Stabilization of nonlinear delay systems: A tutorial on recent results

Stabilization of nonlinear systems under delays is a central and challenging problem in control theory. It is also of considerable interest in engineering, because delay systems are prevalent in aerospace, biological,marine robotic, network control, and many other applications. Input delays naturally arise due to transport phenomena, time consuming information processing, and sensor designs, and they can produce complicated systems that are beyond the scope of standard frequency-domain or Lyapunov function methods. This has led to large control theoretic and engineering literatures on stabilization problems, spanning more than 40 years, based on backstepping, Lyapunov-Krasovskii functionals, prediction, and sampling controllers. In addition to input delays, there may also be state delays in the vector fields that define the system. This tutorial summarizes some recent work on stabilization under input or state delays and suggests future research directions.