On necessary optimality conditions for systems governed by a two point boundary value problem 1

The paper concerns a necessary optimality condition in form of a Pontryagin Minimum Principle for a system governed by a linear two point boundary value problem with homogeneous Dirichlet conditions, whereby the control vector occurs in all coefficients of the differential equation. Without any convexity assumption the optimality condition is derived using a needle-like variation of the optimal control. In case of convex local control constraints the optimality condition implies the linearized minimum principle, which the authors have proved. An example shows that for this linearized optimality condition the convexity of the set of all admissible controls is essential.