Optimal Feedback Control for a Class of Nonlinear Time-Delay Systems

This article addresses the problem of optimal nonlinear feedback control for a class of nonlinear time-delay systems over an infinite horizon, involving a nonlinear-nonquadratic performance functional. To this end, we propose a framework for analyzing and designing nonlinear feedback controllers that minimize such cost functionals for these systems. By applying the Lyapunov functional method, the stability of a class of nonlinear time-delay systems is determined over an infinite horizon. This Lyapunov functional is shown to be the solution to a functional Hamilton–Jacobi–Bellman (HJB) equation. Sufficient conditions for optimality are then provided in the form of a steady-state version of the functional HJB equation, thereby guaranteeing both stability and optimality. In addition, the corresponding results of optimal feedback control problem for a class of nonlinear delay-free systems are further developed. Finally, numerical simulations are carried out to demonstrate the effectiveness of the proposed methods.