On Hyers–Ulam Mittag-Leffler stability of discrete fractional Duffing equation with application on inverted pendulum

Abstract

Abstract A human being standing upright with his feet as the pivot is the most popular example of the stabilized inverted pendulum. Achieving stability of the inverted pendulum has become common challenge for engineers. In this paper, we consider an initial value discrete fractional Duffing equation with forcing term. We establish the existence, Hyers–Ulam stability, and Hyers–Ulam Mittag-Leffler stability of solutions for the equation. We consider the inverted pendulum modeled by Duffing equation as an example. The values are tabulated and simulated to show the consistency with theoretical findings.

Keywords

• Fractional Duffing equation
• Mittag-Leffler function
• Hyers–Ulam stability
• Inverted pendulum