Combining high-order necessary conditions for optimality with nonsmoothness

We present a version of the Pontryagin maximum principle valid for systems of flows rather than for systems governed by ordinary differential equations. The flow maps are required to be differentiate in a generalized sense (using the theory of "generalized differential quotients") which is much weaker than ordinary differentiability and allows the "differentials" to be sets of linear maps rather than single linear maps. The resulting conditions apply to control dynamics with a right-hand side that needs not be smooth, or even Lipschitz, and could even be discontinuous. This is so because the usual adjoint equation, in which there occur derivatives of the reference vector field with respect to the state, is replaced by an integrated form. This form only involves differentials of the reference flow maps, and therefore makes sense as long as these flow maps are differentiable, which can happen even when the reference vector field itself fails to be Lipschitz or even continuous. The resulting "integrated adjoint equation" gives rise to "adjoint vectors" that need not be absolutely continuous, and could be discontinuous and unbounded. Furthermore, this integrated adjoint equation relates the values of the adjoint vector on intervals that could be disjoint and contain singularities in between. This makes it possible to establish necessary conditions for an optimum that yield a global adjoint vector that satisfies various nonsmooth conditions everywhere and at the same time satisfies extra "high-order" requirements, such as the Goh condition, on intervals where the dynamics is sufficiently smooth.