ

+∞

− (λ − ε).

×

(7)

|| φ ||α (µ(A)+ lnργ )t∗ e γZ ∗ t 1 (µ(A)+ lnργ )(t∗ −s) || φ ||α −(λ−ε)s + e [a e γ 0 γ ||φ||α −(λ−ε)(s−r(s)) +(b+ || B ||) × γ e Z +∞ || φ ||α −(λ−ε)(s−ξ) +(c+ || C ||) | h(ξ) | e dξ]ds γ 0 Z t∗ ln γ || φ ||α (µ(A)+ lnργ )t∗ a b+ || B || = e [1 + e(−µ(A)− ρ )s [ e−(λ−ε)s + γ γ γ 0 Z +∞ | h(ξ) | e−(λ−ε)(s−ξ) dξ]ds] ×e−(λ−ε)(s−r(s)) + c+||C|| γ

≤

By the formula for the variation of parameters, the solution of (3) can be presented by: x(t)

= W(t, Z t0)x(0) + W(t, s)[∆Ax(s) + (B + ∆B)x(s − r(s)) 0

+(C + ∆C)

+∞

Z

h(ξ)x(s − ξ)dξ]ds, t ≥ 0,

0

where W(t, t0 ) is the Cauchy matrix of the impulsive linear system (6). Employing Case (1) in Lemma 2.2 and (2), we have:

0

=

|| x(t) ||≤|| Z t W(t, 0) || · || x(0) || + || W(t, s) || [|| ∆A || · || x(s) || 0

+(|| B || + || ∆B ||)× || x(s − r(s)) ||

(λ−ε)τ + b+||B|| γ e

0

≤ +

1 e γ Z t 0

1 e γ

(8)

|| φ ||α

(µ(A)+ lnργ )(t−s)

× || x(s − r(s)) Z ||

+∞

+(c+ || C ||)

≤ [a || x(s) || +(b+ || B ||)

| h(ξ) | · || x(s − ξ) || dξ]ds,

≤

t ≥ 0. Without the less of generality, we assume that || φ ||α > 0. From 0 < γ < 1 and λ > ε, it is easily observed that: ||φ||α −(λ−ε)t ,t γ e

≤ 0.

|| x(t) ||<

≥ 0.

|| x(t) ||<

||φ||α −(λ−ε)t∗ , γ e

||φ||α −(λ−ε)t ,t γ e

< t∗ .

| h(ξ) | e(λ−ε)ξ dξ]ds]

0 t∗

Z

e(−µ(A)−

ln γ ρ −(λ−ε))s

[

0

c+||C|| γ

|| φ ||α (µ(A)+ lnργ )t∗ [1 + e γ

a b+ || B || [ + γ γ

+∞

Z

a γ

| h(ξ) | eηξ dξ]ds]

0 t∗

Z

e(−µ(A)−

ln γ ρ −(λ−ε))s

0

|| φ ||α (µ(A)+ lnργ )t∗ a + (b+ || B ||)e(λ−ε)τ + (c+ || C ||)M e [1 + γ γ lnγ ∗ ×(−µ(A) − lnργ ) − (λ − ε))−1 [e(−µ(A)− ρ −(λ−ε))t − 1]]

lnγ ∗ || φ ||α (µ(A)+ lnργ )t∗ e [1 + e(−µ(A)− ρ −(λ−ε))t − 1] γ || φ ||α −(λ−ε)t∗ . = e γ

(9)

This contradicts (11), and so the estimate (10) holds. Letting ε → 0, we have

(10)

|| x(t) ||≤

If this is not true, by (9) and the piecewise continuities of x(t), then there must exist t∗ > 0 such that: || x(t∗ ) ||≥

0 +∞

ln γ ρ −(λ−ε))s

<

In the following, we shall prove that: ||φ||α −(λ−ε)t ,t γ e

+

e(−µ(A)−

(λ−ε)τ × a+(b+||B||)e γ +(c+||C||)M ds]

0

|| x(t) ||≤|| φ ||α <

|| φ ||α (µ(A)+ lnργ )t∗ [1 + e γ Z ×e(λ−ε)r(s) + c+||C|| γ

t∗

Z

|| φ ||α (µ(A)+ lnργ )t∗ e [1 + ≤ γ

+(|| Z C+∞|| + || ∆C ||) × | h(ξ) | · || x(s − ξ) || dξ]ds (µ(A)+ lnργ )t

| h(ξ) | · || x(s − ξ) || dξ]ds

0

|| φ ||α −λt e , t ≥ 0. γ

The proof is complete. (11)

Letting || · || be 1, 2, ∞−norm or P-norm, we easily obtain the following corollaries. Corollary 3.1. Letting ρ = sup{tk − tk−1 } < ∞. Then the zero

(12)

k∈N

From (8), (12) and (7), we get,

solution of (3) is robustly exponentially stable if there exists a 3

constant 0 < γ < 1 such that || I + Ck ||P ≤ γ and

Employing Corollary 3.2, we can easily obtain the robust exponential stability for the deterministic impulsive control system (5) with infinite delays.

a + b+ || B ||P +(c+ || C ||P )M ln γ + µP (A) + < 0, γ ρ

Corollary 3.5. Letting ρ = sup{tk − tk−1 } > 0 and assume all the

where P = 1, 2 or ∞ and the corresponding matrix norm and matrix measure are given in Section 2.

k∈N

eigenvalues of matrix A have positive real parts. Suppose that there exist a constant 0 < γ < 1 and two symmetrical positive definite matrices P, Q ∈ Rn×n such that:

Combining Lemma 2.1 and Corollary 3.1, we have the following corollary. Corollary 3.2. Letting ρ = sup{tk − tk−1 } < ∞. Then the zero

AT P + PA = Q, s λmax (P) || I + Ck ||2 ≤ γ, λmin (P)

k∈N

solution of (3) is robustly exponentially stable if there exists a constant 0 < γ < 1 and a non-singular matrix D such that || I + DCk D−1 ||2 ≤ γ and a+b+||DBD−1 ||2 +(c+||DCD−1 ||2 )M γ

+ µ2 (DAD−1 ) +

ln γ ρ

ln γ λmax (Q) || PB ||2 M || PC ||2 + + + < 0. ρ 2λmin (P) γλmin (P) γλmin (P)

< 0.

Remark 3.1. It should be noted that under these conditions the stability of continuous system (1) can’t be confirmed. Therefore, regardless of any form of the stability of (1), we can stabilize the system (1) by employing impulsive effects || I +Ck ||≤ γ and a+b+||B||+(c+||C||)M + µ(A) + lnργ < 0. So our results can be γ more widely applied to stabilize the unstable continuous systems with time delays and uncertainties by using impulsive control. In fact, we always can find ρ so small that a+b+||B||+(c+||C||)M γ +µ(A) +

ln γ ρ

Then the zero solution of (5) is robustly exponentially stable. Proof. Since P is symmetrical and positive definite, there exists a non-singular matrix D satisfying P = DT D. From the definition of P−norm we have, || I + DCk D−1 ||2

x,0

< 0 holds since ln γ < 0.

s ≤

Remark 3.2. Theorem 3.1 has considered of uncertain impulsive control system with infinite delays. More importantly, the criteria don’t require the assumption that the eigenvalues of the parameter matrix A all have positive (negative) real parts. Therefore, our results are rather general and have great power in applications.

|| I + hA ||P −1 h→0 h λmax (Q) xT (AT P + PA)x = sup ≤ . 2xT Px 2λmin (P) x,0 Similarly, we can deduce that

k∈N

exists a constant γ ≥ 1 satisfy || I + Ck ||≤ γ and

Z

+∞

ln γ < 0, θ

|| DBD−1 ||2 =|| B ||P ≤

0

a + b+ || DBD−1 ||2 +(c+ || DCD−1 ||2 )M γ ln γ λmax (Q) || PB ||2 || PC ||2 M ≤ + + + < 0, ρ 2λmin (P) γλmin (P) γλmin (P)

system (3) is robustly exponentially stable.

ln γ ρ

Corollary 3.3. Letting θ = inf {tk − tk−1 } > 0. Then the zero k∈N

solution of (3) is robustly exponentially stable if there exists a constant γ ≥ 1 such that || I + Ck || p ≤ γ and ln γ < 0. θ

Similarly, employing Corollary 3.4, we can obtain the following corollary. The proof is similar to the one in Corollary 3.5 and we omit it here.

k∈N

solution of (3) is robustly exponentially stable if there exists a constant γ ≥ 1 and a non-singular matrix D such that || I + DCk D−1 ||2 ≤ γ and γ[a + b+ || DBD + lnθγ < 0.

||2 +(c+ || DCD

−1

+ µ2 (DAD−1 ) +

where a = b = c = 0. So the conclusion holds.

Corollary 3.4. Letting θ = inf {tk − tk−1 } > 0. Then the zero

−1

|| PC ||2 || PB ||2 , || DCD−1 ||2 =|| C ||P ≤ λmin (P) λmin (P)

Consequently, we have

| h(s) | eηs ds. Then the zero solution of the

γ[a + b+ || B ||P +(c+ || C ||P )M] + µP (A) +

λmax (P) || I + Ck ||2 ≤ γ. λmin (P)

µ2 (DAD−1 ) = µP (A) = lim+

Theorem 3.2. Let θ = inf {tk − tk−1 } > 0. Suppose that there

γ[a + b+ || B || +(c+ || C ||)M] + µ(A) +

|| (I + Ck )x ||P || x ||P

Furthermore, since Q = AT P + PA is symmetrical and positive definite, then

According to Case (2) in Lemma 2.2, we have the following results for the case γ ≥ 1. The proof is similar to the one in Theorem 3.1 and we omit it here.

where M =

= || I + Ck ||P = sup

Corollary 3.6. Letting θ = inf {tk − tk−1 } > 0 and assume all k∈N

||2 )M] + µ2 (DAD ) −1

the eigenvalues of matrix A have negative real parts. Suppose that there exist a constant γ ≥ 1 and two symmetrical positive 4

definite matrices P, Q ∈ Rn×n such that:

where W(t, t0 ) is the Cauchy matrix of the impulsive linear system (6). Employing Case (1) in Lemma 2.2 and (2), we have:

AT P + PA = −Q, s λmax (P) || I + Ck ||2 ≤ γ, λmin (P)

|| x(t) ||≤|| Z t W(t, 0) || · || x(0) || + || W(t, s) || [|| ∆A || · || x(s) || +(|| B || + || ∆B ||) 0

× || xµ1 (s − r(s)) || Z

λmin (Q) γ || PB ||2 γM || PC ||2 ln γ − + + < 0. θ 2λmax (P) λmin (P) λmin (P)

+(|| C || + || ∆C ||) ≤

Then we consider the following nonlinear uncertain impulsive control system with infinite delays:

(15) Without the less of generality, we also assume that || φ ||α > 0. Then it is easily observed that

(13)

|| x(t) ||≤|| φ ||α <

|| x(t) ||<

k∈N

exists a constant 0 < γ < 1 satisfy || I + Ck ||≤ γ and

|| x(t) ||<

| h(s) | eµ2 ηs ds. Then the zero solution of the

(17)

||φ||α −(λ−ε)t∗ , γ e

||φ||α −(λ−ε)t ,t γ e

< t∗ .

(18) (19)

|| φ ||α (µ(A)+ lnργ )t∗ || x(t∗ ) ||≤ e Z t∗ γ 1 (µ(A)+ lnργ )(t∗ −s) [a || x(s) || +(b+ || B ||) || xµ1 (s + e 0 γ Z +∞ −r(s)) || +(c+ || C ||) | h(ξ) | · || xµ2 (s − ξ) || dξ]ds

system (13) is robustly exponentially stable. µ1 −1

µ2 −1

Proof. Since γa + (b+||B||)N + (c+||C||)MN + µ(A) + γ µ1 γ µ2 then we choose small enough λ ∈ (0, η) such that µ2 −1 (b+||B||)N µ1 −1 eµ1 λτ + (c+||C||)MN γ µ1 γ µ2 + lnργ + λ < 0.

ln γ ρ

< 0,

+ µ(A)

0

|| φ ||α (µ(A)+ lnργ )t∗ ≤ e γZ ∗ t 1 (µ(A)+ lnργ )(t∗ −s) || φ ||α −(λ−ε)s + e [a e γ 0 γ µ 1 α + (b+||B||)||φ|| × e−µ1 (λ−ε)(s−r(s)) γ µ1 Z +∞ || φ ||µα2 −µ2 (λ−ε)(s−ξ) +(c+ || C ||) | h(ξ) | e dξ]ds γµ2 0 || φ ||α (µ(A)+ lnργ )t∗ ≤ e γ Z t∗ ln γ a ×[1 + e(−µ(A)− ρ )s [ e−(λ−ε)s γ 0 µ −1 1 α + (b+||B||)||φ|| × e−µ1 (λ−ε)(s−r(s)) µ γ1 Z +∞ µ2 −1 α + (c+||C||)||φ|| | h(ξ) | e−µ2 (λ−ε)(s−ξ) dξ]ds] γ µ2

Furthermore, for any ε ∈ (0, λ), we have µ1 −1 µ1 (λ−ε)τ

+ (b+||B||)Nγµ1 e + + lnργ + (λ − ε) < 0.

(c+||C||)MN µ2 −1 γ µ2

+ µ(A)

(14)

By the formula for the variation of parameters, the solution of (13) can be presented by: x(t) = W(t, Z 0)x(0) t

≥ 0.

From (15), (19) and (14), we get,

0

+

||φ||α −(λ−ε)t ,t γ e

|| x(t∗ ) ||≥

a (b+ || B ||)N µ1 −1 (c+ || C ||)MN µ2 −1 ln γ + + + µ(A) + < 0, γ γµ1 γµ2 ρ

a γ

(16)

If this is not true, by (16) and the piecewise continuities of x(t), then there must exist t∗ > 0 such that:

Theorem 3.3. Let ρ = sup{tk − tk−1 } < ∞. Suppose that there

+

≤ 0.

0

we can easily obtain:

a γ

||φ||α −(λ−ε)t ,t γ e

In the following, we shall prove that:

where µ1 ≥ 1, µ2 ≥ 1. SupposeZthat there exist constants N > +∞ 0, η > 0 such that || φ ||≤ N and | h(s) | eµ2 ηs ds < ∞, then

where M =

1 (µ(A)+ lnργ )(t−s) 1 (µ(A)+ lnργ )t e || φ ||α + e [a || x(s) || γ 0 γ µ1 +(b+ || B ||)× || x (s − r(s)) || +(c+ || C ||) Z +∞ × | h(ξ) | · || xµ2 (s − ξ) || dξ]ds, t ≥ 0. 0

 x˙(t) = [A + ∆A]x(t) + [B + ∆B]xµ1 (t − r(t))   Z +∞       +[C + ∆C] h(s)xµ2 (t − s)ds,    0      ∆x(tk ) = x(tk+ ) − x(tk− ) = Ck x(tk− ), k ∈ N,       x(t) = φ(t), α ≤ t ≤ 0,

+∞

| h(ξ) | · || xµ2 (s − ξ) || dξ]ds

0 Z t

Then the zero solution of (5) is robustly exponentially stable.

Z

+∞

W(t, s)[∆Ax(s) + (B + ∆B)xµ1 (s − r(s))

0

+(C + ∆C)

Z

+∞

h(ξ)xµ2 (t − s)dξ]ds, t ≥ 0,

0

0

5

≤

|| φ ||α (µ(A)+ lnργ )t∗ e γ Z ∗ t

×[1 +

e(−µ(A)−

0

µ1 −1

α + (b+||B||)||φ|| γ µ1

µ2 −1

α + (c+||C||)||φ|| γ µ2

e(−µ(A)− µ1 −1

α + (b+||B||)||φ|| γ µ1

µ2 −1

α + (c+||C||)||φ|| γ µ2

ln γ ρ −(λ−ε))s

[

a γ

0

µ −1

1 α + (b+||B||)||φ|| γ µ1

e

µ1 −1

Z

t∗

e(−µ(A)−

ln γ ρ −(λ−ε))s

[

0

µ1 (λ−ε)τ

|| φ ||α (µ(A)+ lnργ )t∗ e [1 + ≤ γ

≤

Remark 3.4. In [23, 24], the authors studied the stability of impulsive control systems with infinite delays, but only uniform stability were considered. In this paper, some results on exponential stability of impulsive control systems with infinite delays are presented, and moreover, it can be applied to uncertain impulsive control systems. Thus our development results are more general than those studied in [23, 24]. In addition, impulsive control in [25] are developed to the study of periodic solution of impulsive neural networks with infinite delays and it showed that periodic attractor can be derived via proper impulsive control. It is worth mentioning that the methods used in this paper can be applied to the study of periodic attractor of impulsive neural networks with infinite delays. We will do some further research work in this topic in the near future.

× eµ1 (λ−ε)τ Z +∞ | h(ξ) | eµ2 ηξ dξ]ds]

|| φ ||α (µ(A)+ lnργ )t∗ ≤ e [1 + γ

+ (b+||B||)N γ µ1

a γ

0

t

0

[

× e(λ−ε)(s−µ1 s+µ1 r(s)) Z +∞ | h(ξ) | e(λ−ε)(s−µ2 s+µ2 ξ) dξ]ds]

|| φ ||α (µ(A)+ lnργ )t∗ ≤ e γ Z ∗ [1 +

ln γ ρ −(λ−ε))s

than the ones in Theorems 3.1 and 3.2 since they require the bounded of the trivial function φ. Thus they are complementary to each other.

Z

+ t∗

µ −1

(c+||C||)M||φ||α2 γ µ2

]ds]

(−µ(A)− lnργ −(λ−ε))s

e 0

eµ1 (λ−ε)τ +

a γ

4. Applications

a [ γ

In this section, we present two numerical examples to illustrate that our results can be applied to stabilize the unstable continuous system by using impulsive control.

(c+||C||)MN µ2 −1 ]ds] γ µ2

|| φ ||α (µ(A)+ lnργ )t∗ a (b+ || B ||)N µ1 −1 µ1 (λ−ε)τ e [1 + ( + e γ γ γµ1 ln γ (c+||C||)MN µ2 −1 −1 ) × (−µ(A) − ρ − (λ − ε)) + γ µ2 ×[e(−µ(A)−

ln γ ∗ ρ −(λ−ε))t

Example 4.1. Consider the following uncertain impulsive control system   x˙(t) = [A + ∆A]x(t) + [B + ∆B]x(t − r(t))   Z +∞       +[C + ∆C] h(s)x(t − s)ds, t , tk ,  0      + − −    ∆x(tk ) = x(tk ) − x(tk ) = Ck x(tk ), k ∈ N,

− 1]]

lnγ ∗ || φ ||α (µ(A)+ lnργ )t∗ < e [1 + e(−µ(A)− ρ −(λ−ε))t − 1] γ || φ ||α −(λ−ε)t∗ = e . γ

where r(t) ∈ [0, τ], τ is any given positive constant, h(s) = 0.1e−1.2s , s > 0, k∆Ak ≤ 0.1, k∆Bk ≤ 0.2, k∆Ck ≤ 0.3 and " # " # 1.2 −1.1 −1.3 0.7 A= , B= , 0.7 0.8 −0.9 0.5

This contradicts (18), and so the estimate (17) holds. Letting ε → 0, we have || x(t) ||≤

|| φ ||α −λt e , t≥0 γ

" C=

The proof is complete. According to Case (2) in Lemma 2.2, we have the following results for the case γ ≥ 1. Theorem 3.4. Let θ = inf {tk − tk−1 } > 0. Suppose that there

where M =

+∞

#

" , Ck =

−0.4 0.1 −0.1 −0.5

# .

k∈N

of the system (20) is robustly exponentially stable if there exists constant γ ∈ (0.6660, 1) such that

exists a constant γ ≥ 1 satisfy || I + Ck ||≤ γ and

Z

−0.1 −0.22 0.43 0.65

Property 4.1. Let ρ = sup{tk − tk−1 } < ∞. The zero solution

k∈N

γa+γµ1 (b+ || B ||)N µ1 −1 +γµ2 (c+ || C ||)MN µ2 −1 +µ(A)+

(20)

1.8615 ln γ + < −1.1502. γ ρ

ln γ < 0, θ

Proof. Let η = 0.2, then

| h(s) | eµ2 ηs ds. Then the zero solution of

0

M = 0.1

the system (13) is robustly exponentially stable.

Z

+∞

e−s ds = 0.1.

0

Remark 3.3. It should be noted that the development results in Theorems 3.3 and 3.4 can be applied to the case of nonlinear systems, which are more general than the ones in Theorems 3.1 and 3.2. But the corresponding conditions are more restrictive

Choose a symmetric positive definite matrix P as follows: " # 0.16 −0.08 P= −0.08 0.29 6

(21)

such that P = DT D, where " # 0.4 −0.2 D= . 0 0.5

where Q = AT P + PA. So by (23), it can be deduced that ln γ λmax (Q) ||PB||2 M||PC||2 ρ + 2λmin (P) + γλmin (P) + γλmin (P) ln γ = 20.0385 + ρ + 8.5502 < 0. γ

By a simple calculation, it can be deduced that

Suppose in addition that the impulsive times occur with a frequency of ρ = 0.01. Also, a simple check shows that both of these conditions are satisfied by choosing γ = 0.6. Therefore, robust exponential stability of the zero solution of (22) can be obtained by Corollary 3.5.

|| I + DCk D−1 ||2 = 0.6660, || DBD−1 ||2 = 1.4110, || DCD−1 ||2 = 1.2048, µ2 (DAD−1 ) = 1.1502. Note that a = 0.1, b = 0.2, c = 0.3, it follows from (21) that a+b+||DBD−1 ||2 +(c+||DCD−1 ||2 )M + µ2 (DAD−1 ) γ ln γ 1.8615 = γ + ρ + 1.1502 < 0.

+

ln γ ρ

5. Conclusion

Especially, the impulsive times occur with a frequency of ρ = 0.08 and a simple check shows that both of these conditions are satisfied by choosing γ = 0.7. So by Corollary 3.2, one may obtain that the zero solution of (20) is robustly exponentially stable.

In this paper, the robust stability problem of uncertain impulsive control systems with infinite delays is concerned. By employing the formula for the variation of parameters and estimating the Cauchy matrix, several criteria on robust exponential stability are derived, which are less restrictive than those in the earlier literature.Two examples are given to illustrate the effectiveness of the theoretical results. In addition, we should point out that the method used in this paper can be applied to the study of periodic solution of impulsive neural networks with infinite delays via impulsive control. We will do further research in this direction.

Example 4.2. Consider the following deterministic impulsive control system   x˙(t) = Ax(t) + Bx(t − r(t))   Z +∞       +C h(s)x(t − s)ds, t , tk , (22)  0      + − −    ∆x(tk ) = x(tk ) − x(tk ) = Ck x(tk ), k ∈ N,

6. Funding

where r(t) is the time-varying delay function with 0 ≤ r(t) ≤ τ, h(s) = e−1.5s , s > 0 and with the following parameter matrices:      −1 0 1   1 −2 2      A =  2 3 −1  , B =  −2 −1 −1  ,     2 0 4 5 −1 1     −1.1 0.1  2 0 1  0    −1.1 0.1 C =  −1 1 −2  , Ck =  0    0.2 0 −1.2 −3 0 1

This work was jointly supported by National Natural Science Foundation of China (No. 11301308), China Postdoctoral Science Foundation founded project (2014M561956,2015T80737) and Research Fund for International Cooperation Training Programme of Excellent Young Teachers of Shandong Normal University,National Natural Science Foundation of China (No. 61471226, No. 61201441), research funding from Shandong Province (JQ201516), and research funding from Jinan City (No. 201401221, No. 20120109).

    .

Property 4.2. The zero solution of the system (22) is robustly exponentially stable if there exists constant γ ∈ (0.5801, 1) such that 20.0385 ln γ + < −8.5502. (23) γ ρ

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Proof. Let η = 0.5, then M=

Z

+∞

e−s ds = 1.

0

Choose a symmetric positive definite matrix P as follows:    0.1929 −0.0893 −0.0786    0.0714  . P =  −0.0893 0.2500   −0.0786 0.0714 0.1643 We directly calculate the following parameters: || PB ||2 = 0.6518, || PC ||2 = 1.3240, || I + Ck ||2 = 0.3000, qλmin (P) = 0.0986,

λmax (P) = 0.3687, λmax (Q) = 1.6861,

λmax (P) λmin (P)

|| I + Ck ||2 = 0.5801, 7

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