Nonsmooth optimization of hydrothermal problems

In this paper the authors present a necessary condition for minimum of a functional J(z) := ∫0T L(t, z(t), z'(t)) dt in the case in which the function L is continuous but not of class C1. This situation arises in problems of optimization of hydrothermal systems with pumped-storage plants. In such problems, the function Lz' (t, z, ċ) is discontinuous in z' = O, which is the borderline point between the power generation zone (z' > 0) and the pumping zone (z' < 0). The problem can be naturally formulated in the framework of nonsmooth analysis, using the generalized (or Clarke's) gradient.