New Estimates of Lyapunov-Type Functions and Stability Analysis for a Class of Generalized Fractional Derivative with Non-Singular Kernel and Applications

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Résumé

In this paper, we propose an estimation of a class of generalized weighted fractional derivatives with non-singular kernel of Hattaf in the Caputo sense, applied to certain functions useful in the construction of Lyapunov-type candidate functions.We derive inequalities associated with the Caputo-Fabrizio fractional derivative in the Caputo sense, as well as with the Atangana-Baleanu fractional derivative in the Caputo sense.Moreover, we also establish an extension of the direct Lyapunov method to analyze the asymptotic stability of fractional differential equations with the generalized weighted fractional derivative of Hattaf in the sense of Caputo.Furthermore, we apply these results to prove the asymptotic stability of some examples of systems of differential equations governed by the generalized fractional derivative of Hattaf.In addition, for each example we carry out numerical simulations to illustrate our theoretical results.

Mots-clés

non-singular kernel, Caputo fractional derivative, Caputo-Fabrizio derivative, Atangana-Baleanu derivative, fractional differential equations, asymptotic stability, Lyapunov direct method, inequality (mathematics)