Stabilization of second order nonlinear equations with variable delay

TL;DR

This paper presents a method to stabilize unstable motions in second order nonlinear non-autonomous systems with variable delays by combining proportional state control and derivative feedback, applicable even with delay perturbations.

Contribution

It introduces a stabilization approach using combined proportional and derivative feedback for systems with variable delays, applicable to complex models like the sunflower equation.

Findings

Effective stabilization of unstable motions achieved

Delay bounds are sufficient for stabilization

Applicable to systems with infinite equilibrium points

Abstract

For a wide class of second order nonlinear non-autonomous models, we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal, allows to stabilize unstable motions of the system. The delays are variable, which leads to more flexible controls permitting delay perturbations; only delay bounds are significant for stabilization by a delayed control. The results are applied to the sunflower equation which has an infinite number of equilibrium points.