For Hilfer derivatives of the product of two functions, we present equations and inequalities, generalizing well‐known results for Caputo and Riemann‐Liouville derivatives. Using the Laplace transformation, we introduce a generalized distributed Mittag‐Leffler‐Hilfer stability and show two results for like‐Lyapunov stability. We also extend equations and inequalities for the product of two functions of Hilfer derivatives of distributed order. Finally, we give some consequences and examples that illustrate the theory.
In this article two different controllers for the stabilization of a fractional-order discrete system in the left Caputo discrete delta operator sense are given. The first one acts by a fractional proportional pulse control, the second acts by a fractional feedback control. These controllers are applied to fractional-order chaotic discrete dynamical systems to obtain their stability and also we show a comparison with the integer order dynamical systems stability. Some simulations are presented for the fractional logistic map and the fractional Henon map.INDEX TERMS Caputo delta difference, chaos, discrete fractional calculus, Mittag-Leffler.LUIS ALBERTO QUEZADA-TÉLLEZ received the Ph.D. degree in engineering sciences from Universidad Iberoamericana, where he is also a Professor with the Department of Physics and Mathematics. He has served as a Teacher in several institutions of higher education, both public and private. He is also an Associate Member of the Center for Complexity Sciences, UNAM. He has participated in multiple lectures and has various publications in national and international journals. His areas of research interest include nonlinear systems, chaos theory, control theory, bifurcations, and vehicular traffic models.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.