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

Abstract

Our aim in this paper is to investigate the asymptotic behavior of solutions of the perturbed linear fractional differential system. We show that if the original linear autonomous system is asymptotically stable then under the action of small (either linear or nonlinear) nonautonomous perturbations the trivial solution of the perturbed system is also asymptotically stable.

On asymptotic properties of solutions to fractional differential equations

“…(ii) Let x ∈ B C d (0, r * ) and ξ ∈ B Cw (0, r). According to (12) in Proposition 18, we obtain that…”

“…Then, there exists a positive constant C 3 such that∞ 0 τ α−1 |E α,α (λτ α )| dτ < C 3 .Proof. See[12, Theorem 3(ii)].…”

Converse theorems in Lyapunov’s second method and applications for fractional order systems

“…The eigenvalues of matrix A ∈ R n×n belong to Λ α := {λ ∈ C \ {0} : | arg(λ)| > απ 2 } . Lyapunov stability of x = 0 at t = 0 has been proved using the explicit solution to system (5.1) (see e.g., [2]).…”

“…However, (5.1) is asymptotically stable if the eigenvalues of A belong to Λ α (see e.g. [2]). Then, condition This is intuitive since the stability used for the latter is weaker than stability for integer order systems.…”

Robustness and Convergence of Fractional Systems and their Applications to Adaptive Schemes

“…In recent papers [1,2,3,4] (1.1) where we have implicitly assumed that the initial time for the derivative t = 0, is the same that the time for the arbitrary initial condition x 0 ∈ R n . They were able to prove ([1, Theorem 5]) that if Q : [0, ∞) → R n×n is a continuous matrix function such that sup t≥0 t 0 τ α−1 ||E α,α (A, τ )[Q(t − τ )]||dτ < 1, where E α,α (A, t) := E α,α (t α A) is the two-parameter Mittagc 2017 Diogenes Co., Sofia pp.…”