Homogeneous tangent vectors and high-order necessary conditions for optimal controls

An extension of the classical Pontryagin maximum principle for Mayer problems without terminal constraints, subject to affine control systems , is proved. In connection with a suitable dilation on the state space ℝ ⁿ , we introduce a class of “homogeneous tangent vectors” which provide a nonlinear, high-order approximation of the attainable set in the case where the usual linear approximation proves to be inadequate. By studying control variations which generate homogeneous tangent vectors, we derive new necessary conditions for optimality that are particularly effective for basically nonlinear optimal control problems where other high-order tests provide no conclusive information. This research was partially supported by Istituto Nazionale di Alta Matematica “F. Severi.”