We provide a duality result linking the value function for a control problem with supremum cost H under an isoperimetric inequality G ≤ gmax, and the value function for the same controlled dynamics with cost G and state constraint H ≤ 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 recent works dealing with non-pharmaceutically controlled epidemics under peak or budget restrictions.