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Electronic Journal of Qualitative Theory of Differential Equations 2007, No.6, 1-14;http://www.math.u-szeged.hu/ejqtde/

## systems

Sheren A. Abd El-Salam e.mail: shrnahmed@maktoob.com

Ahmed M. A. El-Sayed e.mail: amasayed@hotmail.com

Faculty of Science, Alexandria University, Alexandria, Egypt

Abstract

The fractional calculus (integration and differentiation of fractional-order) is a one of the singular integral and integro-differential operators. In this work a class of fractional-order non-autonomous systems will be considered. The stability (and some other prop-erties concerning the existence and uniqueness) of the solution will be proved.

Key words: Fractional calculus, fractional-order non-autonomous systems, stability, asymp-totic stability.

### Introduction

LetL1[a, b] denotes the space of all Lebesgue integrable functions on the interval [a, b],

0≤a < b <∞.

Definition 1.1 The fractional (arbitrary) order integral of the function f ∈ L1[a, b] of

orderβ ∈R+ is defined by (see  and  - )

Iaβ f(t) =

Z t

a

(t − s)β −1

Γ(β) f(s)ds,

where Γ(.) is the gamma function.

Definition 1.2The Riemann-Liouville fractional-order derivative off(t) of orderα∈(0,1) is defined as (see  and  - )

∗Daα f(t) = d dt I

1 − α

a f(t), t ∈ [a, b].

Definition 1.3 The (Caputo) fractional-order derivative Dα of order α (0,1] of the

functiong(t) is defined as (see  - )

Daα g(t) = Ia1 − α d

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Now consider the non-autonomous linear system:

x′(t) = A(t) x(t), (1)

with the initial condition

x(t0) = x0, t ≥ t0,

where A(t) is a continuous n by n matrix on the half-axis t ≥ 0. We know that (see ) the solution of system (1) is given by:

x(t, t0, x0) = X(t) X−1(t0) x0, t ≥ 0,

where X(t) is an arbitrary fundamental matrix of the system (1) defined on the whole half-axist≥0.

Now we shall present the main definitions (see ) related to the concepts of stability of the solutionx= 0 of (1).

Definition 1.4The solutionx= 0 of (1) will be called stable if to anyε >0, t0 ≥0 there

correspondsδ(ε, t0)>0 such that||x(t, t0, x0)||< εfort≥t0 as soon as ||x0||< δ.

Definition 1.5 The solution x = 0 of (1) will be called uniformly stable if δ(ε, t0) from

definition 1.4 can be chosen independent oft0: δ(ε, t0)≡δ(ε).

Definition 1.6The solutionx= 0 of (1) will be called asymptotically stable if it is stable in the sense of definition 1.4 and there existsγ(t0)>0 such that lim

t→∞||x(t, t0, x

0)||= 0 for

everyx(t, t0, x0) with||x0||< γ.

Definition 1.7The solution x= 0 of (1) will be called uniformly asymptotically stable if it is uniformly stable in the sense of definition 1.5 and moreover, for anyε >0 there exists

T(ε) > 0 such that ||x(t, t0, x0)|| < ε for every t ≥ t0+T(ε) and all x0 with ||x0|| < γ0,

whereγ0 is independent of t0.

In other words,x= 0 of (1) will be called uniformly asymptotically stable if it is uniformly stable and lim||x(t, t0, x0)|| = 0 as t−t0 → +∞ uniformly with respect to (t0, x0), t0 ≥

0,||x0||< γ0.

Theorem 1.1

LetX(t) be a fundamental matrix of the system (1). A necessary and sufficient condition for the stability of the solutionx= 0 is the boundedness ofX(t) on t≥0:

||X(t)|| ≤ M, t ≥ 0.

A necessary and sufficient condition for the asymptotic stability of the solutionx= 0 is

lim

t→∞||X(t)|| = 0.

Theorem 1.2

LetX(t) be a fundamental matrix of the system (1). A necessary and sufficient condition for the uniform stability of the solution x = 0 is the existence of a number M > 0 such that:

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A necessary and sufficient condition for the uniform asymptotic stability of the solution

x= 0 is the existence of two positive numbersM and η such that:

||X(t) X−1(t0)|| ≤ M exp[−η (t − t0)], t ≥ t0 ≥ 0.

Here in this work, we study the stability (and some other properties concerning the existence and uniqueness) of the solutions of the non-autonomous linear systems:

Dtα0 x(t) = A(t) x(t) + f(t), α ∈ (0,1]

and

x′(t) = A(t) d

dt I α

t0 x(t) + f(t), α ∈ (0,1],

with the initial condition

x(t0) = x0.

Also the special cases:

Dtα0 x(t) = A x(t), α ∈ (0,1], x(t0) = x0

and

x′(t) = A d dt I

α

t0 x(t), α ∈ (0,1], x(t0) = x

0

will be studied.

### Existence of solution

Here the spaceB[t0, T] denotes the space of allnvector functionsysuch thate−N t|yi(t)| ∈ L1[t0, T], T <∞, N >0, and the spaceC∗[t0, T] denotes the space of allnvector functions

x such thate−N t|xi(t)| ∈C[t0, T], T <∞, N >0, while the space AC∗[t0, T] denotes the

space of all nvector absolutely continuous functions, in addition the norm onB[t0, T] will

be denoted by ||.||1, that is, for y ∈B[t0, T],||y||1 =Pni=1||yi||1 =Pni=1

Rt t0e

−N s|y

i(s)|ds,

while the norm on C∗[t0, T] will be denoted by ||.||2, that is, for x ∈ C∗[t0, T],||x||2 =

Pn

i=1||xi||2 = Pni=1supte−N t|xi(t)|. Throughout this paper we define an n×n matrix

functionA(t) = (aij(t)), i, j = 1,2, ..., n such that A: [t0, T]→R, T <∞, also define

||A∗|| := ||a∗ij|| = sup

t

|aij(t)| and ||Ae|| := ||eaij|| = sup t

|a′ij(t)|

Consider firstly the problem:

t0 x(t) = A(t) x(t) + f(t), α ∈ (0,1], x(t0) = x0. (2)

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Letf(t)∈AC∗[t0, T]. IfA(t) ∈AC∗[t0, T], then there exists a unique solution of problem

(2).

Proof. Problem (2) is equivalent to the equation

y(t) = x0 d

Operating byItα0 on both sides of the last equation, we obtain

I y(t) = x0 Itα0 A(t) + Itα0 A(t) I y(t) + Itα0 f(t),

Operating byIt10− α on both sides of the last equation, we obtain

It10−α x′(t) = It10−α d

Which proves the equivalence.

(5)
(6)

Consider secondly the problem then there exists a unique solution of problem (5).

Proof. Problem (5) is equivalent to the equation

y(t) = x0 (t − t0)

Which proves the equivalence.

(7)

therefore

||F yi−F zi||1 ≤

||a∗ij||

Nα ||yj − zj||1, n

X

i=1

||F yi − F zi||1 ≤

n

X

i=1

||a∗ij||

Nα ||yj − zj||1,

||F y − F z||1 ≤ ||

A∗||

Nα ||y − z||1.

Now chooseN large enough such that||A∗||< Nα, then we get

||F y − F z||1 < ||y − z||1,

therefore the mapF :B →Bis contraction and (7) has a unique fixed pointy∈B[t0, T], therefore we deduce that the problem (5) has a unique solution x∈AC∗[t0, T].

### Stability of non-autonomous systems

In this section we study the stability of the solution of the initial-value problems (2) and (5).

Theorem 3.1

The solution of the initial-value problem (2) is uniformly stable Proof. Lety(t) be a solution of

y(t) = x0 d dt I

α

t0 A(t) + d dt I

α

t0 A(t) I y(t) + d dt I

α t0 f(t)

= A(t0) x0 (t−t0) α−1

Γ(α) + x

0 Iα t0 A

(t) + Iα t0 A

(t) I y(t) + Iα

t0 A(t) y(t) + d dt I

α t0 f(t),

and letye(t) be a solution of the above linear system such thatye(t0) =xe0, then

yi(t) − yei(t) = (x0j − xe0j) aij(t0)

(t − t0)α−1

Γ(α) + (x

0

j − xe0j)Itα0 a ′ ij(t)

+ Itα0 a′ij(t)I (yj(t) − yej(t)) + Itα0 aij(t) (yj(t) − yej(t)),

e−N t |yi(t)−yei(t)| ≤ e−N t |x0j − xej0| |aij(t0)|

(t − t0)α−1 Γ(α)

+ e−N t |x0j − xe0j|

Z t

t0

(t − s)α−1 Γ(α) |a

ij(s)|ds

+ e−N t

Z t

t0

(t − s)α−1 Γ(α) |a

′ ij(s)|

Z s

t0

|yj(θ) − yej(θ)|dθ ds

+ e−N t

Z t

t0

(t − s)α−1

(8)
(9)

+ ||eaij||

(10)
(11)

≤ ||x0i − xe0i||2 +

||a∗ij||

− ||a∗ ij||

!

||x0j − xe0j||2

≤ N

α

− ||a∗ ij||

!

||x0j − xe0j||2.

Therefore

n

X

i=1

||xi − xei||2 ≤

n

X

i=1

Nα Nα − ||a

ij||

!

||x0j − xe0j||2,

||x − xe||2 ≤

Nα Nα − ||A||

||x0 − xe0||2

Therefore, if ||x0 − xe0||2 < δ(ε), then||x − xe||2 < ε, which complete the proof of the

theorem.

### Autonomous systems

Now we study the problems:

Dtα0 x(t) = A x(t), α ∈ (0,1], x(t0) = x0 (8)

and

x′(t) = A d dt I

α

t0 x(t), α ∈ (0,1], x(t0) = x

0 (9)

which are the special cases of the initial-value problems (2) and (5) whenA(t) = A, where

Ais a real constant matrix and f(t) is the zero vector.

Theorem 4.1

The solution of the initial-value problem (8) or (9) is given by the formula

x(t) =

X

k=0

(A Itα0)k x0

=

X

k=0

Ak (t t

0)kα

Γ(1 + kα) x

0

Proof. Firstly we prove that the two problems are equivalent to each other, indeed: Let

x(t) be a solution of (8), then

It10−α dx

dt = A x(t),

operating byItα0 on both sides of the last relation, we get

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differentiating both sides, we obtain (9) and whent=t0 , we obtain x(t0) =x0.

we complete the proof.

Theorem 4.2

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+ eaij(t− t0) |x0

j − xe0j|

= | −

X

k=0

akij

Γ(1 +k)

1 − Γ(1 +k)

Γ(1 +kα) (t−t0)

kα−k (tt

0)k| |x0j −xe0j|

+ eaij(t−t0) |x0

j − xe0j|.

Letk α = k − β, α ∈ (0,1], β > 0, then

|xi(t) − xei(t)| ≤ | − ∞

X

k=0

akij

Γ(1 +k)

1 − Γ(1 +k)

Γ(1 +k − β) (t−t0)

−β (tt

0)k| |x0j −xe0j|

+ eaij(t −t0) |x0

j − xe0j|

≤ |

X

k=0

akij (t − t0)k

Γ(1 + k) | |x

0

j − xe0j | + eaij(t− t0) |x0j − ex0j|

= eaij(t −t0) |x0

j − xe0j| + eaij(t−t0) |x0j − xe0j|

= 2eaij(t− t0) |x0

j − ex0j|.

SinceAis a real constant matrix with the characteristic roots all having negative real parts, then there exists positive constantsK andσ such that

eAt ≤ K e−σt,

therefore

e−N t |xi(t) − xei(t)| ≤ 2 K e−σ (t−t0) e−N t |x0i − ex0i|

≤ 2 K e−σ (t−t0) e−N t0 |x0

i − xe0i|,

||xi − xei||2 ≤ 2 K e−σ (t−t0) ||x0i − xe0i||2,

n

X

i=1

||xi − xei||2 ≤ 2 K e−σ (t−t0)

n

X

i=1

||x0i − xe0i||2,

||x − xe||2 ≤ 2 K e−σ (t−t0) ||x0 − xe0||2.

Therefore, we deduce that the solution is uniformly asymptotically stable.

### References

 Corduneanu, C. Principles of Differential and Integral equations, Allyn and Bacon, Inc., Boston, Massachusetts (1971).

 Miller, K. S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New York (1993).

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 Podlubny, I. and EL-Sayed, A. M. A. On two definitions of fractional calculus,Preprint UEF 03-96 (ISBN 80-7099-252-2), Slovak Academy of Science-Institute of Experimen-tal phys. (1996).

 Podlubny, I. Fractional Differential Equations, Acad. press, San Diego-New York-London (1999).

 Samko, S., Kilbas, A. and Marichev, O. L.Fractional Integrals and Derivatives, Gordon and Breach Science Publisher, (1993).

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