ABSTRACT
This chapter outlines the proof of a version of the Maximum Principle of finite-dimensional Control Theory that requires weaker technical assumptions than the “classical” and “nonsmooth” formulations. In Lojasiewicz’s result, only the reference vector field has to be locally Lipschitz with respect to the state variable — with the usual integral bounds of Carathedory type for the Lipschitz constants—while the other vector fields are only assumed to be continuous with respect to the state variable. Most importantly, the chapter incorporates high-order point variations, thereby making the necessary conditions much stronger than those of the usual “smooth” and “nonsmooth” versions. This is done by abstractly defining a class of “variations” much larger than that of the usual needle variations of the classical Maximum Principle. The geometric formulation of the Maximum Principle as a separation theorem includes a transversality condition, and this requires the choice of an appropriate concept of “tangent cone” to a set at a point.
