Stability Analysis of Fractional-Order Periodic Piecewise Nonlinear Systems

Abstract

This paper investigates the stability problem of fractional-order periodic piecewise nonlinear systems. As a preparation, the existence and uniqueness of solutions of the system are derived under the Lipschitz condition and expressed in an equivalent integral expression based on the memorability of fractional derivative. We further propose an upper bound of the system states by using the Laplace transform technique. With the help of piecewise Lyapunov functions and class- $\mathcal {K}$ functions, some sufficient conditions are given to verify the Mittag-Leffler and asymptotic stability of the investigated system mixed with stable and unstable subsystems. Furthermore, asymptotic stability criteria are proposed for a general case, i.e., all subsystems are unstable. Three numerical simulation examples are finally provided to illustrate the effectiveness of the proposed results.

Keywords

• Asymptotic stability
• fractional-order periodic piecewise nonlinear systems
• Mittag-leffler stability
• piecewise Lyapunov function