Reachable Space by an ODE 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.
 A: In general, there is no language that expresses the reachable set of states of an ODE. This has been proven to be undecidable. The decidability barrier stops at some constrained types of dynamics (constant dynamics or rectangular inclusions of the derivatives with particular assumptions). 
Consequently, to determine the reachable set for a given time $t$ starting from an initial set is not possible. This is not possible for a chosen value of $t$ either. 
This being said, one can compute an over(or under)-approximation of the reachable set of states instead of the exact set. There are nice formally verified tools that compute over(under)-approximations of reachable sets up till a time horizon $T$ such as CORA (matlab based), SpaceEx. If self-containment is achieved (i.e. the over-approximation becomes included in itself when time moves forward) then you found yourself a set where all reachable states remain within for any time $t$.
This is often not the case, especially in chaotic systems, thus one needs to use methods that could be specific to matrices $A$ and $B$ to find some barrier certificate functions that hold the reachable set. This task is challenging and not trivial and remains an  active research area. 
A: 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.
