Lyapunov functions for fractional order h-difference systems

TL;DR

This paper develops new stability criteria for fractional order h-difference systems using Lyapunov functions, extending the discrete fractional Lyapunov method with quadratic and polynomial forms.

Contribution

It introduces novel propositions for fractional order h-difference operators and Lyapunov functions, enabling stability analysis of such systems with general quadratic and polynomial Lyapunov functions.

Findings

New propositions for fractional order h-difference operators.

Stability proofs using quadratic and polynomial Lyapunov functions.

Illustrative examples demonstrating the results.

Abstract

This paper presents some new propositions related to the fractional order $h$-difference operators, for the case of general quadratic forms and for the polynomial type, which allow proving the stability of fractional order $h$-difference systems, by means of the discrete fractional Lyapunov direct method, using general quadratic Lyapunov functions, and polynomial Lyapunov functions of any positive integer order, respectively. Some examples are given to illustrate these results.