Opuscula Math. 43, no. 1 (2023), 67-79
https://doi.org/10.7494/OpMath.2023.43.1.67

Opuscula Mathematica

# Nonoscillation of damped linear differential equations with a proportional derivative controller and its application to Whittaker-Hill-type and Mathieu-type equations

Kazuki Ishibashi

Abstract. The proportional derivative (PD) controller of a differential operator is commonly referred to as the conformable derivative. In this paper, we derive a nonoscillation theorem for damped linear differential equations with a differential operator using the conformable derivative of control theory. The proof of the nonoscillation theorem utilizes the Riccati inequality corresponding to the equation considered. The provided nonoscillation theorem gives the nonoscillatory condition for a damped Euler-type differential equation with a PD controller. Moreover, the nonoscillation of the equation with a PD controller that can generalize Whittaker-Hill-type equations is also considered in this paper. The Whittaker-Hill-type equation considered in this study also includes the Mathieu-type equation. As a subtopic of this work, we consider the nonoscillation of Mathieu-type equations with a PD controller while making full use of numerical simulations.

Keywords: nonoscillation, proportional derivative controller, Riccati technique, Mathieu equation, Whittaker-Hill equation.

Mathematics Subject Classification: 34C10, 26A24, 34B30.

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• Kazuki Ishibashi
• https://orcid.org/0000-0003-1812-9980
• Department of Electronic Control Engineering, National institute of Technology (KOSEN), Hiroshima College, Toyota-gun 725-023, Japan
• Communicated by Josef Diblík.
• Revised: 2022-10-21.
• Accepted: 2022-11-04.
• Published online: 2022-12-30.