Lyapunov Stability Theory with Some Applications

In this research, a detailed overview of Lyapunov stability theorems of linear and nonlinear systems is presented. The Lyapunov first and second methods are investigated and the stability analysis of fractional differential systems is highlighted. A new Lemma for the Caputo fractional derivative is reviewed and a class of fractional-order gene regulatory networks is investigated. Besides the stabilization of continuous-time fractional for positive linear systems is reviewed. An elementary Lemma which estimates the fractional derivatives of Volterra-type Lyapunov functions is also put forward, in order to see how it can satisfy the uniform asymptotic stability of Caputo-type epidemic systems.