Stability of fractional‐order nonlinear systems by Lyapunov direct method

Abstract

In this paper, by using a characterization of functions having fractional derivative and properties of positive solutions to a Volterra integral equation, we propose a rigorous fractional Lyapunov function candidate method to analyze the stability of fractional-order nonlinear systems. First, we prove an inequality concerning the fractional derivatives of convex Lyapunov functions without the assumption of the existence of derivative of pseudo-states. Second, we establish fractional Lyapunov functions to fractional-order systems without the assumption of the global existence of solutions. Our theorems fill the gaps and strengthen results in some existing papers. * httuan@math.ac.vn,

On asymptotic properties of solutions to fractional differential equations

“…It improves and strengthens a recent result by Tuan and Hieu [30,Theorem 3]. In particular, we removed the condition c > b in the statement of [30,Theorem 3(c)]. Moreover, we obtained the Mittag-Leffler stability of the trivial solution instead of the weakly asymptotic stability.…”

“…The authors combined effective fractional derivative inequalities [2, inequalities (6) and (16)], [15, inequality (24)], [7, inequality (10)] and mains results in [21,22] to obtain the estimation of solutions to FDEs. However, in those papers there are some shortcomings of that approach and some flaws in the proofs, which were shown in [30]. Recently, using other tools the authors of [30] were able to avoid the shortcomings and flaws mentioned above and proposed a rigorous method of fractional Lyapunov candidate functions to study the weakly asymptotical stability for FDEs.…”

Stability and Stabilization for a Class of Semilinear Fractional Differential Systems

“…where M, M 3 , and ω are constants appeared in (13) and (8), then equation (9), with the initial condition x(0) � x 0 and x′(0) � x 1 , has a unique solution in [0, T].…”

“…e modelling and stability of the water jet mixed-flow pump fractional-order shafting system has also been studied [11]. e stability of fractional-order nonlinear systems with 0 < α < 1 was derived in [6,12,13], according to the Lyapunov approach. Based on the uncertain Takagi-Sugeno fuzzy model, the stability problems of nonlinear fractional-order systems were studied, whereas the sidingmode control approach was used to investigate the stabilization and synchronization problems of the nonlinear fractional-order system (e.g., [14][15][16][17][18]).…”

Generalized Mittag-Leffler Input Stability of the Fractional Differential Equations

“…In [9], Song et al analyze the stability of the fractional differential equations with time variable impulses. In [10], Tuan et al propose a novel methodology for studying the stability of the fractional differential equations using the Lyapunov direct method. In [11], Makhlouf studies the stability with respect to part of the variables of nonlinear Caputo fractional differential equations.…”