Given is a $M$-dimensional torus and a known $M'$-dimensional involutive distribution $\triangle$, $M' < M$, on this torus. (Furthermore, $M'$ is known as a function of $M$.) If ${\bf x}$ is a point on the torus, the question is whether there is a theory for finding the set ${\cal O}({\bf x})$ of all the points on the torus reachable from ${\bf x}$ by a flow induced by $\triangle$. (In control-theoretic terms, ${\cal O}({\bf x})$ is the attainable set of ${\bf x}$ under the $\triangle$.)
The Frobenius theorem implies that $\triangle$, being involutive, is therefore integrable. So ${\cal O}( {\bf x})$ is a submanifold of the torus and is the maximal manifold of $\triangle$.
I am trying, however, to determine something more; i.e., whether ${\cal O}({\bf x})$ intersects a given, known sub-torus of the torus.
What theory would you recommend I study so as to explore this question?