Linearized Asymptotic Stability for Fractional Differential Equations

We prove the theorem of linearized asymptotic stability for fractional differential equations. More precisely, we show that an equilibrium of a nonlinear Caputo fractional differential equation is asymptotically stable if its linearization at the equilibrium is asymptotically stable. As a consequence we extend Lyapunov's first method to fractional differential equations by proving that if the spectrum of the linearization is contained in the sector $\{\lambda \in \C : |\arg \lambda|>\frac{\alpha \pi}{2}\}$ where $\alpha>0$ denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable.


Introduction
In recent years, fractional differential equations have attracted increasing interest due to the fact that many mathematical problems in science and engineering can be modeled by fractional differential equations, see e.g., [5,6,12].
One of the most fundamental problems in the qualitative theory of fractional differential equations is stability theory. Following Lyapunov's seminal 1892 thesis [7], two methods are expected to also work for fractional differential equations: • Lyapunov's First Method: The method of linearization of the nonlinear equation along an orbit, the study of the resulting linear variational equation by means of Lyapunov exponents (exponential growth rates of solutions), and the transfer of asymptotic stability from the linear to the nonlinear equation (the so-called theorem of linearized asymptotic stability).
• Lyapunov's Second Method: The method of Lyapunov functions, i.e., of scalar functions on the state space which decrease along orbits.
There have been many publications on Lypunov's second method for fractional differential equations and we refer the reader to [10] or [9] for a survey.
In this paper we develop Lyapunov's first method for the trivial solution of a fractional differential equation of order α ∈ (0, 1) where A ∈ R d×d and f : R d → R d is a continuously differentiable function satisfying that f (0) = 0 and Df (0) = 0 (in fact, we only require a slightly weaker assumption on f ). The asymptotic stability of (the trivial solution of) its linerization C D α 0+ x(t) = Ax(t) is known to be equivalent to its spectrum lying in the sector {λ ∈ C : | arg λ| > απ 2 }, see [5,Theorem. 7.20]. What remains to be shown is that the asymptotic stability of (2) implies the asymptotic stability of the trivial solution of (1) which is our main result Theorem 5 on linearized asymptotic stability for fractional differential equations.
The linearization method is a useful tool in the investigation of stability of equilibria of nonlinear systems: it reduces the problem to a much simpler problem of stability of autonomous linear systems which can be solved explicitly, hence it gives us a criterion for stability of the equilibrium of the nonlinear system. Our theorem does the same service to the investigation of stability of nonlinear fractional differential equations as its classical counterpart does for the investigation of stability of nonlinear ordinary differential equations.
Note that there are several people dealing with the stability of fractional differential equations similar to our problem: in [1] our Theorem 5 is stated but without a complete proof; the main literature we are aware of are four papers [2,13,16,17] where the authors formulated a theorem on linearized stability under various assumptions but all these four papers contain serious flaws in the proofs of the theorem which make the proofs incorrect, a detailed discussion can be found in Remark 11.
The structure of this paper is as follows: In Section 2, we recall some background on fractional calculus and fractional differential equations. Section 3 is devoted to the main theorem about linear asymptotic stability for fractional differential equations. Section 4 contains an application of our main result (Theorem 5) and discusses a stabilization by linear feedback of a fractional Lotka-Volterra system. We conclude this introductory section by introducing some notation which is used throughout the paper.
Let R d be endowed with the max norm, i.e., x = max(|x 1 |, . . . , |x d |) for all x = (x 1 , . . . , x d ) T ∈ R d , let R ≥0 be the set of all nonnegative real numbers and C ∞ (R ≥0 , R d ), · ∞ denote the space of all continuous functions ξ : It is well known that C ∞ (R ≥0 , R d ), · ∞ is a Banach space.

Preliminaries
We start this section by briefly recalling a framework of fractional calculus and fractional differential equations. We refer the reader to the books [5,6] for more details.
Then, the Riemann-Liouville integral operator of order α is defined by where the Euler Gamma function Γ : (0, ∞) → R is defined as see e.g., [5]. The Caputo fractional derivative C D α a+ x of a function x ∈ C m ([a, b]), m := ⌈α⌉ is the smallest integer larger or equal α, which was introduced by Caputo (see e.g., [5]), is defined by T . Since f is Lipschitz continuous, [5, Theorem 6.5] implies unique existence of solutions of initial value problems (1), We now recall the notions of stability and asymptotic stability of the trivial solution of (1), cf. [5, Definition 7.2, p. 157].
Definition 1. The trivial solution of (1) is called: • unstable if it is not stable.
• attractive if there exists δ > 0 such that lim t→∞ ϕ(t, x 0 ) = 0 whenever The trivial solution is called asymptotically stable if it is both stable and attractive.
For f = 0, system (1) reduces to a linear time-invariant fractional differential equation As shown in [5], where the Mittag-Leffler matrix function E α,β (A), for β ∈ R and a matrix A ∈ R d×d is defined as In the following theorem, we recall a spectral characterization on asymptotic stability of the trivial solution of (3).
In the remaining part of this section, we establish some estimates involving the Mittag-Leffler functions. These estimates will be used to prove the contraction property of the Lyapunov-Perron operator introduced in the next section. For this purpose, let γ(ε, θ), ε > 0, θ ∈ (0, π] denote the contour consisting of the following three parts: The contour γ(ε, θ) divides the complex plane (z) into two domains, which we denote by G − (ε, θ) and G + (ε, θ). These domains lie correspondingly on the left and on the right side of the contour γ(ε, θ).
(ii) There exists a positive constant C(α, λ) such that Using the identity leads to Let t 0 := 1 which together with (4) implies that Consequently, for all t ≥ t 0 In what follows, we treat separately two cases t ≤ t 0 and t > t 0 , where t 0 is defined as in the statement (i).
see, e.g., [12, pp. 24]. Therefore, we get that Case 2: t > t 0 : From (i), we see that (5) Using a similar statement as in Case 1, we obtain that . The proof is complete.

Linearized Asymptotic Stability for Fractional Differential Equations
We now state the main result of this paper and use the abbreviation ℓ f (r) to denote the Lipschitz constant Theorem 5 (Linearized Asymptotic Stability for Fractional Differential Equations). Consider the nonlinear fractional differential equation (1). Let λ 1 , . . . ,λ m denote the eigenvalues of A and assume that Suppose that the nonlinear term f : R d → R d is a locally Lipschitz continuous function satisfying that Then, the trivial solution of (1) is asymptotically stable.
Before going to the proof of this theorem, we need two preparatory steps: • Transformation of the linear part: The aim of this step is to transform the linear part of (1) to a matrix which is "very close" to a diagonal matrix. This technical step reduces the difficulty in the estimation of the operators constructed in the next step.
• Construction of an appropriate Lyapunov-Perron operator: In this step, our aim is to present a family of operators with the property that any solution of the nonlinear system (1) can be interpreted as a fixed point of these operators. Furthermore, we show that these operators are contractive and hence the fixed points of these operators can be estimated and can be shown to tend to zero when time goes to infinity.
We are now presenting the details of these preparatory steps.

Transformation of the linear part
Using [14, Theorem 6.37, pp. 146], there exists a nonsingular matrix T ∈ C d×d transforming A into the Jordan normal form, i.e., where for i = 1, . . . , n the block A i is of the following form where η i ∈ {0, 1}, λ i ∈ {λ 1 , . . . ,λ m }, and the nilpotent matrix N d i ×d i is given by Let us notice that by this transformation we go from the field of real numbers out to the field of complex numbers, and we may remain in the field of real numbers only if all eigenvalues of A are real. For a general real-valued matrix A we may simply embed R into C, consider A as a complex-valued matrix and thus get the above Jordan form for A. Alternatively, we may use a more cumbersome real-valued Jordan form (for discussion of a similar issue for FDE see also Diethelm [5,). For simplicity we use the embedding method and omit the discussion on how to return back to the field of real numbers. Note also that this kind of technique is well known in the theory of ordinary differential equations.
Remark 6. Note that the map x → diag(δ 1 N d 1 ×d 1 , . . . , δ n N dn×dn )x is a Lipschitz continuous function with Lipschitz constant δ. Thus, by (6) we have Remark 7. The type of stability of the trivial solution of equations (1) and (7) are the same, i.e., they are both stable, attractive or unstable.

Construction of an appropriate Lyapunov-Perron operator
In this subsection, we concentrate only on equation (7). We are now introducing a Lyapunov-Perron operator associated with (7). Before doing this, we discuss some conventions which are used in the remaining part of this section: The space R d can be written as R d = R d 1 × · · · × R dn . A vector x ∈ R d can be written component-wise as x = (x 1 , . . . , x n ).
For any is called the Lyapunov-Perron operator associated with (7). The role of this operator is stated in the following theorem.
Theorem 8. Let x ∈ R d be arbitrary and ξ : R ≥0 → R d be a continuous function satisfying that ξ(0) = x. Then, the following statements are equivalent: (i) ξ is a solution of (7) satisfying the initial condition x(0) = x.
(ii) ξ is a fixed point of the operator T x .
Proof. The assertion follows from the variation of constants formula for fractional differential equations, see e.g., [6].
Next, we provide some estimates on the operator T x . The main ingredient to obtain these estimates is the preparatory work in Proposition 4.
So far, we have proved that the Lyapunov-Perron operator is well-defined and Lipschitz continuous. Note that the Lipschitz constant C(α, λ) is independent of the constant δ which is hidden in the coefficients of system (7). From now on, we choose and fix the constant δ as follows δ := 1 2C(α,λ) . The remaining difficult question is now to choose a ball with small radius in C ∞ (R ≥0 , R d ) such that the restriction of the Lyapunov-Perron operator to this ball is strictly contractive. A positive answer to this question is given in the following technical lemma. Lemma 10. The following statements hold: (i) There exists r > 0 such that (ii) Choose and fix r > 0 satisfying (10). Define and Proof. (i) By Remark 6, lim r→0 ℓ h (r) ≤ δ. Since δC(α, λ) = 1 2 , the assertion (i) is proved.
(ii) Let x ∈ R d be arbitrary with x ≤ r * . Let ξ ∈ B C∞ (0, r) be arbitrary. According to (9) in Proposition 9, we obtain that which proves that T x (B C∞ (0, r)) ⊂ B C∞ (0, r). Furthermore, by Proposition 4 and part (i) for all x ∈ B R d (0, r * ) and ξ, ξ ∈ B C∞ (0, r) we have which concludes the proof.
Proof of Theorem 5. Due to Remark 7, it is sufficient to prove the asymptotic stability for the trivial solution of system (7). For this purpose, let r * be defined as in (11). Let x ∈ B R d (0, r * ) be arbitrary. Using Lemma 10 and the Contraction Mapping Principle, there exists a unique fixed point ξ ∈ B C∞ (0, r) of T x . This point is also a solution of (7) with the initial condition ξ(0) = x. Since the initial value problem for Equation (7) has unique solution, this shows that the trivial solution 0 is stable. To complete the proof of the theorem, we have to show that the trivial solution 0 is attractive. Suppose that ξ(t) = ((ξ) 1 (t), . . . , (ξ) n (t)) is the solution of (7) which satisfies ξ(0) = x for an arbitrary x = (x 1 , . . . , x n ) ∈ B R d (0, r * ). From Lemma 10, we see that ξ ∞ ≤ r. Put a := lim sup t→∞ ξ(t) , then a ∈ [0, r]. Let ε be an arbitrary positive number. Then, there exists T (ε) > 0 such that for any t ≥ T (ε).
Due to the assumption ℓ h (r)C(α, λ) < 1, we get that a = 0 and the proof is complete.

Applications
In this section, we revisit the problem of stablization by linear feedback of the following fractional Lotka-Volterra system:    C D α 0+ x 1 (t) = x 1 (t)(h + ax 1 (t) + bx 2 (t)), C D α 0+ x 2 (t) = x 2 (t)(−r + cx 1 (t)), where the parameters h, r are positive, see e.g., [1,16]. This system can be rewritten as follows where In the following lemma, we first prove instability of the trivial solution for system (12). Finally, we show that by using a suitable state-feedback controller, the controlled system becomes stable.
Lemma 12. The following statements hold: (i) The trivial solution of (12) is unstable.