Lyapunov Direct Method for Nonlinear Hadamard-Type Fractional Order Systems

Abstract

In this paper, a rigorous Lyapunov direct method (LDM) is proposed to analyze the stability of fractional non-linear systems involving Hadamard or Caputo–Hadamard derivatives. Based on the characteristics of Hadamard-type calculus, several new inequalities are derived for different definitions. By means of the developed inequalities and modified Laplace transform, the sufficient conditions can be derived to guarantee the Hadamard–Mittag–Leffler (HML) stability of the systems. Lastly, two illustrative examples are given to show the effectiveness of our proposed results.

Keywords

• Hadamard derivative
• Caputo—Hadamard derivative
• inequalities
• Lyapunov direct method
• asymptotical stability