Abstract
Maximum Principle type necessary conditions are obtained for optimal control problems governed by discontinuous right-hand side state equations, without differentiability assumptions.
First, we define a new set-valued “Wrapper”, an extention of the concept of derivative, which is a good working tool for this kind of problems. Secondly, we construct a convex set similar to the cone of attainability proposed by Pontryagin et al. in the proof of their Maximum Principle. Lastly, we investigate certain properties of the set which play an important role in deriving the necessary conditions for the present problems.
Difficulties which arise from nondifferentiability assumptions and the existence of “separating hyperplanes” dividing the state space into a finite number of subregions are discussed in detail. The results presented here are considered to be an extension of those obtained by Masubuchi and Kanoh (1968) in the case where ordinary derivatives exist.
