New criterion for finite-time stability of fractional delay systems

Journal Pre-proof New criterion for finite-time stability of fractional delay systems

Feifei Du, Jun-Guo Lu

PII: DOI: Reference:

S0893-9659(20)30041-0 https://doi.org/10.1016/j.aml.2020.106248 AML 106248

To appear in:

Applied Mathematics Letters

Received date : 23 November 2019 Revised date : 19 January 2020 Accepted date : 19 January 2020 Please cite this article as: F. Du and J.-G. Lu, New criterion for finite-time stability of fractional delay systems, Applied Mathematics Letters (2020), doi: https://doi.org/10.1016/j.aml.2020.106248. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2020 Elsevier Ltd. All rights reserved.

Journal Pre-proof

New Criterion for finite-time stability of fractional delay systems Feifei Du, Jun-Guo Lu∗ Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China

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Abstract In this paper, a new fractional Gronwall inequality with time delay is developed. Based on this inequality, a new criterion for finite-time stability of fractional delay systems is derived. Two numerical examples are given to show that proposed results are less conservative than the existing ones.

Keywords: Finite-time stability, fractional delay systems, Gronwall inequality.

1. Introduction

During the past decades, much attention has been received on fractional calculus and a number of excellent results [1, 2] has been obtained. Since Lazarevic’s seminal work [3], an increasing interest has been attracted on finite-time 5

stability analysis of fractional delay systems and numerous methods are developed to study finite-time stability of fractional delay systems, for example, delayed Mittag-Leffler matrix function [4], method of steps [5], linear matrix inequality [6], Gronwall inequality [7], H¨ older inequality [8, 9]. However, all the mentioned results above to guarantee the finite-time stability of the systems are sufficient ones, which are still conservative.

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Inspired by the above discussions, finite-time stability of fractional delay

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systems is investigated in this paper. The main contributions are presented as ∗ Corresponding

author Email address: [email protected] (Jun-Guo Lu)

Preprint submitted to Applied Mathematics Letters

January 23, 2020

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follows. (1) A new fractional Gronwall inequality with time delay is developed. Compared with the existing one [10, Theorem 2.4] which needs to divide order 15

λ ∈ (0, 1) into λ ∈ (0, 1/2) and λ ∈ (1/2, 1), our inequality is uniform and convenient to verify. (2) A new criterion for finite-time stability of fractional delay systems is derived, which is less conservative than the existing ones [4, 5]. 2. Preliminaries and problem descriptions In this section, preliminaries on Caupto fractional derivative, finite-time stability and problem descriptions are given. If a vector x ∈ Rn , then we define Pn n×n kxk1 = , then we define the induced norm k · k1 as i=1 |xi |. If A ∈ R Pn kAk1 = max1≤j≤n i=1 |ai,j |. In the subsequent passage, all the notations k · k

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20

just means k · k1 .

Definition 1 ([11]). For x(t) ∈ C n ([0, +∞), R), the Caputo derivative of order λ for x is defined by c

Dλ x(t) =

Z

0

where λ ∈ (n − 1, n], n ∈ N+ .

25

t

(t − s)n−λ−1 (n) x (s)ds, Γ(n − λ)

Consider the following fractional delay systems   c Dλ x(t) = Ax(t − τ ), t ∈ [0, T ],  x(t) = φ(t), t ∈ [−τ, 0],

(1)

where x : [0, T ] → Rn is the state, 0 < λ < 1, A ∈ Rn×n is a constant matrix,

τ > 0 is a constant delay, φ : [−τ, 0] → Rn is a continuous vector function.

Definition 2 ([3]). The system (1) is finite-time stable w.r.t. {T, δ, ε} with 0 < δ < ε if and only if kφk ≤ δ implies kx(t)k ≤ ε, for any t ∈ [0, T ], where

3. Main results

In this section, a new fractional Gronwall inequality with time delay is de-

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kφk = sup−τ ≤r≤0 kφ(r)k.

veloped. Based on this inequality, a new criterion for finite-time stability of fractional delay systems is derived. 2

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Lemma 1. Let u, a, b and k be nonnegative continuous functions defined on 35

[t0 , T ], ϕ be nonnegative continuous function defined on [t0 − τ, t0 ], and suppose   u(t) ≤ a(t) + b(t) R t k(s)up (s − τ )ds
1/p , t0  u(t) ≤ ϕ(t), t ∈ [t − τ, t ], 0

t ∈ [t0 , T ],

(2)

0

where p ≥ 1 and τ > 0 are constants. Then

1/p Z t k(s)ϕp (s − τ )ds u(t) ≤ a(t) + b(t)

(3)

t0

t0 +τ

Z

t

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for t ∈ [t0 , t0 + τ ] and Z u(t) ≤a(t) + b(t)

t0

+

Z

p

k(s)ϕ (s − τ )ds exp

t

t0 +τ

k(s)2p−1 ap (s − τ ) exp

Z

s

t

k(s)2

t0 +τ

 b (s − τ )ds

p−1 p

(4)

 1/p k(r)2p−1 bp (r − τ )dr ds

for t ∈ [t0 + τ, T ].

Furthermore, if a(t), b(t), ϕ(t) are nondecreasing and ϕ(t0 ) = a(t0 ), then u(t) ≤ M (t),

where

t ∈ [t0 , T ],

  R 1/p  t   a(t) 1 + b(t) k(s)ds , t ∈ [t0 , t0 + τ ],   t0   (       R t +τ a(t) 1 + bp (t) t00 k(s)ds + 1 M (t) =    1/p )     Rt  p−1 p  · exp 2 b (t) t0 +τ k(s)ds − 1 ,  

(5)

t ∈ [t0 + τ, T ].

Proof . If t ∈ [t0 , t0 + τ ], then t − τ ∈ [t0 − τ, t0 ]. From (2), we have

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Z t 1/p p u(t) ≤ a(t) + b(t) k(s)ϕ (s − τ )ds , t0

t ∈ [t0 , t0 + τ ].

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Define a function v(t) by v(t) =

Z

t

t0

k(s)up (s − τ )ds,

3

t ∈ [t0 , T ].

(6)

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It is obvious that v(t) is nonnegative and nondecreasing for t ∈ [t0 , T ]. It follows from (2) and (6) that u(t) ≤ a(t) + b(t)v 1/p (t),

t ∈ [t0 , T ].

(7)

For t ∈ [t0 + τ, T ], we have v 0 (t) = k(t)up (t − τ )

(8)

≤ k(t)[a(t − τ ) + b(t − τ )v 1/p (t − τ )]p ≤ k(t)[a(t − τ ) + b(t − τ )v 1/p (t)]p

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[12, Cor. 8.1.4]

≤

k(t)2p−1 ap (t − τ ) + k(t)2p−1 bp (t − τ )v(t).

Applying [13, Lemma 1.1], we have Z Z t0 +τ v(t) ≤ k(s)ϕp (s − τ )ds exp t0

+

Z

 k(s)2p−1 bp (s − τ )ds

t

t0 +τ

t

k(s)2

p−1 p

a (s − τ ) exp

t0 +τ

Z

 b (r − τ )dr ds.

t

k(r)2

(9)

p−1 p

s

From (7) and (9), we have (4).

If a(t), b(t) and ϕ(t) are nondecreasing and a(t0 ) = ϕ(t0 ), then for t ∈ [t0 + τ, T ],

Z u(t) ≤a(t) + a(t) +

Z

t0 +τ

p

b (t)k(s)ds exp

t0

t

p−1 p

k(s)2

b (t) exp

t0 +τ

Z =a(t) + a(t) Z

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+ exp (

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≤a(t) 1 +

Z

t

k(r)2

p

b (t)k(s)ds exp

t0

t

t0 +τ



Z

k(s)2

t0 +τ

k(s)ds + 1

t0



4

t

(10)

 1/p b (t)dr ds k(s)2

t0 +τ 1/p



p−1 p

· exp 2

 b (t)ds

p−1 p

p−1 p

Z

 k(r)2p−1 bp (t)dr − 1

bp (t)

t

t0 +τ

s

t0 +τ

Z

b (t)

Z

t

t0 +τ

 b (t)ds

p−1 p

 1/p ) k(s)ds − 1 .

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For t ∈ [t0 , t0 + τ ],

1/p Z t k(s)ϕp (s − τ )ds u(t) ≤ a(t) + b(t) t0 "  # Z 1/p

t

≤ a(t) 1 + b(t)

(11)

.

k(s)ds

t0

From (10) and (11), we obtain the inequality (5). 40

Remark 1. Let τ = 0, it is easy to see that [13, Theorem 4.2] is a special case of Lemma 1.

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Motivated by [14], we give the following fractional Gronwall inequality with time delay with the help of Lemma 1.

Theorem 1. Assume a(t), b(t) and k(t) are continuous, nonnegative function on [t0 , T ], and u(t) is a continuous, nonnegative function on [t0 , T ] with   u(t) ≤ a(t) + b(t) R t (t − r)λ−1 k(r)u(r − τ )dr, t ∈ [t , T ], 0 Γ(λ) t0  u(t) ≤ ϕ(t), t ∈ [t − τ, t ]. 0

Then

(12)

0

Z t 1/q q q u(t) ≤ a(t) + B(t) k (s)ϕ (s − τ )ds

(13)

t0

for t ∈ [t0 , t0 + τ ] and  Z t0 +τ Z u(t) ≤a(t) + B(t) k q (s)ϕp (s − τ )ds exp t0

+

Z

t0 +τ

t

t0 +τ

t

k q (s)2p−1 ap (s − τ ) exp

Z

s

 k q (s)2p−1 B p (s − τ )ds (14)

 1/p t k q (r)2p−1 B p (r − τ )dr ds

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for [t0 + τ, T ], where p, q > 0 such that λ > 1/p and 1/p + 1/q = 1, B(t) =

b(t)(t − t0 )λ−1/p , t ∈ [t0 , T ], Γ(λ)(q(λ − 1) + 1)1/q

Furthermore, if a(t), b(t), ϕ(t) are nondecreasing and a(t0 ) = ϕ(t0 ), then ˜ (t), u(t) ≤ M 5

t ∈ [t0 , T ],

(15)

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where   R 1/p  t p   a(t) 1 + B(t) k (s)ds , t ∈ [t0 , t0 + τ ],   t0   (       R t +τ a(t) 1 + B p (t) t00 k p (s)ds + 1 ˜ (t) = M    1/p )     Rt  p−1 p p  , · exp 2 B (t) t0 +τ k (s)ds − 1  

t ∈ [t0 + τ, T ].

Proof . Let p, q such that λ > 1/p and 1/p + 1/q = 1. For t ∈ [t0 , T ], using the

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H¨older inequality, we obtain Z b(t) t (t − r)λ−1 k(r)u(r − τ )dr. u(t) ≤ a(t) + Γ(λ) t0 Z t 1/q  Z t 1/p b(t) q(λ−1) p ≤ a(t) + (t − r) dr (k(r)u(r − τ )) dr . Γ(λ) t0 t0 Z t 1/p b(t)(t − t0 )λ−1+1/q p ≤ a(t) + (k(r)u(r − τ )) dr . Γ(λ)(q(λ − 1) + 1)1/q t0 Let A(t) = a(t), B(t) =

b(t)(t−t0 )λ−1+1/q , Γ(λ)(q(λ−1)+1)1/q

u(t) ≤ A(t) + B(t)

Z

t

t0

and K(t) = k p (t). Then we obtain

1/p K(s)up (s − τ )ds ,

t ∈ [t0 , T ].

Using Lemma 1, we obtain the inequalities (13), (14) and (15). Theorem 2. The system (1) is finite-time stable w.r.t {T, δ, ε} if ˆ (t) ≤ ε, M

t ∈ [0, T ],

(16)

where p, q > 0 such that λ > 1/p and 1/p + 1/q = 1,

and

tλ−1/p , t ∈ [0, T ], Γ(λ)(q(λ − 1) + 1)1/q

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B(t) =

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     δ 1 + B(t)kAkt1/p , t ∈ [0, τ ],   (       δ 1 + B p (t)kAkp τ + 1 ˆ M (t) =   1/p )      p−1 p p  · exp 2 B (t)kAk (t − τ ) − 1 ,   6

t ∈ [τ, T ].

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Proof . It is easy to show the system (1) is equivalent to the following system   x(t) = x(0) + 1 R t (t − r)λ−1 [Ax(r − τ )]dr, t ∈ [0, T ], Γ(λ) 0 (17)  x(t) = φ(t), t ∈ [−τ, 0].

45

Taking the norm on both sides of (17), we have   kx(t)k ≤ kφk + 1 R t (t − r)λ−1 kAkkx(r − τ )kdr, t ∈ [0, T ], Γ(λ) 0  kx(t)k ≤ kφk, t ∈ [−τ, 0].

(18)

Let t0 = 0, u(t) = kx(t)k, a(t) = ϕ(t) = kφk, b(t) = 1, k(t) = kAk. It is

obvious that a(t), b(t), ϕ(t) are nondecreasing and a(0) = ϕ(0).

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Using Theorem 1, we obtain the criterion (16).

Remark 2. If φ(t) in the systems (1) is a constant, then the finite-time stability criterion of the systems (1) given in [4, Theorem 4.1] is δEλ (kAktλ ) ≤ , t ∈ [0, T ].

(19)

Remark 3. Let A0 = 0, A1 = A. According to [5, (4.8)], a criterion for finitetime stability of the systems (1) can be obtained directly.

50

4. Numerical examples

In this section, two numerical examples are given to show that proposed results are less conservative than the existing ones [4, 5].

Example 1. Consider the following fractional delay system      0.1 0  c 0.95   x(t − 0.3), t ∈ [0, 1], D x(t) =  0 0.9     φ(t) = (0.1, 0.1)T , t ∈ [−0.3, 0],

(20)

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  0.1 0 , λ = 0.35, τ = 0.2, T = 1. Let p = 1.2, δ = 0.2. where A =  0 0.9 Obviously, kAk = 0.9, λ = 0.95 > 1/p = 5/6. According to the criterion (16) in Theorem 2 and (19) in [4], the largest possible bounds ε of the system (20) are

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drawn in Fig. 1 respectively. From Fig. 1, it is easy to see that for T > 0.3 our

results are less conservative than the one [4]. 7

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0.55 criterion (19) derived from [4] criterion (16) in this paper 0.5

0.45

ε

0.4

0.35

0.3

0.25

0

0.2

0.4

0.6

0.8

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0.2

1

T (s)

Fig. 1. The largest possible bounds ε of the system (20).

Example 2. Consider the following fractional delay system      0.1 0   c D0.7 x(t) =   x(t − 0.3), t ∈ [0, 5], 0 0.2     φ(t) = (0.1, 0.1)T , t ∈ [−0.3, 0],

(21)

  0.1 0 , λ = 0.7, τ = 0.2, T = 5. Let p = 3, δ = 0.2. Obviously, where A =  0 0.2 kAk = 0.2, λ = 0.7 > 1/p = 1/3. According to the criterion (16) in Theorem 60

2 and the one in [5, (4.8)], the largest possible bounds ε of the system (21) are drawn in Fig. 2 respectively. From Fig. 2, it is easy to see that for T > 1.5 our results are less conservative than the one [5].

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (No.61374030 and No.61533012) and the China Postdoctoral Science Foundation (No. 2018M641991).

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8

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1 criterion (16) in this paper criterion (4.8) derived from [5]

0.9 0.8

ε

0.7 0.6 0.5 0.4 0.3

0

1

2

3

4

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0.2

5

T (s)

Fig. 2. The largest possible bounds ε of the system (21).

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Journal Pre-proof Credit Author Statement

Feifei Du: Writing- Reviewing and Editing, Funding acquisition, Visualization, Writing-Original Draft, Conceptualization, Software, Formal analysis, Investigation

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Jun-Guo Lu: Supervision, Funding acquisition