Hopf bifurcation control for a class of delay differential systems with discrete-time delayed feedback controller

Area of Science:

• Control Theory
• Dynamical Systems
• Applied Mathematics

Background:

• Unstable delay differential equations pose significant control challenges.
• Existing control methods may not fully address systems with discrete-time feedback and sampling periods.

Purpose of the Study:

• To develop and analyze continuous-time delayed feedback controllers (C-TDFC) and discrete-time delayed feedback controllers (D-TDFC) for unstable delay differential equations.
• To establish effective control ranges ensuring asymptotical stabilization.
• To investigate the application of Hopf bifurcation theory to delay differential equations with D-TDFC.

Main Methods:

• Development and analysis of C-TDFC and D-TDFC.
• Application of Hopf bifurcation theory to analyze stability with D-TDFC.
• Derivation of effective control ranges for both controller types.
• Estimation of bounds on the sampling period for D-TDFC.

Main Results:

• An effective control range for C-TDFC ensuring asymptotical stability was determined.
• A corresponding effective control range for D-TDFC was obtained, approximating the C-TDFC range for small sampling periods.
• A bound on the sampling period for D-TDFC was estimated.

Conclusions:

• The study successfully presents and analyzes both C-TDFC and D-TDFC for stabilizing unstable delay differential equations.
• The findings provide practical control ranges and insights into the impact of sampling periods.
• The theoretical results were validated through application to a physiological system.