Linearizing and stabilizing discontinuous feedbacks for delay systems as stabilizers in the sample-and-hold sense

In this paper nonlinear systems with an arbitrary number of delays, of discrete and distributed type, are considered. It is assumed that there exists a state feedback (continuous or not), by which the dynamics in closed-loop becomes linear (with delays) and stable. Stability, however, is here to be intended in a “virtual” sense, since discontinuous feedbacks may well be hard to be implemented in practice in continuous time. Therefore, the implementation by sampling and holding is investigated. The following result is proved: for any large ball and small ball of the origin, there exist a suitable small sampling period and a time T, such that all the trajectories of the system in closed-loop with the sampled-data state feedback, starting with initial state in the above large ball, are uniformly bounded and driven into the small ball and kept in for all times greater than T. That is, it is proved that any linearizing and (virtually) stabilizing feedback (continuous or not) is a stabilizer in the sample-and-hold sense. The possibility of admitting discontinuities in the feedback can simplify the search of continuous time linearizing (virtual) stabilizers. Examples showing the effectiveness of the proposed result are given.