<tex>n</tex>-Dimensional controllability with<tex>(n-1)</tex>controls

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> be a connected real-analytic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -dimensional manifold, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f, g₁, ... ,gₙ₋₁</tex> be complete real-analytic vector fields on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> which are linearly independent at some point of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u₁, ... , uₙ₋₁</tex> be real-valued controls. Consider the controllability of the system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ẋ(t)=f(x(t)) + ∑min{i=1}max{n-1} uᵢ(t)gᵢ(x(t)), x(0)=x₀ ∈ M</tex> . Necessary and sufficient conditions are given so that this system is controllable on any simply connected domain <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</tex> contained in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> on which <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g₁,... ,gₙ₋₁</tex> are linearly independent. These conditions depend on the computation of Lie brackets at those points where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f, g₁, ... ,gₙ₋₁</tex> are linearly dependent.