Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities

Oscar Martínez-Fuentes, Fidel Meléndez-Vázquez, Guillermo Fernández-Anaya and José Francisco Gómez-Aguilar
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Oscar Martínez-Fuentes: School of Engineering, Universidad Anáhuac Veracruz, Campus Xalapa, Circuito Arco Sur s/n, Col. Lomas Verdes, Xalapa 91098, Mexico
Fidel Meléndez-Vázquez: Departamento de Física y Matemáticas, Universidad Iberoamericana Ciudad de México, Prol. Paseo de la Reforma 880, Lomas de Santa Fe, Alvaro Obregon, Mexico City 01219, Mexico
Guillermo Fernández-Anaya: Departamento de Física y Matemáticas, Universidad Iberoamericana Ciudad de México, Prol. Paseo de la Reforma 880, Lomas de Santa Fe, Alvaro Obregon, Mexico City 01219, Mexico
José Francisco Gómez-Aguilar: CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira s/n, Col. Palmira, Cuernavaca 62490, Mexico

Mathematics, 2021, vol. 9, issue 17, 1-29

Abstract: In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we determine a form of calculating its fractional derivative; this result is essential due to its connection to the fractional derivative of Lyapunov functions. In addition, some other new results are developed, leading to Lyapunov-like theorems and a Lyapunov direct method that serves to prove asymptotic stability in the sense of the operators with general analytic kernels. The FOB-stability concept is introduced, which generalizes the classical Mittag–Leffler stability for a wide class of systems. Some inequalities are established for operators with general analytic kernels, which generalize others in the literature. Finally, some new stability results via convex Lyapunov functions are presented, whose importance lies in avoiding the calculation of fractional derivatives for the stability analysis of dynamical systems. Some illustrative examples are given.

Keywords: nonlinear fractional-order systems; general analytic kernels; FOB-stability; generalized lyapunov stability; inequalities (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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