On the Polytope Escape Problem for Continuous Linear Dynamical Systems

TL;DR

This paper proves the decidability of the Polyhedral Escape Problem for continuous linear systems by reducing it to linear programming with algebraic coefficients, using spectral and Diophantine approximation techniques.

Contribution

It introduces a polynomial-time reduction of the escape problem to linear programming with algebraic coefficients, establishing its place in the complexity class XR.

Findings

The problem is decidable for continuous linear dynamical systems.

The reduction places the problem in the complexity class XR.

Spectral and Diophantine approximation methods are key to the solution.

Abstract

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.