Hysteresis-based supervisory control with application to non-pharmaceutical containment of COVID-19

The recent COVID-19 outbreak has motivated an extensive development of non-pharmaceutical intervention policies for epidemics containment. While a total lockdown is a viable solution, interesting policies are those allowing some degree of normal functioning of the society, as this allows a continued, albeit reduced, economic activity and lessens the many societal problems associated with a prolonged lockdown. Recent studies have provided evidence that fast periodic alternation of lockdown and normal-functioning days may effectively lead to a good trade-off between outbreak abatement and economic activity. Nevertheless, the correct number of normal days to allocate within each period in such a way to guarantee the desired trade-off is a highly uncertain quantity that cannot be fixed a priori and that must rather be adapted online from measured data. This adaptation task, in turn, is still a largely open problem, and it is the subject of this work. In particular, we study a class of solutions based on hysteresis logic. First, in a rather general setting, we provide general convergence and performance guarantees on the evolution of the decision variable. Then, in a more specific context relevant for epidemic control, we derive a set of results characterizing robustness with respect to uncertainty and giving insight about how a priori knowledge about the controlled process may be used for fine-tuning the control parameters. Finally, we validate the results through numerical simulations tailored on the COVID-19 outbreak.