Local high order approximations of control systems and some of their applications (Q1912509)

There are only a few separate results which are devoted to the third order approximation of optimality conditions for nonlinear control systems. This paper proceeds from a differential geometric approach and is devoted to the description of admissible directions of a motion and conditions of optimality of orders as high as desired. Many invariant properties of control systems may be characterized in terms of Lie brackets of vector fields defining a control system and the values of some Lie algebra elements taken at a fixed point will be the values of tangent vectors. The reachable set around a fixed trajectory is analyzed in detail using convex cone of Legendre vectors and their expressions in terms of Lie brackets. The computations are done explicitly and the paper is ended with two applications for getting optimality conditions and controllability properties by using high order approximations developped previously. It is highly recommended to researchers interested in nonlinear control theory and especially involved in geometric methods.