On Sontag’s formula for the input-to-state practical stabilization of retarded control-affine systems

In this paper input-to-state practically stabilizing control laws for retarded, control-affine, nonlinear systems with actuator disturbance are investigated. The developed methodology is based on Artstein’s theory of control Lyapunov functions and related Sontag’s formula, extended to retarded systems. If the actuator disturbance is bounded, then the controller yields the solution of the closed-loop system to achieve an arbitrarily fixed neighborhood of the origin, by increasing a control tuning parameter. The considered systems can present an arbitrary number of discrete as well as distributed time-delays, of any size, as long as they are constant and, in general, known.