Nonlinear delay systems, Lie algebras and Lyapunov transformations

In this paper we consider nonlinear delay systems of the form ẋ(t) = A(x(t), x(t – δ))x(t) + B(x(t), x(t – δ))u(t) and obtain new stability and control results by replacing the system by a sequence of linear, time‐varying systems and using an explicit representation for the solution of such systems in terms of a Lie algebra associated with the system. In the case of nilpotent Lie algebras we use a Lyapunov transformation to prove stability.