Finite-time stability analysis of fractional differential systems with variable coefficients

Abstract

In this paper, we study finite-time stability of fractional differential systems with variable coefficients, which includes the homogeneous and nonhomogeneous delayed cases. Based on the theories of fractional differential equations, we obtain three theorems on the finite-time stability, which give some sufficient conditions on finite-time stability, respectively, for homogeneous systems without and with time delay and for the nonhomogeneous system with time delay.

A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis

“…Definition 3. [27] The fractional system given by (1) satisfying the initial state (2) is finite-time stable with respect to {0, J T , δ, ε, h} with t ∈ J T and δ ≤ ε if and only if the inequality Φ < δ implies that X(t) < ε for each t ∈ J T , where X(t) is the unique solution of IP (1), (2).…”

“…In the works [24,25] the so-called 1-norm is used (i.e., for W = {w ij } i,j∈ n ∈ R n×n the matrix norm [26,27] is used the spectral norm as well as in our work. A direct comparison shows that the condition (43) in our work based on the estimate ( 18) is more accurate in compare with the condition ( 9) in Theorem 4.1 [24] proved via the integral representation approach and condition (16) in Theorem 3.2 in [27] proved by Gronwall's approach, even in the partial cases considered in these works. Please note that for the partial case when Φ is a constant both conditions (43) and ( 9) in [24] coincide.…”

“…A historical overview of this theme can be obtained from the survey of Dorato [20]. Concerning the more recent works devoted to the different approaches to study the finite-time stability, we refer to the works [21][22][23][24][25][26][27][28][29][30].…”

“…The aim of our work, motivated by remarkable works [24][25][26][27], is twofold. First, we obtain a priori estimates using the two most popular approaches and then compare the precisions of the obtained via them estimates.…”

Finite-time stability of multiterm fractional nonlinear systems with multistate time delay

“…Zhang and Niu [42] discussed the exponential stability of a class of nonlinear delay-integrodifferential equations. In [43], the analysis of FTS of fractional systems with variable coefficients with α ∈ (0, 1) was examined using certain sufficient inequalities which were obtained by applying the Hölder and generalized Gronwall inequalities. Zhang et al [44] discussed the stability concept for fractional nonlinear systems with order from (0, 2).…”