Stability Analysis on Nabla Discrete Distributed-Order Dynamical System

Abstract

This paper addresses the problems of the stability of a nabla discrete distributed-order dynamical system (NDDS). Firstly, based on a proposed generalized definition of discrete integral, some related definitions of nabla discrete distributed-order calculus are given. Then, several useful inequalities in sense of nabla discrete fractional-order difference are extended to distributed-order cases. Meanwhile, on basis of the proposed inequalities and Lyapunov direct method, some sufficient conditions guaranteeing the asymptotic stability of the origin of NDDS are established under both the Caputo and Riemann–Liouville sense. Finally, some designed simulation examples are given to validate the correctness and practicability of the obtained results.

Keywords

• nabla discrete distributed-order calculus
• asymptotic stability
• nabla Laplace transformation
• Lyapunov direct method