High Order Algebraic Conditions for Controllability

Abstract

Let X,Y²,...,Y^m be analytic vector fields on an analytic, n — dimensional, manifold M, and ϑ the control system

$$\mathop x\limits^. = X(x) + \sum\nolimits_{i = 2}^m {{u_i}(t){Y^i}(x),x(o) = p \in M,(\mathop x\limits^. = dx/dt)} $$

((1))

where, unless stated otherwise, an admissible control is a Lebesgue measurable function u with components |u_i(t)| ≤ 1. a(t,p, ϑ) will denote the set of all points attainable at time t ≥ 0 by solutions of ϑ corresponding to all admissible controls; T^X (•)p will denote the solution of ϑ corresponding to u = 0 . Our problem is to determine necessary and sufficient conditions that T^X(t)p ∈ int. a(t,p, ϑ) ∀t > 0 .

This research was supported by the National Science Foundation under grant GP 27957.