On Control Lyapunov–Razumikhin Functions, Nonconstant Delays, Nonsmooth Feedbacks, and Nonlinear Sampled-Data Stabilization

A methodology for the design of stabilizers in the sample-and-hold sense for nonlinear retarded systems is provided. The methodology is based on control Lyapunov-Razumikhin functions. Fully nonlinear retarded systems with an arbitrary number of arbitrary time-varying time-delays of discrete and distributed type are covered by the theory developed here, as long as the map describing the dynamics is uniformly in time Lipschitz on bounded sets of the Banach state space and the Euclidean input space. The standard assumption that all involved time-delay signals are globally Lipschitz is introduced. It is assumed, for the system at hand, that there exist a control Lyapunov-Razumikhin function and a suitably defined induced steepest descent state feedback. Moreover, such state feedback has to satisfy a suitably weak Lipschitz property, uniformly in time and in any bounded subset of the Banach state space. Such weak Lipschitz property has the advantage to allow for discontinuities in the feedback, which arise, for instance, in sliding-mode control methodologies. The following fact is shown: the above steepest descent state feedback is a stabilizer in the sample-and-hold sense, that is, such feedback, if applied by suitably fast sampling and holding, guarantees practical semiglobal stability of the closed-loop system, with arbitrary small final target ball of the origin. The problem of nonavailability in the buffer device of past values of the system internal variables is solved by spline approximation methods, even in the case discontinuities arise in the state feedback.