A Pontryagin Maximum Principle for systems of flows

We present a generalization of the Pontryagin Maximum Principle, in which the usual adjoint equation, which contains derivatives of the system vector fields with respect to the state, is replaced by an integrated form, containing only differentials of the reference flow maps. In this form, the conditions of the maximum principle make sense for a number of control dynamical laws whose right-hand side can be nonsmooth, nonlipschitz, and even discontinuous. The “adjoint vectors” that are solutions of the “adjoint equation” no longer need to be absolutely continuous, and may be discontinuous and unbounded. We illustrate this with two examples: the “reflected brachistochrone problem” (RBP), and the derivation of Snell’s law of refraction from Fermat’s minimum time principle. In the RBP, where the dynamical law is Hölder continuous with exponent 1/2, the adjoint vector turns out to have a singularity, in which one of the components goes to infinity from both sides, at an interior point of the interval of definition of the reference trajectory. In the refraction problem, where the dynamical law is discontinuous, the adjoint vector is bounded but has a jump discontinuity.