  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.
