Qualitative analyses of attainability set of nonlinear controllable systems

Let us consider a nonlinear system x = f (t, x) + g (t, x)u (1) where the vector-function f(t, x) is defined and continuous in the domain D = {(t, x) | t ≥ 0, x ∈ Rⁿ}. We presuppose in addition, that f(t, x) is a homogeneous vector-function of order m > 1 of the argument x. It has the continuous and bounded partial derivatives ∂f_i(t, x)/∂x_j, ij = 1, n in the domain G = {(t, x) |t ≥ 0, ||x|| ≤ H} for any H > 0.