Global controllability of nonlinear systems in two dimensions

LetM be a connected real-analytic 2-dimensional manifold. Consider the system(t) = f(x(t)) + u(t)g(x(t)),x(0) =x₀ M, wheref andg are real-analytic vector fields onM which are linearly independent at some point ofM, andu is a real-valued control. Sufficient conditions on the vector fieldsf andg are given so that the system is controllable fromx₀. Suppose that every nontrivial integral curve ofg has a pointp wheref andg are linearly dependent,g(p) is nonzero, and that the Lie bracket [f,g] andg are linearly independent atp. Then the system is controllable (with the possible exception of a closed, nowhere dense set which is not reachable) from any pointx₀ such that the vector space dimension of the Lie algebraL_A generated byf,g and successive Lie brackets is 2 atx₀.Research supported in part by the National Science Foundation under NSF Grant MCS76-05267-A01 and by the Joint Services Electronics Program under ONR Contract 76-C-1136.