Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions

El-Sayed Ahmed, Gaafar, Gaafar Fatma

Description

In this paper, we establish sufficient conditions for the existence of a unique solution for a class of nonlinear non-autonomous system of Riemann-Liouville fractional differential systems with different constant delays and non-local condition is. The stability of the solution will be proved. As an application, we also give some examples to demonstrate our results.

Sujets

• Delay differential equation

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 8
• Langue: English

Extrait

El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47
http://www.advancesindifferenceequations.com/content/2011/1/47
RESEARCH Open Access
Stability of a nonlinear non-autonomous
fractional order systems with different delays and
non-local conditions
1* 2Ahmed El-Sayed and Fatma Gaafar
* Correspondence: Abstract
[e-mail]
1
Faculty of Science, Alexandria In this paper, we establish sufficient conditions for the existence of a unique solution
University, Alexandria, Egypt for a class of nonlinear non-autonomous system of Riemann-Liouville fractional
Full list of author information is
differential systems with different constant delays and non-local condition is. Theavailable at the end of the article
stability of the solution will be proved. As an application, we also give some
examples to demonstrate our results.
Keywords: Riemann-Liouville derivatives, nonlocal non-autonomous system,
timedelay system, stability analysis
1 Introduction
Here we consider the nonlinear non-local problem of the form
αD x (t) = f (t,x (t),...,x (t)) +g (t,x (t −r ),...,x (t −r )),t ∈ (0,T), T < ∞, (1)i i 1 n i 1 1 n n
x(t)= (t)for t < 0 and lim (t)=0, (2)
−t→0
1−αI x(t)| =0, (3)t=0
awhere D denotes the Riemann-Liouville fractional derivative of order aÎ (0, 1), x(t)
=(x (t), x (t), ..., x (t))’,where ‘ denote the transpose of the matrix, and f, g :[0, T]×1 2 n i i
nR ® R are continuous functions,F(t)=(j(t)) are given matrix and O is the zeroi n×1
matrix, r ≥ 0, j = 1, 2, ..., n, are constant delays.j
Recently, much attention has been paid to the existence of solution for fractional
differential equations because they have applications in various fields of science and
engineering. We can describe many physical and chemical processes, biological systems,
etc., by fractional differential equations (see [1-9] and references therein).
In this work, we discuss the existence, uniqueness and uniform of the solution of
stability non-local problem (1)-(3). Furthermore, as an application, we give some
examples to demonstrate our results.
For the earlier work we mention: De la Sen [10] investigated the non-negative
solution and the stability and asymptotic properties of the solution of fractional differential
dynamic systems involving delayed dynamics with point delays.
© 2011 El-Sayed and Gaafar; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 Page 2 of 8
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El-Sayed [11] proved the existence and uniqueness of the solution
u(t)oftheproblem
c α c βD u(t)+C D u(t −r)= Au(t)+Bu(t −r), 0 ≤ β ≤ α ≤ 1,a a
u(t) = g(t), t ∈ [a −r,a], r > 0
by the method of steps, where A, B, C are bounded linear operators defined on a
Banach space X.
El-Sayed et al. [12] proved the existence of a unique uniformly stable solution of the
non-local problem
n n
αD x (t)= a (t)x (t)+ b (t)x (t −r)+h (t), t > 0,i ij j ij j j i
j=1 j=1
βx(t)= (t)for t < 0, lim (t)= O and I x(t)| = O, β ∈ (0,1).t=0
−t→0
Sabatier et al. [6] delt with Linear Matrix Inequality (LMI) stability conditions for
fractional order systems, under commensurate order hypothesis.
Abd El-Salam and El-Sayed [13] proved the existence of a unique uniformly stable
solution for the non-autonomous system
c α 0D x(t)= A(t)x(t)+f(t), x(0) = x , t > 0,a
c αwhere D is the Caputo fractional derivatives (see [5-7,14]), A(t)and f(t) are contin-a
uous matrices.
Bonnet et al. [15] analyzed several properties linked to the robust control of
fractional differential systems with delays. They delt with the BIBO stability of both
retarded and neutral fractional delay systems. Zhang [16] established the existence of a
unique solution for the delay fractional differential equation
αD x(t) = A x(t)+A x(t −r)+f(t), t > 0, x(t) = φ(t), t ∈ [−r,0],0 1
by the method of steps, where A , A are constant matrices and studied the finite0 1
time stability for it.
2 Preliminaries
Let L [a, b] be the space of Lebesgue integrable functions on the interval [a, b], 0 ≤ a1
b<b < ∞ with the norm .||x|| = |x(t)|dtL1 a
Definition 1 The fractional (arbitrary) order integral of the function f(t)Î L [a, b]of1
+order aÎ R is defined by (see [5-7,14,17])
α−1t (t −s)αI f(t)= f(s)ds,a (α)a
where Γ (.) is the gamma function.
Definition 2 The Caputo fractional (arbitrary) order derivatives of order a, n <a <n
+ 1, of the function f(t) is defined by (see [5-7,14]),
t1c α n−α n n−α−1D f(t)= I D f(t)= (t −s) f(s)ds, t ∈ [a,b],a a (n − α) aEl-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 Page 3 of 8
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Definition 3 The Riemann-liouville fractional (arbitrary) order derivatives of order a,
n <a <n + 1 of the function f (t) is defined by (see [5-7,14,17])
n n td 1 dα n−α n−α−1D f(t)= I f(t)= (t −s) f(s)ds, t ∈ [a,b],a an ndt (n − α) dt a
The following theorem on the properties of fractional order integration and
differentiation can be easily proved.
+
Theorem 1 Let a, bÎ R . Then we have
α β α+βα(i) I : L → L , and if f(t)Î L then .1 1 I I f(t)= I f(t)a 1 a aa
α nlimI = I(ii) , n = 1,2,3,... uniformly.a aα→n
−α(t −a)c α α(iii) D f(t)= D f(t) − f(a), aÎ (0,1), f (t) is absolutely continuous.
(1 − α)
df
c α α(iv) , aÎ (0,1), f (t) is absolutely continuous.lim D f(t)= = lim D f(t)a
α→1 dt α→1
3 Existence and uniqueness
Let X=(C (I), || . || ), where C (I) is the class of all continuous column n-vectorsn 1 n
n −Ntfunction. For xÎ C [0, T], the norm is defined by ||x|| = sup {e |x (t)|},1 in t∈[0,T]i=1
where N>0.
n
Theorem 2 Let f , g :[0, T]× R ® R be continuous functions and satisfy thei i
Lipschitz conditions
n
|f (t,u ,...,u ) −f (t,v ,...,v ) ≤ h |u −v |,i 1 n i 1 n ij j j
j=1
n
|g (t,u ,...,u ) −g (t,v ,...,v )|≤ k |u −v |,i 1 n i 1 n ij j j
j=1
n n n nand h = |h | = max |h |, k = |k | = max |k |.i ∀j ij i ∀j iji=1 i=1 i=1 i=1
Then there exists a unique solution ×Î X of the problem (1)-(3).
Proof Let tÎ (0, T). Then equation (1) can be written as
d 1−αI x (t)= f (t,x (t),...,x (t))+g (t,x (t −r ),...,x (t −r ).i i 1 n i 1 1 n n
dt
Integrating both sides, we obtain
t
1−α 1−αI x (t)−I x (t)| = {f (t,x (t),...,x (t))+g (t,x (t−r ),...,x (t−r ))}ds.i i t=0 i 1 n i 1 1 n n
0
From (3), we get
t
1−αI x (t)= {f (t,x (t),...,x (t))+g (t,x (t −r ),...,x (t −r ))}ds.i i 1 n i 1 1 n n
0
a
Operating by I on both sides, we obtain
α+1Ix (t) = I {f (t,x (t),...,x (t))+g (t,x (t −r ),...,x (t −r ))}.i i 1 n i 1 1 n n
El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 Page 4 of 8
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Differentiating both side is, we get
αx (t) = I {f (t,x (t),...,x (t))+g (t,x (t −r ),...,x (t −r ))}, i=1,2,...,n. (4)i i 1 n i 1 1 n n
Now let F : X® X, defined by
αFx = I {f (t,x (t),...,x (t))+g (t,x (t −r ),...,x (t −r ))}.i i 1 n i 1 1 n n
then
α|Fx −Fy | = |I {f (t,x (t),...,x (t)) −f (t,y (t),...,y (t))i i i 1 n i 1 n
+g (t,x (t −r ),...,x (t −r )) −g (t,y (t −r ),...,y (t −r ))}|i 1 1 n n i 1 1 n n
α−1t (t −s)
≤ |f (s,x (s),...,x (s)) −f (s,y (s),...,y (s))|dsi 1 n i 1 n
(α)0
α−1t (t −s)
+ |g (s,x (s −r ),...,x (s −r )) −g (s,y (s −r ),...,y (s −r ))|dsi 1 1 n n i 1 1 n n
(α)0
α−1 nt (t −s)
≤ h |x (s) −y (s)|dsij j j
(α)0
j=1
nt α−1(t −s)
+ k |x (s −r ) −y (s −r )|dsij j j j j
(α)0 j=1
and
n t α−1 (t −s)−Nt −N(t−s) −Nse |Fx −Fy|≤ h e |x (s) −y (s)|dsei i i j j
(α)0j=1
n α−1t (t −s) −N(t−s+r ) −N(s−r )j j+k e e |x (s −r ) −y (s −r)|dsi j j j j
(α)rjj=1
n t α−1 (t −s)−Nt −N(t−s)≤ h sup{e |x (t) −y (t)|} e dsi j j
(α)t 0j=1
n α−1t (t −s)−Nt −Nr −N(t−s)j+k sup{e |x (t) −y (t)|}e e dsi j j
(α)t rjj=1
n Nt α−1 −u 1 u e−Nt≤