Abstract
This article addresses new applications of the generalized Mittag-Leffler input stability to the fractional-order electrical circuits. We consider the fractional-order electrical circuits in the context of the generalized Caputo-Liouville derivative. We propose the Lyapunov characterizations of the fractional differential equations. A new numerical discretization, including the fractional differential equations represented by the generalized Caputo derivative, has been successfully applied to the fractional electrical circuits. To support the results, we have proposed the graphics generated by our numerical discretization. The graphics of the solutions have been analyzed and interpreted in the context of generalized Mittag-Leffler input stability and the generalized Mittag-Leffler stability. The generalized Mittag-Leffler input stability is a new stability notion for the fractional differential equations recently introduced in the literature.
Highlights
For the applications of the fractional operators with exponential kernel and the Mittag-Leffler kernel, we advise readers to check the following works [3], [4]. These recent years, modeling electrical circuits using fractional operators have been introduced in the literature. The motivations of these introductions are due to the fact the fractional operators describe more realistically the real-world problems, and another reason is the fractional operators take into account the memory effect
The illustrations of these two Lemmas will be done when we investigate on the generalized Mittag-Leffler input stability of the RL, LC, RC, and RLC electrical circuits
NUMERICAL DISCRETIZATION AND THE FIGURES WITH INTERPRETATIONS OF THE FRACTIONAL-ORDER CIRCUITS we propose a novel numerical discretization of the fractional-order differential equations described by the generalized Caputo-Liouville derivative
Summary
F RACTIONAL calculus is a new arena focusing in many domains, like physics [16], [19], [23], [35], [37], mechanics [23], fluid models [27], science and engineering [16], [24], mathematical modeling in biology [36], mathematical physics [16], [17], [20], [30], mathematical modeling [3], [31], [32], [33] and others [25], [29], [35]. For the applications of the fractional operators with exponential kernel and the Mittag-Leffler kernel, we advise readers to check the following works [3], [4] These recent years, modeling electrical circuits using fractional operators have been introduced in the literature. SENE: GENERALIZED MITTAG-LEFFLER INPUT STABILITY OF THE FRACTIONAL-ORDER ELECTRICAL CIRCUITS work on the RLC electrical circuit with Caputo derivative and have suggested the analytical solutions of the considered models. In [21], Sene and Gomez-Aguilar have proposed the analytical solution of the fractional-order RL, RC, LC, and RLC electrical circuits by considering different types of fractional operators. We mainly consider the fractional-order RL, RC, LC, and RLC electrical circuits represented by the generalized Caputo-Liouville derivative. The main objective is to analyze the generalized Mittag-Leffler input stability of the considered fractional-order electrical circuits. The importance of the Lyapunov direct approach can be explained by the fact the analytical solutions of the fractional differential equations are not all time trivial, and the Lyapunov functions give alternative ways to study the stability notions
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