Linearized asymptotic stability for fractional differential equations

Abstract

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 {λ ∈ C : | arg λ| > απ 2 } where α > 0 denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable. * ndcong@math.ac.vn,

On asymptotic properties of solutions to fractional differential equations

“…Our task is to study the asymptotic behavior of solutions to (8) around the origin. In [10], the authors give a linearized stability theorem for the trivial solution of (8) as follows.…”

“…After that in [33,6,35], the authors formulated theorems on linearized stability. Unfortunately, as showed in [10,Remark 3.7], these papers contain some serious flaws in the proof of the linearization theorems. Using other tools, the authors of [10] improved the assertions presented in the papers mentioned above and gave a powerful stability criterion.…”

“…Unfortunately, as showed in [10,Remark 3.7], these papers contain some serious flaws in the proof of the linearization theorems. Using other tools, the authors of [10] improved the assertions presented in the papers mentioned above and gave a powerful stability criterion. However, the convergence rate of solutions to an equilibrium point is still unavailable.…”

Asymptotic Stability of Linear Fractional Systems with Constant Coefficients and Small Time-Dependent Perturbations

“…Proof. We follow the lines of the proof of Theorem 5 in [9] with some modifications to adapt to our case. Let ε > 0 be arbitrary.…”

“…In this paper we will show that this is also the case for fractional differential equations. Note that if the unperturbed system (5) is asymptotically stable and the nonlinear perturbation f having Lipschitz constant uniformly small in a neighborhood of the origin, then the trivial solution of the nonlinear perturbed system (1) is also asymptotically stable, see [9].…”