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Advances in Mathematical Physics
Volume 2017, Article ID 1938513, 11 pages
Research Article

PID Controller Singularly Perturbing Impulsive Differential Equations and Optimal Control Problem

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Paholyothin Road, Klong Luang, Rangsit, Pathumthani 12121, Thailand

Correspondence should be addressed to Wichai Witayakiattilerd;

Received 27 July 2017; Revised 8 October 2017; Accepted 17 October 2017; Published 12 November 2017

Academic Editor: Alkesh Punjabi

Copyright © 2017 Wichai Witayakiattilerd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study singular perturbation of impulsive system with a proportional-integral-derivative controller (PID controller) and solve an optimal control problem. The perturbation system comprises two important variables, a fast variable and a slow variable. Because of the complexity of the system, it is difficult to find its exact solution. This paper presents an approximation method for solving it. The aim of the approximation method is to reduce the complexity of the system by eliminating the fast variable. The solution of the method is expressed in an integral form, and it is called an approximated mild solution of the perturbed system. An example is provided to illustrate our result.