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One of the most important branches of nonlinear control theory is theso-called sliding-mode. Its aim is the design of a (nonlinear) feedback lawthat brings and maintains the state trajectory of a dynamic system on a givensliding surface. Here, dynamics becomes completely independent of the modelparameters and can be tuned accordingly to the desired target. In this paper westudy this problem when the feedback law is subject to strong structuralconstraints. In particular, we assume that the control input may take valuesonly over two bounded and disjoint sets. Such sets could be also non perfectlyknown a priori. An example is a control input allowed to switch only betweentwo values. Under these peculiarities, we derive the necessary and sufficientconditions that guarantee sliding-mode control effectiveness for a class oftime-varying continuous-time linear systems that includes all the stationarystate-space linear models. Our analysis covers several scientific fields. It isonly apparently confined to the linear setting and allows also to study animportant set of nonlinear models. We describe fundamental examples related toepidemiology where the control input is the level of contact rate among peopleand the sliding surface permits to control the number of infected. For popularepidemiological models we prove the global convergence of control schemes basedon the introduction of severe restrictions, like lockdowns, to containepidemic. This greatly generalizes previous results obtained in the literatureby casting them within a general sliding-mode theory.

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