$\mathbb {L}^{\infty }$/$\mathbb {L}^{1}$ duality results in optimal control problems

We consider optimal control problems which consist in minimizing the $L^\infty$ norm of an output function under an isoperimetric or $L^{1}$ inequality. These problems typically arise in control applications when one looks to minimizing the maximum trajectory deviation or “peak” under a budget constraint. We show a duality with more classical problems which amount to minimizing the $L^{1}$ cost under the state constraint given by an upper bound on the $L^\infty$ norm of the output. More precisely, we provide a result linking the value functions of these two problems, as functions of the levels of the two kind of constraints. This is obtained for initial conditions at which lower semi-continuity of the value functions can be guaranteed, and is completed with optimality considerations. When the duality holds, we show that the two problems have the same optimal controls. Furthermore, we provide structural assumptions on the dynamics under which the semi-continuity of the value functions can be established. We illustrate theses results on non-pharmaceutically controlled epidemics models under peak or budget restrictions.