Sliding-mode theory under feedback constraints and the problem of epidemic control

One of the most important branches of nonlinear control theory is the so-called sliding-mode. Its aim is the design of a (nonlinear) feedback law that brings and maintains the state trajectory of a dynamic system on a given sliding surface. Here, dynamics becomes completely independent of the model parameters and can be tuned accordingly to the desired target. In this paper we study this problem when the feedback law is subject to strong structural constraints. In particular, we assume that the control input may take values only over two bounded and disjoint sets. Such sets could be also non perfectly known a priori. An example is a control input allowed to switch only between two values. Under these peculiarities, we derive the necessary and sufficient conditions that guarantee sliding-mode control effectiveness for a class of time-varying continuous-time linear systems that includes all the stationary state-space linear models. Our analysis covers several scientific fields. It is only apparently confined to the linear setting and allows also to study an important set of nonlinear models. We describe fundamental examples related to epidemiology where the control input is the level of contact rate among people and the sliding surface permits to control the number of infected. For popular epidemiological models we prove the global convergence of control schemes based on the introduction of severe restrictions, like lockdowns, to contain epidemic. This greatly generalizes previous results obtained in the literature by casting them within a general sliding-mode theory.