Let $\dot{x}(t) = Ax(t) + Bu(t)$ be an $n$-dimensional first order ODE where $u(t) \in \mathcal{P}$ for some convex polytope $\mathcal{P}$, for every $t \in \mathbb{R}$. Assume $x(0) = 0$. Is there a way to know the reachable space by such an ODE. That is, can we find the minimal space inside which $x(t)$ live?
Thank you.
Suppose the reachable subspace $\mathcal{R}$ is $k$ dimensional for $u(t) \in \mathbb{R}^m$. Then, there always exists $k$ linearly independent vectors in the reachable subspace for $u(t) \in \mathcal{P}$, which are also in $\mathcal{R}$. Because the dimension of reachable subspace is not related to the bounds of $u(t)$.
Note that you cannot reach all $\mathbb{R}^k$ for $u(t) \in \mathcal{P}$, you can reach only a bounded subspace(?) of it. However, I think finding the bounds of this space for a given polytope is nontrivial.