Optimal Control of Nonsmooth Systems with Classically Differentiable Flow Maps

We present a version of the Pontryagin Maximum Principle for control dynamics with a possibly non smooth, nonlipschitz and even discontinuous righthand side. The usual adjoint equation, where state derivatives occur, is replaced by an integrated form, containing only differentials of the reference flow maps. The resulting “integrated adjoint equation” leads to “adjoint vectors” that need not be absolutely continuous, and could be discontinuous and unbounded. We illustrate this with the “reflected brachistochrone problem,” for which the adjoint vectors have a singularity at an interior point of the interval of definition of the reference trajectory.