Optimal synthesis for a class of L $\infty$ optimal control problems in the plane with L 1 constraint on the input

For a particular class of planar dynamics that are linear with respect to the control variable, we show that the feedback strategy ''null-singular null'' is minimizing the maximum of a coordinate over infinite horizon, under a L 1 budget constraint on the control. Moreover, we characterize the optimal cost as a function of the budget. The proof is based on an unusual use of the clock form. This result generalizes the one obtained formerly for the SIR epidemiological model to more general Kolmogorov dynamics, that we illustrate on other biological models.