On stability of a harmonic oscillator with a delayed feedback system

On stability of a harmonic oscillator with a delayed feedback system

Li, Vincent (TU Delft Electrical Engineering, Mathematics and Computer Science; TU Delft Delft Institute of Applied Mathematics)

van Horssen, Wim (mentor)
Spandaw, Jeroen (graduation committee)
Budko, Neil (graduation committee)

Delft University of Technology

Applied Mathematics

2019-07-03

In this thesis the stability type of y=0 is being considered for the delay differential equation y''(t) + ay(t) + by(t-1) = 0 with a and b real numbers. It is already known that y=0 is stable when b=0 and a>0 and unstable when b=0 and a<0. The aim of this project is to determine the stability of y=0 for all values of a and b. First, the general stability theory for delay differential equations was highlighted before giving an in-depth stability analysis of the equation y''(t) + ay(t) + by(t-1) = 0. It turns out that a theorem of Pontryagin (1908 -1988) is really helpful for answering these stability questions. Due to this theorem all values for a and b are determined such that y=0 is asymptotically stable for y''(t) + ay(t) + by(t-1) = 0. However, this does not cover the stability type of y=0 for all values of a and b. So more analysis was done in order to give a full answer of the stability problem. The full answer was not achieved as there are still values for a and b where the stability is unknown. Finally, numerical solutions of y''(t) + ay(t) + by(t-1) = 0 are shown to confirm the results that are obtained.

Delay differential equation
Exponential polynomial
Harmonic oscillator

http://resolver.tudelft.nl/uuid:2ebc68be-f1fb-416b-9725-614be2f044bd

Student theses

bachelor thesis

© 2019 Vincent Li