  The Polyhedral Escape Problem for continuous linear dynamical systems
consists of deciding, given an affine function $f: \mathbb{R}^{d} \rightarrow
\mathbb{R}^{d}$ and a convex polyhedron $\mathcal{P} \subseteq \mathbb{R}^{d}$,
whether, for some initial point $\boldsymbol{x}_{0}$ in $\mathcal{P}$, the
trajectory of the unique solution to the differential equation
$\dot{\boldsymbol{x}}(t)=f(\boldsymbol{x}(t))$,
$\boldsymbol{x}(0)=\boldsymbol{x}_{0}$, is entirely contained in $\mathcal{P}$.
We show that this problem is decidable, by reducing it in polynomial time to
the decision version of linear programming with real algebraic coefficients,
thus placing it in $\exists \mathbb{R}$, which lies between NP and PSPACE. Our
algorithm makes use of spectral techniques and relies among others on tools
from Diophantine approximation.
