  We provide a duality result linking the value function for a control problem
with supremum cost H under an isoperimetric inequality G $\le$ gmax, and the
value function for the same controlled dynamics with cost G and state
constraint H $\le$ hmax. This duality is proven for initial conditions at which
lower semi-continuity of the value functions can be guaranteed, and is
completed with optimality considerations. Furthermore, we provide structural
assumptions on the dynamics under which such regularity can be established. As
a by-product, we illustrate the partial equivalence between recentworks dealing
with non-pharmaceutically controlled epidemics under peak or budget
restrictions.
