A sufficient condition for locally controlled invariance of a manifold for general non linear systems

This paper presents a sufficient condition for a manifold ¿ to be locally controlled invariant at x₀ ¿ ¿ which reduces, in the cases of linear and non linear affine systems, to well known results in the literature: essentially the result says that a manifold ¿ ¿ ¿ⁿ is locally controlled at x₀ ¿ ¿ if first of all we can find a control u₀ ¿ ¿^m such that F(x₀, u₀) ¿ T_x0¿ (this condition is evidently necessary), second F(x,u) continues to stay in something larger than T_x¿ (namely T_x¿ + ¿_uF(x, u)(¿^m)) in a neighborhood of (x₀, u₀) in ¿ × ¿^m.