# Stability for Caputo Fractional Differential Systems

Authors: Sung Kyu Choi; Bowon Kang; Namjip Koo
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We introduce the notion ofh-stability for fractional differential systems. Then we investigate the boundedness and h-stability of solutions of Caputo fractional differential systems by using fractional comparison principle and fractional Lyapunov direct method. Furthermore, we give examples to illustrate our results.


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## Body

## 1. Introductions and Preliminaries

Lakshmikantham et al. [1–5] investigated the basic theory of initial value problems for fractional differential equations involving Riemann-Liouville differential operators of order 0<q<1. They followed the classical approach of the theory of differential equations of integer order in order to compare and contrast the differences as well as the intricacies that might result in development [6, Vol. I]. Li et al. [7] obtained some results about stability of solutions for fractional-order dynamic systems using fractional Lyapunov direct method and fractional comparison principle. Choi and Koo [8] improved on the monotone property of Lemma  1.7.3 in [5] for the case g(t,u)=λu with a nonnegative real number λ. Choi et al. [9] also investigated Mittag-Leffler stability of solutions of fractional differential equations by using the fractional comparison principle.In this paper we introduce the notion ofh-stability for fractional differential equations. Then, we investigate the boundedness and h-stability of solutions of Caputo fractional differential systems by using fractional comparison principle and fractional Lyapunov direct method. Furthermore, we give some examples to illustrate our results.For the basic notions and theorems about fractional calculus, we mainly refer to some books [5, 10, 11].We recall the notions of Mittag-Leffler functions which were originally introduced by Mittag-Leffler in 1903 [12]. Similar to the exponential function frequently used in the solutions of integer-order systems, a function frequently used in the solutions of fractional order systems is the Mittag-Leffler function, defined as
(1)Eα(z)=∑k=0∞‍zkΓ(kα+1),
where α>0 and Γ is the Gamma function [11]. The Mittag-Leffler function with two parameters has the following form:
(2)Eα,β(z)=∑k=0∞‍zkΓ(kα+β),
where α>0 and β>0. For β=1, we have Eα(z)=Eα,1(z). Also, E1,1(z)=ez.Note that the exponential functioneat possesses the semigroup property (i.e., ea(t+s)=eateas for all t,s≥0), but the Mittag-Leffler function Eα(atα) does not satisfy the semigroup property unless α=1 or a=0 [13].We recall briefly the notions and basic properties about fractional integral operators and fractional derivatives of functions [5, 10]. Let J=[t0,∞)⊂ℝ+=[0,∞).Definition 1 (see [5]).
The Riemann-Liouville fractional integral of orderα>0 of a function f∈L1(J,ℝ) is defined as
(3)It0αf(t)=1Γ(α)∫t0t‍(t-s)α-1f(s)ds,
where t0∈ℝ (provided that the integral exists in the Lebesgue sense).Definition 2 (see [5]).
The Riemann-Liouville fractional derivative of orderα>0 of a continuous function f:J→ℝ is given by
(4)Dt0αf(t)=1Γ(n-α)(dndtn∫t0t‍(t-s)n-α-1f(s)ds),hhhhhhhhhhhhhhhihn-1<α<n,n∈ℕ,
provided that the right side is pointwise defined on J.If0<α<1, then the Riemann-Liouville fractional derivative of order α of a function f reduces to
(5)Dt0αf(t)=1Γ(1-α)ddt∫t0t‍(t-s)-αf(s)ds.Note that the Riemann-Liouville fractional derivatives have singularity att0 and the fractional equations in the Riemann-Liouville sense require initial conditions at some point different from t0. To overcome this issue, Caputo [14] defined the fractional derivative in the following way.Definition 3 (see [10]).
Letα be a positive real number such that n-1<α≤n for n∈ℕ. The Caputo fractional derivative of order α of a function f is defined by
(6)CDt0αf(t)=1Γ(n-α)∫t0t‍(t-s)n-α-1f(n)(s)ds,
where f(n)(s)=dn/dsnf(s).When0<α<1, then the Caputo fractional derivative of order α of f reduces to
(7)CDt0αf(t)=1Γ(1-α)∫t0t‍f′(s)(t-s)αds.When0<α<1, we have
(8)CDt0αf(t)=Dt0α[f(t)-f(t0)]=Dt0αf(t)-f(t0)Γ(1-α)(t-t0)-α.
In particular, if f(t0)=0, then we have
(9)CDt0αf(t)=Dt0αf(t).Hence, we can see that the Caputo derivative is defined for functions for which the Riemann-Liouville derivative exists. Also, we note that the Mittag-Leffler functionsEα(z) and Eα,α(z) satisfy the more general differential relations
(10)CDt0αEα(λ(t-t0)α)=λEα(λ(t-t0)α),Dt0αEα,α(λ(t-t0)α)=λEα,α(λ(t-t0)α),
respectively, for λ∈ℝ.We can obtain the following asymptotic property forEα(λtα) and Eα,α+1(λtα) from the result [10, page 51].Lemma 4 (see [10]).
Whenα>0, then Eα,β(z) has different asymptotic behavior at infinity for 0<α<2 and α≥2. (1)
If0<α<2 and μ is a real number such that
(11)πα2<μ<min{π,πα},
then, for p∈ℕ∖{1}, the following asymptotic expansions are valid:
(12)Eα,β(z)=1αz(1-β)/αexp(z1/α)-∑k=1p‍1Γ(β-αk)1zk+O(1zp+1),
with |z|→∞,|arg(z)|≤μ; and
(13)Eα,β(z)=-∑k=1p‍1Γ(β-αk)1zk+O(1zp+1),
with |z|→∞,μ≤|arg(z)|≤π.(2)
Whenα≥2, then, for p∈ℕ∖{1}, the following asymptotic estimate holds:
(14)Eα,β(z)=1α∑n‍(z1/αexp[2nπiα])1-βexp[exp(2nπiα)z1/α]-∑k=1p‍1Γ(β-αk)1zk+O(1zp+1),
with |z|→∞,|arg(z)|≤απ/2, and where the first sum is taken over all integer n such that
(15)|arg(z)+2πn|≤απ2.Lemma 5.
Let0<α≤1 and λ<0. Then, Eα,α(λtα) and Eα,α+1(λtα) tend monotonically to zero as t→∞.Proof.
If we setβ=α and z=λtα in Lemma 4, then it follows from Lemma 4 that for p=2 we have
(16)Eα,α(λtα)=-∑k=12‍1Γ(α-αk)1λktkα+O(1λ3t3α)=-1Γ(-α)1λ2t2α+O(1λ3t3α)⟶0,ast⟶∞,μ≤|arg(λ)|≤π.
Thus, we have
(17)Eα,α(λtα)⟶0,ast⟶∞,|arg(λ)|>πα2.
For β=α+1, we also have
(18)Eα,α+1(λtα)⟶0,ast⟶∞,|arg(λ)|>πα2,
by the above similar argument. This completes the proof.Corollary 6.
Let0<α<1 and |arg(λ)|>πα/2. Then, one has
(19)tαEα,α+1(λtα)=-1λ-1Γ(1-α)λ2tα+O(1λ3t2α)ast⟶∞.


## 2. Main Results

Let0<q<1 and p=1-q. Denote by Cp([t0,T],ℝn) the function space
(20)Cp([t0,T],ℝn)={x∈C((t0,T],ℝn)∣x(t)(t-t0)p∈C([t0,T],ℝn)x∈C((t0,T],ℝn)∣x(t)(t-t0)p}.LetΩ⊂ℝn be a domain and f∈C([t0,t0+a]×Ω,ℝn). We consider the Caputo fractional differential system with the initial value
(21)CDt0qx=f(t,x),x(t0)=x0,
where f(t,0)=0. If x∈Cp([t0,t0+a],ℝn) satisfies (21), it also satisfies the Volterra fractional integral equation
(22)x(t)=x0+1Γ(q)∫t0t‍(t-s)q-1f(s,x(s))ds,hhhhhhhhhhhhhhhhht0≤t≤t0+a,
and vice versa.In the sequential we assume that the solutionx(t) of (21) exists globally on J=[t0,∞). See [5, Theorem 2.10.1] for the existence and uniqueness result.Next, we consider the nonhomogeneous linear fractional differential equation with Caputo fractional derivative(23)CDt0qx=λx+h(t),x(t0)=x0,
where h∈Cp(J,ℝ) is Hölder continuous with exponent q. Then, we get the unique solution of (23) as
(24)x(t)=x0Eq(λ(t-t0)q)+∫t0t‍(t-s)q-1Eq,q(λ(t-s)q)h(s)ds
for each t∈J.Lemma 7 (see [9, Lemma 3.2]).
If one setsh(t)≡d in (23) with a constant d, then the solution of (24) reduces to
(25)x(t)=x0Eq(λ(t-t0)q)+d(t-t0)q×Eq,q+1(λ(t-t0)q),t∈J.Remark 8.
Ifh(t)≡0, then it follows from Lemma 7 that
(26)x(t)=x0Eq(λ(t-t0)q),t∈J.
We can obtain the following Caputo fractional differential inequality of Gronwall type by Lemma7.Lemma 9.
Suppose thatm∈Cp(ℝ+,ℝ) satisfies
(27)CDt0qm(t)≤λm(t)+d,m(t0)=m0,t≥t0≥0,
where λ,d∈ℝ. Then one has
(28)m(t)≤m(t0)Eq(λ(t-t0)q)+d(t-t0)qEq,q+1(λ(t-t0)q),t≥t0≥0.Proof.
There exists a nonnegative functionn(t) satisfying
(29)CDt0qm(t)-λm(t)-d+n(t)=0,t≥t0≥0.
It follows from Lemma 7 that
(30)m(t)=m(t0)Eq(λ(t-t0)q)+d(t-t0)qEq,q+1(λ(t-t0)q)-n(t)*tq-1Eq,q(λ(t-t0)q),t≥t0,
where * denotes the convolution operator of nonnegative functions n(t) and tq-1Eq,q(λ(t-t0)q). Since n(t)*tq-1Eq,q(λ(t-t0)q) is nonnegative for each t≥t0, then we have
(31)m(t)≤m(t0)Eq(λ(t-t0)q)+d(t-t0)qEq,q+1(λ(t-t0)q),t≥t0≥0.
This completes the proof.Remark 10.
If we setq=1 and d=0 in Lemma 9, then we have
(32)m(t)≤m(t0)E1,1(λ(t-t0))=m(t0)eλ(t-t0),t≥t0≥0.
We can obtain the following result about fractional integral inequality. It is adapted from the comparison principle regarding nonstrict inequalities in [2, 5].Lemma 11 (see [8, Lemma 2.11]).
Let0<q<1 and g∈C(J×ℝ,ℝ+). Suppose that w,v∈C(J,ℝ+) satisfy the fractional integral inequality:
(33)v(t)-It0qg(t,v(t))<w(t)-It0qg(t,w(t)),
where It0qg(t,v(t))=1/(Γ(q))∫t0t‍(t-s)q-1g(s,v(s))ds and g(t,u) is monotonic nondecreasing in u for each t≥t0. If v(t0)<w(t0), then one has v(t)<w(t) on J.Pinto [15] introduced h-stability which is an important extension of the notions of exponential stability and uniform Lipschitz stability for differential equations.We will give the notion ofh-stability for Caputo fractional differential systems.Definition 12.
The zero solutionx=0 of (21) is said to be (1)
anh-system if there exist a constant c≥1 and a positive continuous function h:J→ℝ such that
(34)|x(t)|≤c|x(a)|h(t)h(a)-1,t≥a≥t0,
for |x(a)|≤δ. Here h(a)-1=1/h(a).(2)
h-stable if h is bounded.We recall the stability in the sense of Mittag-Leffler [8, 16].Definition 13.
The zero solutionx=0 of (21) is said to be a Mittag-Leffler system if
(35)|x(t)|≤{m(x(t0))Eq(λ(t-t0)q)}b,t≥t0,
where λ∈ℝ,b>0,m(0)=0,m(x)≥0, and m(x) are locally Lipschitz on x∈B⊆ℝn with Lipschitz constant m0.
The zero solutionx=0 of (21) is called Mittag-Leffler stable if the constant λ in (35) is nonpositive.Note that the Mittag-Leffler stability impliesh-stability, but the converse does not hold in general. See Remark 19 for the example.We can obtain the following result adapted from Theorem  3.4 in [8].Theorem 14.
Suppose that the functionf of (21) satisfies
(36)|f(t,x)|≤g(t,|x|),
where g∈C(J×ℝ,ℝ+) is monotonic increasing in u for each t∈J with g(t,0)=0. One considers the Caputo fractional differential equation
(37)CDt0qu(t)=g(t,u),u(t0)=u0,t≥t0.
If the zero solution u=0 of (37) is an h-system, then the zero solution x=0 of (21) is also an h-system whenever u0>|x(t0)|.Proof.
The equation (21) is equivalent to the following Volterra fractional integral equation:
(38)x(t)=x(t0)+It0qf(t,x(t)),t≥t0.
Then, we obtain
(39)|x(t)|=|x(t0)|+|It0qf(t,x(t))|≤|x(t0)|+It0q|f(t,x(t))|≤|x(t0)|+It0qg(t,|x(t)|),t≥t0.
Thus we have
(40)|x(t)|-It0qg(t,|x(t)|)≤|x(t0)|<u0=u(t)-It0qg(t,u(t)),t≥t0,
where u0=u(t0). By Lemma 11, we have x(t)<u(t) for all t≥t0.  Since u=0 of (37) is an h-system, there exist a constant c1≥1 and a positive continuous function h:J→ℝ such that
(41)|u(t)|≤c1|u(a)|h(t)h(t0)-1,t≥a≥t0,
for |u(a)|≤δ. Thus, we see that
(42)|x(t)|<u(t)≤c1|u(a)|h(t)h(a)-1=c|x(a)|h(t)h(a)-1,t≥a≥t0,
where u(a)=|x(a)|d with d>1 and c=c1d. This completes the proof.Corollary 15.
Suppose that all conditions of Theorem14 hold. The asymptotic stability of (37) implies the corresponding asymptotic stability of (21).We can obtain an upper bound of solutions for Caputo fractional differential equations via fractional Gronwall’s inequality. The following result is adapted from Theorem  5.1 in [7] and Theorem  3.15 in [9].Lemma 16.
Suppose thatD⊂ℝn is a domain containing the origin and q∈(0,1). Let V:ℝ+×D→ℝ be a continuously differentiable function and locally Lipschitz with respect to x satisfying
(43)α1|x|a≤V(t,x)≤α2|x|ab,(44)CDt0qV(t,x)≤-α3|x|ab+L,t≥t0,
where L∈ℝ and α1,α2,α3,a,b are positive constants. Then one has
(45)|x(t)|≤{α2α1|x0|abEq(-α3α2(t-t0)q)+Lα1(t-t0)qEq,q+1(-α3α2(t-t0)q)}1/a,hhhhhhhhhhhhhhhhhhhihhhhhht≥t0,
where x(t)=x(t,t0,x0) is any solution of (21).Proof.
It follows from (43) and (44) that
(46)CDt0qV(t,x)≤-α3|x|ab+L≤-α3α2V(t,x)+L,t≥t0.
It follows from Lemma 9 that
(47)V(t,x(t))≤V(t0,x(t0))Eq(-α3α2(t-t0)q)+L(t-t0)qEq,q+1(-α3α2(t-t0)q),t≥t0.
Substituting (47) into (43) yields
(48)|x(t)|≤{α2α1|x0|abEq(-α3α2(t-t0)q)+Lα1(t-t0)qEq,q+1(-α3α2(t-t0)q)}1/a,t≥t0.
This complete the proof.We can obtain the boundedness of solutions for Caputo fractional differential equations via the fractional Lyapunov direct method.Theorem 17.
Under the same assumptions of Lemma16, all solutions of (21) are eventually bounded on J.Proof.
Letx(t)=x(t,t0,x0) be any solution of (21). Then it follows from Lemma 16 that
(49)|x(t)|≤{α2α1|x0|abEq(-α3α2(t-t0)q)+Lα1(t-t0)qEq,q+1(-α3α2(t-t0)q)}1/a,
for each t≥t0. In view of Lemma 5 and Corollary 6, we note that Eq(-(α3/α2)(t-t0)q) tends monotonically zero as t→∞ and (t-t0)qEq,q+1(-(α3/α2)(t-t0)q) is eventually bounded on [t0,∞). Hence, there exist a positive constant M0=M(|x0|) and t1>t0 such that
(50)|x(t)|≤M0,t≥t1.
This completes the proof.We can obtain the following result [7, Theorem 5.1] about Mittag-Leffler stability of (21) as a corollary of Lemma 16.Corollary 18.
If one setsL=0 in the assumption of  Lemma 16, then the zero solution x=0 of (21) is Mittag-Leffler stable.


## 3. Examples

In this section we give tow examples which illustrate some results in the previous section.Example 1 (see [8]).
To illustrate Theorem14, we consider the Caputo fractional differential equation
(51)CDt01/2x(t)=xt(1+x2),t≥t0>0,
where f(t,x)=x/t1+x2. Then the zero solution x=0 of (51) is h-stable.Proof.
The functionf satisfies
(52)|f(t,x)|≤g(t,|x|)=|x|t,t>0,
and the solution of the Caputo fractional differential equation
(53)CD01/2u(t)=1tu,u(t0)=u0
is given by u(t)=ct-1/2e-1/t,t>0. We have
(54)u(t)=u0t0e1/t01te1/t=u0h(t)h(t0)-1,t≥t0>0,
where h(t)=t-1/2e-1/t. Thus, the zero solution u=0 of (53) is h-stable. Hence, the zero solution x=0 of (51) is h-stable by Theorem 14.Remark 19.
We note that the fractional differential equation (53) given in the proof of Example 1 is h-stable but not Mittag-Leffler stable.Proof.
Letu(t) be any solution of (53). Then, it follows from [8, Example 2.2] that u(t) is neither monotonic nondecreasing in t nor monotonic nonincreasing in t. Furthermore, we easily see that
(55)limt→0+u(t)=0=limt→∞u(t).
Suppose that (53) is Mittag-Leffler stable; that is, there exist positive constants λ and b satisfying
(56)|u(t)|≤{m(u(t0))E1/2(-λ(t-t0)1/2)}b,t≥t0>0,
where m(0)=0,m(x)≥0, and m(x) is locally Lipschitz in x. Since E1/2(-λ(t-t0)1/2) is monotonic nonincreasing in t [17], we see that the right-hand function of (56) also is monotonic decreasing in t. This contradicts the fact that u(t) has neither monotonic nondecreasing property nor monotonic nonincreasing property.Next, we will give an example to illustrate Theorem17.Example 2.
Let0<q<1. We consider the Caputo fractional differential equation
(57)CD0q|x(t)|=-|x(t)|+f(t),t≥0,
where f∈C(ℝ+,ℝ) is bounded by a constant d. Let V(t,x)=|x|. Then it follows that
(58)CD0qV(t,x)=CD0q|x|=-|x|+f(t)≤-V(t,x)+d,t≥0.
Let α1=α2=α3=a=b=1. Applying them in  Lemma 16 gives
(59)|x(t)|≤|x(0)|Eq(-tq)+dtqEq,q+1(-tq),t≥0.
Hence, all solutions of (57) are eventually bounded by Theorem 17.



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