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An explicit criterion for finite-time stability of linear nonautonomous systems with delays
Applied Mathematics Letters 30 (2014) 12–18
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Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
An explicit criterion for finite-time stability of linear nonautonomous systems with delays Le Van Hien ∗ Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Road, Hanoi, Viet Nam
article
info
Article history: Received 24 September 2013 Received in revised form 13 December 2013 Accepted 13 December 2013 Keywords: Finite-time stability Nonautonomous systems Time-varying delay Metzler matrix
abstract In this paper, the problem of finite-time stability of linear nonautonomous systems with time-varying delays is considered. Using a novel approach based on some techniques developed for linear positive systems, we derive new explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. These conditions are shown to be relaxed for the Lyapunov asymptotic stability. A numerical example is given to illustrate the effectiveness of the obtained result. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction The stability of time delay systems has been one of the most attractive research topics during the past decades [1–5]. While the concept of Lyapunov stability, recognized as infinite time behavior, has been well investigated and developed, the concept of finite-time stability (FTS) (or short-time stability) has been extensively studied in recent years (see, [6–14] and the references therein). Roughly speaking, a system is finite-time stable if, for a given bound on the initial condition, its state trajectories do not exceed a certain threshold during a pre-specified time interval [7]. It is noted that, a system may be finite-time stable but not Lyapunov asymptotic stable, and vice versa [6,7,13] (see, also, Remark 2.1 in this paper). Although the Lyapunov asymptotic stability (LAS) has been successfully applied in many models, FTS is a useful concept to study in many practical systems in the vivid world [8–10,12,13]. However, most of the existing results in the literature so far have been devoted to linear autonomous (i.e. time-invariant) systems. For linear time-invariant systems with constant delay, some finite-time stability conditions have been derived in terms of feasible linear matrix inequalities based on the main approach is the Lyapunov–Krasovskii functional method [6,10–14]. There has been no result concerned with the FTS of nonautonomous systems (time-varying systems) with timevarying delays. Moreover, it should be noted that, the proposed conditions for FTS of time-varying systems based on the Lyapunov functional approach have usually been derived in terms of Lyapunov or Riccati matrix differential equations [7–9] which lead to indefinite matrix inequalities with lack of efficient computational tools to solve them. Therefore, an alternative approach when dealing with the FTS of time-varying systems with delays is clearly needed, which has motivated our present investigation. In this paper, we consider the problem of FTS of linear nonautonomous systems with discrete and distributed timevarying delays. By utilizing some techniques developed for linear positive systems, we derive new explicit conditions in
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L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18
13
terms of matrix inequalities ensuring that, for each given bound on the initial conditions, the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. Our conditions are derived in terms of some types of the Metzler matrix which can be easily verified. The novel feature of the result obtained in this paper is twofold. Firstly, the system considered in this paper is time-varying subjected to interval, nondifferentiable delays, which means that the lower and the upper bounds for the time-varying delays are available but the delay functions are not necessary to be differentiable. This allows that the time-delays can be fast time-varying functions. Secondly, by a novel approach without using the Lyapunov–Krasovskii functional method, we derive an explicit criterion for the FTS of the system in terms of Metzler matrix inequalities which is intuitive and easy to verify. Notations. For a given positive integer n, we denote n := {1, 2, . . . , n}. Rn denotes the n-dimensional space with the norm ∥x∥∞ = maxi∈n |xi |. The set of real m × n-matrices is denoted by Rm×n . For u = (ui ), v = (vi ) in Rn , u ≥ v iff ui ≥ vi , ∀i ∈ n; u ≫ v iff ui > vi , ∀i ∈ n. We denote a vector e = (1 . . . 1)T ∈ Rn .
2. Problem statement and preliminaries Consider the following linear nonautonomous system with time-varying delays x˙ (t ) = A(t )x(t ) + D(t )x(t − τ (t )) + G(t ) x(t ) = φ(t ),
t
t −κ(t )
x(s)ds,
t ≥ 0,
(2.1)
t ∈ [−d, 0],
where x(t ) ∈ Rn is the state; A(t ) = (aij (t )) ∈ Rn×n , D(t ) = (dij (t )) ∈ Rn×n and G(t ) = (gij (t )) ∈ Rn×n are the system matrices; τ (t ), κ(t ) are time-varying delays satisfying 0 ≤ τ ≤ τ (t ) ≤ τ , 0 ≤ κ(t ) ≤ κ , t ≥ 0; φ(t ) = (φi (t )) ∈ C ([−d, 0], Rn ), where d = max{τ , κ}, is the initial condition. Let us denote |φi | = sup−d≤t ≤0 |φi (t )| and ∥φ∥∞ = maxi∈n |φi |. Definition 2.1. For given a time T > 0 and positive numbers r1 < r2 , system (2.1) is said to be finite-time stable with respect to (r1 , r2 , T ) if for any initial condition φ(t ) ∈ C ([−d, 0], Rn ), ∥φ∥∞ ≤ r1 implies that ∥x(t , φ)∥∞ < r2 for all t ∈ [0, T ]. Remark 2.1. It should be noted that, FTS and LAS are independent concepts in the following sense: a system which is FTS may be not LAS, and vice versa. This will be illustrated in the following example. Example 2.1. Consider the following delay differential equations x˙ (t ) = −1.2x(t ) + x˙ (t ) = −0.8x(t ) +
t +2 t +1 t t +6
x(t − 1),
t ≥ 0,
(2.2)
x(t − 1),
t ≥ 0.
(2.3)
Eq. (2.2) is globally LAS. However, this equation is not FTS with respect to r1 = 1, r2 = 1.25 and T = 10. Conversely, Eq. (2.3) is FTS with respect to r1 = 1, r2 = 1.5 and T = 10 but (2.3) is not LAS, even every non-zero solution of (2.3) goes to infinity as time tends to infinity. The state trajectories of (2.2) and (2.3) with initial condition φ(t ) = 1, t ∈ [−1, 0], are presented in Figs. 1 and 2, respectively. The main purpose of this paper is to find conditions for the stability of system (2.1) over a finite time interval [0, T ]. By utilizing some techniques developed for positive systems, some new explicit conditions are derived in terms of the Metzler matrix for the FTS of system (2.1). 3. Main results Let A(t ) = (aij (t )), D(t ) = (dij (t )) and G(t ) = (gij (t )) be given matrices with continuous elements. We make the following assumptions which are usually used for time-varying systems (see, for example, [3]). For given T > 0, assume that A1. aii (t ) ≤ aii , i ∈ n, |aij (t )| ≤ aij , i ̸= j, i, j ∈ n, t ∈ [0, T ]. A2. |dij (t )| ≤ dij , |gij (t )| ≤ g ij , t ∈ [0, T ], i, j ∈ n. We denote A = γ I + e−γ τ D + κ G.
aij , D =
dij and G =
g ij . For a nonnegative scalar γ , let us define the matrix Mγ = A −
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L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18
Fig. 1. A state trajectory of Eq. (2.2).
Fig. 2. A state trajectory of Eq. (2.3).
We are now in a position to state our main result as follows. Theorem 3.1. Let assumptions A1, A2 hold. Then for given 0 < r1 < r2 and T > 0, system (2.1) is finite-time stable with respect to (r1 , r2 , T ) if there exist γ ≥ 0, positive numbers λ1 , λ2 and a vector ξ ∈ Rn satisfying the following conditions
Mγ ξ ≪ 0,
(3.1a)
λ1 e ≤ ξ ≤ λ2 e, λ2 r2 < e−γ (T +d) . λ1 r1
(3.1b) (3.1c)
L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18
15
Remark 3.1. It is worth noting that, if condition (3.1a) is satisfied with γ = 0 then system (2.1) is exponentially stable in the sense of Lyapunov (see the Appendix). Then we obtain a similar result in [3]. Moreover, in this case, system (2.1) is finite-time stable with respect to (r1 , r2 , T ) for any 0 < r1 < r2 , T > 0. Proof. It is necessary to prove for the case γ > 0. Let ξ ∈ Rn satisfies (3.1a), (3.1b). Then we have
A − γ I + e−γ τ D + κ G ξ ≪ 0
and thus n
aij + dij e−γ τ + κ g ij ξj ≤ γ ξi ,
∀i ∈ n.
(3.2)
j=1
In the following, we will use x(t ) to denote the solution x(t , φ) if it does not make any confusion. At first, from (2.1) we have D+ |xi (t )| = sgn(xi (t ))˙xi (t )
≤ aii (t )|xi (t )| +
n
|aij (t )∥xj (t )| +
j=1,j̸=i
≤ aii |xi (t )| +
n
n
|dij (t )∥xj (t − τ (t ))| +
j =1
aij |xj (t )| +
j=1,j̸=i
n
n
|gij (t )|
j =1
dij |xj (t − τ (t ))| +
j =1
n j =1
t −κ(t )
|xj (s)|ds
t
g ij
t
t −κ(t )
|xj (s)|ds,
∀t ≥ 0, i ∈ n,
(3.3)
where D+ denotes the Dini upper-right derivative. Next, let us consider the functions vi (t ), i ∈ n, as follows eγ d
vi ( t ) =
λ1
∥φ∥∞ ξi eγ t ,
t ≥ −d.
(3.4)
It is noted that, for all t ≥ 0 and j ∈ n, we have
vj (t − τ (t )) =
eγ d
λ1
∥φ∥∞ ξj eγ (t −τ (t )) ≤
eγ d
λ1
∥φ∥∞ ξj eγ t e−γ τ ≤ e−γ τ vj (t )
(3.5)
and
t
vj (s)ds =
t −κ(t )
eγ d
λ1
∥φ∥∞ ξj
1 − e−γ κ(t ) γ t e ≤ κvj (t )
(3.6)
γ
−γ κ −γ κ by using the fact that 1−eγ ≤ limγ ↓0+ 1−eγ = κ for any κ ≥ 0. Therefore, from (3.2), (3.4)–(3.6), we obtain
n
aii vi (t ) +
aij vj (t ) +
j=1,j̸=i
γt
≤ ηe
n
dij vj (t − τ (t )) +
j =1
aii ξi +
n
aij ξj +
j=1,j̸=i
≤ ηeγ t
n j=1
n
−γ τ
ξj dij e
+
j =1
n
n
g ij
t t −κ(t )
vj (s)ds
κ g ij ξj ,
j =1
aij + dij e−γ τ + κ g ij ξj
j =1
≤ ηγ ξi eγ t ,
∀ t ≥ 0 , i ∈ n,
(3.7)
eγ d
where η = λ ∥φ∥∞ . Thus, it follows from (3.7) that 1
v˙ i (t ) ≥ aii vi (t ) +
n j=1,j̸=i
aij vj (t ) +
n j =1
dij vj (t − τ (t )) + g ij
t t −κ(t )
vj (s)ds,
t ≥ 0.
(3.8)
We will prove that |xi (t )| ≤ vi (t ), ∀t ∈ [0, T ], i ∈ n. Let ρi (t ) = |xi (t )| − vi (t ), t ≥ −d. Noticing that, for t ∈ [−d, 0], we have |xi (t )| ≤ |φi | ≤ ηξi eγ t = vi (t ), and hence, ρi (t ) ≤ 0, for all t ∈ [−d, 0], i ∈ n. Assume that there exists an index
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L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18
i ∈ n and t1 ∈ (0, T ] such that ρi (t1 ) = 0, ρi (t ) > 0, t ∈ (t1 , t1 + δ) for some δ > 0 and ρj (t ) ≤ 0, ∀t ∈ [−d, t1 ], j ∈ n. Then D+ ρi (t1 ) > 0. However, from (3.3) and (3.8), it follows for t ∈ [0, t1 ) that n
D ρi (t ) ≤ aii ρi (t ) + +
aij ρj (t ) +
j=1,j̸=i
n
dij ρj (t − τ (t )) + g ij
t
t −κ(t )
j =1
ρj (s)ds ≤ aii ρi (t ),
and therefore, D+ ρi (t1 ) ≤ 0, which yields a contradiction. This shows that ρi (t ) ≤ 0 for all t ≥ 0, i ∈ n, and thus, we readily γd
obtain |xi (t )| ≤ eλ ∥φ∥∞ ξi eγ t , ∀t ≥ 0, i ∈ n. Consequently, 1
∥x(t )∥∞ ≤
eγ d
λ1
∥φ∥∞ ∥ξ ∥∞ eγ t ≤
λ 2 eγ d ∥φ∥∞ eγ t , λ1
t ∈ [0, T ].
(3.9)
λ
If ∥φ∥∞ ≤ r1 then, by (3.9), ∥x(t )∥∞ ≤ λ2 r1 eγ (T +d) < r2 , ∀t ∈ [0, T ]. This shows that system (2.1) is finite-time stable with 1 respect to (r1 , r2 , T ). The proof is completed. Remark 3.2. Since Mγ is a Metzler matrix (see [3,15] for more details), condition (3.1a) can be checked by various criteria. Condition (3.1b) can be relaxed in the following sense, if ξ ∈ Rn satisfies (3.1a) then ξ also satisfies (3.1b) with λ1 = mini∈n ξi , λ2 = ∥ξ ∥∞ . However, in order to use the parameter searching method in Matlab, we impose condition (3.1b) for improving FTS parameters r1 , r2 , T and the delay size d. Remark 3.3. When the finite-time stability parameters r1 , r2 , T are pre-specified, one seeks to find the maximum allowable delay (MAD) dmax . The following optimization problem can be formulated to find MAD min γ > 0, λ > 0 subject to
Mγ ξ ≪ 0
e ≤ ξ ≤ λe.
Then the MAD can be determined by dmax =
1
γ
ln
r2
λ r1
− T.
4. An illustrative example Consider the following system x˙ (t ) = A(t )x(t ) + D(t )x(t − τ (t )) + G(t )
t t −κ(t )
x(s)ds,
t ≥ 0,
(4.1)
where A(t ) =
−4 sin2 2t
|cos t | , −4
D(t ) =
sin2 t 0
√ cos 2 t cos2 t
and τ (t ) = |sin 4t |, κ(t ) = |cos t |. System (4.1) satisfies A1, A2 for all T > 0 and we have A =
,
−4 1
G(t ) =
1 −4
√ sin t
,D =
sin2 3t
1 0
1 1
0
,
|cos 2t |
,G =
1 1
0 1
. It should be noted
that system (4.1) does not satisfy the Lyapunov stability conditions proposed in [3]. More precisely, in this case the matrix M = A + D + κ G is not invertible, and hence does not satisfy conditions of Theorem 2.5 in [3]. However, Mγ = M − γ I
2 ξ1 < ξ2 < 2 ξ1 . Let us satisfies (3.1a) for any γ > 0 and the domain of the solution ξ ∈ R2 of (3.1a) is defined by 2+γ take r1 = 1, r2 = 1.25 and γ = 0.01 then system (4.1) is finite-time stable with respect to (r1 , r2 , T ) for any finite time 0 < T < Tmax = 21.3144. 2+γ
Remark 4.1. Since the matrix Mγ satisfies (3.1a) for any γ > 0, limγ ↓0+ γ1 ln r2 = +∞ for all 0 < r1 < r2 , then for given 1 r T > 0, there exists γ > 0 such that 1 < r2 e−γ (T +d) . Therefore, system (4.1) is finite-time stable with respect to (r1 , r2 , T ) 1 for any 0 < r1 < r2 , T > 0. However, system (4.1) is not LAS as shown in Fig. 3. r
5. Conclusion This paper has dealt with the problem of finite-time stability for a class of linear nonautonomous systems with timevarying delays. An explicit criterion for the finite-time stability of the system has been proposed in terms of matrix inequalities for a type of Metzler matrix. The effectiveness of the proposed conditions has been illustrated by a numerical example.
L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18
17
Fig. 3. A state trajectory of system (4.1).
Acknowledgments The authors would like to thank the editors and anonymous reviewers for their constructive comments. This work was supported by the NAFOSTED of Vietnam (101.01-2011.51) and the Ministry of Education and Training of Vietnam (B2013.17.42). Appendix. Proof for the claim in Remark 3.1 Let ξ ∈ Rn , ξ ≫ 0, such that (A + D + κ G)ξ ≪ 0. Then there exists η > 0 such that n
aij + dij + κ g ij ξj ≤ −η,
∀ i ∈ n.
j=1
Therefore, there exists γ ∗ > 0 satisfying
γ ξi +
n
eγ τ − 1 dij +
j=1
eγ κ − 1
γ
− κ g ij ξj − η ≤ 0,
∀i ∈ n, γ ∈ (0, γ ∗ ].
By the same arguments used in (3.3)–(3.9), we obtain
|xi (t )| ≤
1
ξmin
∥φ∥∞ ξi e−γ t ,
∀i ∈ n, t ≥ 0,
where ξmin = mini∈n ξi . Therefore, ∥x(t )∥∞ ≤ exponentially stable. The proof is completed.
∥ξ ∥∞ ∥φ∥∞ e−γ t , t ξmin
≥ 0, which shows that system (2.1) is Lyapunov
References [1] T. Erneux, Applied Delay Differential Equations, Springer, New York, 2009. [2] V.N. Phat, L.V. Hien, An application of Razumikhin theorem to exponential stability for linear non-autonomous systems with time-varying delay, Appl. Math. Lett. 22 (2009) 1412–1417. [3] P.H.A. Ngoc, On exponential stability of nonlinear differential systems with time-varying delay, Appl. Math. Lett. 25 (2012) 1208–1213. [4] B. Song, J.H. Park, Z.G. Wu, Y. Zhang, New results on delay-dependent stability analysis for neutral stochastic delay systems, J. Franklin Inst. 350 (2013) 840–852. [5] A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: application to time-delay systems, Automatica 49 (2013) 2860–2866. [6] F. Amato, M. Ariola, C. Cosentino, Finite-time stabilization via dynamic output feedback, Automatica 42 (2006) 337–342. [7] F. Amato, R. Ambrosino, M. Ariola, C. Cosentino, Finite-time stability of linear time-varying systems with jumps, Automatica 45 (2009) 1354–1358. [8] G. Garcia, S. Tarbouriech, J. Bernussou, Finite-time stabilization of linear time-varying continuous systems, IEEE Trans. Automat. Control 54 (2009) 364–369. [9] F. Amato, M. Ariola, C. Cosentino, Finite-time control of discrete-time linear systems: analysis and design conditions, Automatica 46 (2010) 919–924. [10] S. He, F. Liu, Observer-based finite-time control of time-delayed jump systems, Appl. Math. Comput. 217 (2010) 2327–2338. [11] W. Xiang, J. Xiao, M.N. Iqbal, Robust finite-time bounded observer design for a class of uncertain non-linear Markovian jump systems, IMA J. Math. Control Inform. 29 (2012) 551–572. [12] Y. Zhang, C. Liu, X. Mu, Robust finite-time stabilization of uncertain singular Markovian jump systems, Appl. Math. Model. 36 (2012) 5109–5121.
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[13] X. Lin, H. Du, S. Li, Y. Zou, Finite-time boundedness and finite-time l2 gain analysis of discrete-time switched linear systems with average dwell time, J. Franklin Inst. 350 (2013) 911–928. [14] L. Hou, G. Zong, Y. Wu, Observer-based finite-time exponential l2 − l∞ control for discrete-time switched delay systems with uncertainties, Trans. Inst. Meas. Control 35 (2013) 310–320. [15] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
An explicit criterion for finite-time stability of linear nonautonomous systems with delays Le Van Hien ∗ Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Road, Hanoi, Viet Nam
article
info
Article history: Received 24 September 2013 Received in revised form 13 December 2013 Accepted 13 December 2013 Keywords: Finite-time stability Nonautonomous systems Time-varying delay Metzler matrix
abstract In this paper, the problem of finite-time stability of linear nonautonomous systems with time-varying delays is considered. Using a novel approach based on some techniques developed for linear positive systems, we derive new explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. These conditions are shown to be relaxed for the Lyapunov asymptotic stability. A numerical example is given to illustrate the effectiveness of the obtained result. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction The stability of time delay systems has been one of the most attractive research topics during the past decades [1–5]. While the concept of Lyapunov stability, recognized as infinite time behavior, has been well investigated and developed, the concept of finite-time stability (FTS) (or short-time stability) has been extensively studied in recent years (see, [6–14] and the references therein). Roughly speaking, a system is finite-time stable if, for a given bound on the initial condition, its state trajectories do not exceed a certain threshold during a pre-specified time interval [7]. It is noted that, a system may be finite-time stable but not Lyapunov asymptotic stable, and vice versa [6,7,13] (see, also, Remark 2.1 in this paper). Although the Lyapunov asymptotic stability (LAS) has been successfully applied in many models, FTS is a useful concept to study in many practical systems in the vivid world [8–10,12,13]. However, most of the existing results in the literature so far have been devoted to linear autonomous (i.e. time-invariant) systems. For linear time-invariant systems with constant delay, some finite-time stability conditions have been derived in terms of feasible linear matrix inequalities based on the main approach is the Lyapunov–Krasovskii functional method [6,10–14]. There has been no result concerned with the FTS of nonautonomous systems (time-varying systems) with timevarying delays. Moreover, it should be noted that, the proposed conditions for FTS of time-varying systems based on the Lyapunov functional approach have usually been derived in terms of Lyapunov or Riccati matrix differential equations [7–9] which lead to indefinite matrix inequalities with lack of efficient computational tools to solve them. Therefore, an alternative approach when dealing with the FTS of time-varying systems with delays is clearly needed, which has motivated our present investigation. In this paper, we consider the problem of FTS of linear nonautonomous systems with discrete and distributed timevarying delays. By utilizing some techniques developed for linear positive systems, we derive new explicit conditions in
∗
Tel.: +84 912494391. E-mail address: [email protected].
0893-9659/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.aml.2013.12.005
L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18
13
terms of matrix inequalities ensuring that, for each given bound on the initial conditions, the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. Our conditions are derived in terms of some types of the Metzler matrix which can be easily verified. The novel feature of the result obtained in this paper is twofold. Firstly, the system considered in this paper is time-varying subjected to interval, nondifferentiable delays, which means that the lower and the upper bounds for the time-varying delays are available but the delay functions are not necessary to be differentiable. This allows that the time-delays can be fast time-varying functions. Secondly, by a novel approach without using the Lyapunov–Krasovskii functional method, we derive an explicit criterion for the FTS of the system in terms of Metzler matrix inequalities which is intuitive and easy to verify. Notations. For a given positive integer n, we denote n := {1, 2, . . . , n}. Rn denotes the n-dimensional space with the norm ∥x∥∞ = maxi∈n |xi |. The set of real m × n-matrices is denoted by Rm×n . For u = (ui ), v = (vi ) in Rn , u ≥ v iff ui ≥ vi , ∀i ∈ n; u ≫ v iff ui > vi , ∀i ∈ n. We denote a vector e = (1 . . . 1)T ∈ Rn .
2. Problem statement and preliminaries Consider the following linear nonautonomous system with time-varying delays x˙ (t ) = A(t )x(t ) + D(t )x(t − τ (t )) + G(t ) x(t ) = φ(t ),
t
t −κ(t )
x(s)ds,
t ≥ 0,
(2.1)
t ∈ [−d, 0],
where x(t ) ∈ Rn is the state; A(t ) = (aij (t )) ∈ Rn×n , D(t ) = (dij (t )) ∈ Rn×n and G(t ) = (gij (t )) ∈ Rn×n are the system matrices; τ (t ), κ(t ) are time-varying delays satisfying 0 ≤ τ ≤ τ (t ) ≤ τ , 0 ≤ κ(t ) ≤ κ , t ≥ 0; φ(t ) = (φi (t )) ∈ C ([−d, 0], Rn ), where d = max{τ , κ}, is the initial condition. Let us denote |φi | = sup−d≤t ≤0 |φi (t )| and ∥φ∥∞ = maxi∈n |φi |. Definition 2.1. For given a time T > 0 and positive numbers r1 < r2 , system (2.1) is said to be finite-time stable with respect to (r1 , r2 , T ) if for any initial condition φ(t ) ∈ C ([−d, 0], Rn ), ∥φ∥∞ ≤ r1 implies that ∥x(t , φ)∥∞ < r2 for all t ∈ [0, T ]. Remark 2.1. It should be noted that, FTS and LAS are independent concepts in the following sense: a system which is FTS may be not LAS, and vice versa. This will be illustrated in the following example. Example 2.1. Consider the following delay differential equations x˙ (t ) = −1.2x(t ) + x˙ (t ) = −0.8x(t ) +
t +2 t +1 t t +6
x(t − 1),
t ≥ 0,
(2.2)
x(t − 1),
t ≥ 0.
(2.3)
Eq. (2.2) is globally LAS. However, this equation is not FTS with respect to r1 = 1, r2 = 1.25 and T = 10. Conversely, Eq. (2.3) is FTS with respect to r1 = 1, r2 = 1.5 and T = 10 but (2.3) is not LAS, even every non-zero solution of (2.3) goes to infinity as time tends to infinity. The state trajectories of (2.2) and (2.3) with initial condition φ(t ) = 1, t ∈ [−1, 0], are presented in Figs. 1 and 2, respectively. The main purpose of this paper is to find conditions for the stability of system (2.1) over a finite time interval [0, T ]. By utilizing some techniques developed for positive systems, some new explicit conditions are derived in terms of the Metzler matrix for the FTS of system (2.1). 3. Main results Let A(t ) = (aij (t )), D(t ) = (dij (t )) and G(t ) = (gij (t )) be given matrices with continuous elements. We make the following assumptions which are usually used for time-varying systems (see, for example, [3]). For given T > 0, assume that A1. aii (t ) ≤ aii , i ∈ n, |aij (t )| ≤ aij , i ̸= j, i, j ∈ n, t ∈ [0, T ]. A2. |dij (t )| ≤ dij , |gij (t )| ≤ g ij , t ∈ [0, T ], i, j ∈ n. We denote A = γ I + e−γ τ D + κ G.
aij , D =
dij and G =
g ij . For a nonnegative scalar γ , let us define the matrix Mγ = A −
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L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18
Fig. 1. A state trajectory of Eq. (2.2).
Fig. 2. A state trajectory of Eq. (2.3).
We are now in a position to state our main result as follows. Theorem 3.1. Let assumptions A1, A2 hold. Then for given 0 < r1 < r2 and T > 0, system (2.1) is finite-time stable with respect to (r1 , r2 , T ) if there exist γ ≥ 0, positive numbers λ1 , λ2 and a vector ξ ∈ Rn satisfying the following conditions
Mγ ξ ≪ 0,
(3.1a)
λ1 e ≤ ξ ≤ λ2 e, λ2 r2 < e−γ (T +d) . λ1 r1
(3.1b) (3.1c)
L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18
15
Remark 3.1. It is worth noting that, if condition (3.1a) is satisfied with γ = 0 then system (2.1) is exponentially stable in the sense of Lyapunov (see the Appendix). Then we obtain a similar result in [3]. Moreover, in this case, system (2.1) is finite-time stable with respect to (r1 , r2 , T ) for any 0 < r1 < r2 , T > 0. Proof. It is necessary to prove for the case γ > 0. Let ξ ∈ Rn satisfies (3.1a), (3.1b). Then we have
A − γ I + e−γ τ D + κ G ξ ≪ 0
and thus n
aij + dij e−γ τ + κ g ij ξj ≤ γ ξi ,
∀i ∈ n.
(3.2)
j=1
In the following, we will use x(t ) to denote the solution x(t , φ) if it does not make any confusion. At first, from (2.1) we have D+ |xi (t )| = sgn(xi (t ))˙xi (t )
≤ aii (t )|xi (t )| +
n
|aij (t )∥xj (t )| +
j=1,j̸=i
≤ aii |xi (t )| +
n
n
|dij (t )∥xj (t − τ (t ))| +
j =1
aij |xj (t )| +
j=1,j̸=i
n
n
|gij (t )|
j =1
dij |xj (t − τ (t ))| +
j =1
n j =1
t −κ(t )
|xj (s)|ds
t
g ij
t
t −κ(t )
|xj (s)|ds,
∀t ≥ 0, i ∈ n,
(3.3)
where D+ denotes the Dini upper-right derivative. Next, let us consider the functions vi (t ), i ∈ n, as follows eγ d
vi ( t ) =
λ1
∥φ∥∞ ξi eγ t ,
t ≥ −d.
(3.4)
It is noted that, for all t ≥ 0 and j ∈ n, we have
vj (t − τ (t )) =
eγ d
λ1
∥φ∥∞ ξj eγ (t −τ (t )) ≤
eγ d
λ1
∥φ∥∞ ξj eγ t e−γ τ ≤ e−γ τ vj (t )
(3.5)
and
t
vj (s)ds =
t −κ(t )
eγ d
λ1
∥φ∥∞ ξj
1 − e−γ κ(t ) γ t e ≤ κvj (t )
(3.6)
γ
−γ κ −γ κ by using the fact that 1−eγ ≤ limγ ↓0+ 1−eγ = κ for any κ ≥ 0. Therefore, from (3.2), (3.4)–(3.6), we obtain
n
aii vi (t ) +
aij vj (t ) +
j=1,j̸=i
γt
≤ ηe
n
dij vj (t − τ (t )) +
j =1
aii ξi +
n
aij ξj +
j=1,j̸=i
≤ ηeγ t
n j=1
n
−γ τ
ξj dij e
+
j =1
n
n
g ij
t t −κ(t )
vj (s)ds
κ g ij ξj ,
j =1
aij + dij e−γ τ + κ g ij ξj
j =1
≤ ηγ ξi eγ t ,
∀ t ≥ 0 , i ∈ n,
(3.7)
eγ d
where η = λ ∥φ∥∞ . Thus, it follows from (3.7) that 1
v˙ i (t ) ≥ aii vi (t ) +
n j=1,j̸=i
aij vj (t ) +
n j =1
dij vj (t − τ (t )) + g ij
t t −κ(t )
vj (s)ds,
t ≥ 0.
(3.8)
We will prove that |xi (t )| ≤ vi (t ), ∀t ∈ [0, T ], i ∈ n. Let ρi (t ) = |xi (t )| − vi (t ), t ≥ −d. Noticing that, for t ∈ [−d, 0], we have |xi (t )| ≤ |φi | ≤ ηξi eγ t = vi (t ), and hence, ρi (t ) ≤ 0, for all t ∈ [−d, 0], i ∈ n. Assume that there exists an index
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i ∈ n and t1 ∈ (0, T ] such that ρi (t1 ) = 0, ρi (t ) > 0, t ∈ (t1 , t1 + δ) for some δ > 0 and ρj (t ) ≤ 0, ∀t ∈ [−d, t1 ], j ∈ n. Then D+ ρi (t1 ) > 0. However, from (3.3) and (3.8), it follows for t ∈ [0, t1 ) that n
D ρi (t ) ≤ aii ρi (t ) + +
aij ρj (t ) +
j=1,j̸=i
n
dij ρj (t − τ (t )) + g ij
t
t −κ(t )
j =1
ρj (s)ds ≤ aii ρi (t ),
and therefore, D+ ρi (t1 ) ≤ 0, which yields a contradiction. This shows that ρi (t ) ≤ 0 for all t ≥ 0, i ∈ n, and thus, we readily γd
obtain |xi (t )| ≤ eλ ∥φ∥∞ ξi eγ t , ∀t ≥ 0, i ∈ n. Consequently, 1
∥x(t )∥∞ ≤
eγ d
λ1
∥φ∥∞ ∥ξ ∥∞ eγ t ≤
λ 2 eγ d ∥φ∥∞ eγ t , λ1
t ∈ [0, T ].
(3.9)
λ
If ∥φ∥∞ ≤ r1 then, by (3.9), ∥x(t )∥∞ ≤ λ2 r1 eγ (T +d) < r2 , ∀t ∈ [0, T ]. This shows that system (2.1) is finite-time stable with 1 respect to (r1 , r2 , T ). The proof is completed. Remark 3.2. Since Mγ is a Metzler matrix (see [3,15] for more details), condition (3.1a) can be checked by various criteria. Condition (3.1b) can be relaxed in the following sense, if ξ ∈ Rn satisfies (3.1a) then ξ also satisfies (3.1b) with λ1 = mini∈n ξi , λ2 = ∥ξ ∥∞ . However, in order to use the parameter searching method in Matlab, we impose condition (3.1b) for improving FTS parameters r1 , r2 , T and the delay size d. Remark 3.3. When the finite-time stability parameters r1 , r2 , T are pre-specified, one seeks to find the maximum allowable delay (MAD) dmax . The following optimization problem can be formulated to find MAD min γ > 0, λ > 0 subject to
Mγ ξ ≪ 0
e ≤ ξ ≤ λe.
Then the MAD can be determined by dmax =
1
γ
ln
r2
λ r1
− T.
4. An illustrative example Consider the following system x˙ (t ) = A(t )x(t ) + D(t )x(t − τ (t )) + G(t )
t t −κ(t )
x(s)ds,
t ≥ 0,
(4.1)
where A(t ) =
−4 sin2 2t
|cos t | , −4
D(t ) =
sin2 t 0
√ cos 2 t cos2 t
and τ (t ) = |sin 4t |, κ(t ) = |cos t |. System (4.1) satisfies A1, A2 for all T > 0 and we have A =
,
−4 1
G(t ) =
1 −4
√ sin t
,D =
sin2 3t
1 0
1 1
0
,
|cos 2t |
,G =
1 1
0 1
. It should be noted
that system (4.1) does not satisfy the Lyapunov stability conditions proposed in [3]. More precisely, in this case the matrix M = A + D + κ G is not invertible, and hence does not satisfy conditions of Theorem 2.5 in [3]. However, Mγ = M − γ I
2 ξ1 < ξ2 < 2 ξ1 . Let us satisfies (3.1a) for any γ > 0 and the domain of the solution ξ ∈ R2 of (3.1a) is defined by 2+γ take r1 = 1, r2 = 1.25 and γ = 0.01 then system (4.1) is finite-time stable with respect to (r1 , r2 , T ) for any finite time 0 < T < Tmax = 21.3144. 2+γ
Remark 4.1. Since the matrix Mγ satisfies (3.1a) for any γ > 0, limγ ↓0+ γ1 ln r2 = +∞ for all 0 < r1 < r2 , then for given 1 r T > 0, there exists γ > 0 such that 1 < r2 e−γ (T +d) . Therefore, system (4.1) is finite-time stable with respect to (r1 , r2 , T ) 1 for any 0 < r1 < r2 , T > 0. However, system (4.1) is not LAS as shown in Fig. 3. r
5. Conclusion This paper has dealt with the problem of finite-time stability for a class of linear nonautonomous systems with timevarying delays. An explicit criterion for the finite-time stability of the system has been proposed in terms of matrix inequalities for a type of Metzler matrix. The effectiveness of the proposed conditions has been illustrated by a numerical example.
L.V. Hien / Applied Mathematics Letters 30 (2014) 12–18
17
Fig. 3. A state trajectory of system (4.1).
Acknowledgments The authors would like to thank the editors and anonymous reviewers for their constructive comments. This work was supported by the NAFOSTED of Vietnam (101.01-2011.51) and the Ministry of Education and Training of Vietnam (B2013.17.42). Appendix. Proof for the claim in Remark 3.1 Let ξ ∈ Rn , ξ ≫ 0, such that (A + D + κ G)ξ ≪ 0. Then there exists η > 0 such that n
aij + dij + κ g ij ξj ≤ −η,
∀ i ∈ n.
j=1
Therefore, there exists γ ∗ > 0 satisfying
γ ξi +
n
eγ τ − 1 dij +
j=1
eγ κ − 1
γ
− κ g ij ξj − η ≤ 0,
∀i ∈ n, γ ∈ (0, γ ∗ ].
By the same arguments used in (3.3)–(3.9), we obtain
|xi (t )| ≤
1
ξmin
∥φ∥∞ ξi e−γ t ,
∀i ∈ n, t ≥ 0,
where ξmin = mini∈n ξi . Therefore, ∥x(t )∥∞ ≤ exponentially stable. The proof is completed.
∥ξ ∥∞ ∥φ∥∞ e−γ t , t ξmin
≥ 0, which shows that system (2.1) is Lyapunov
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