On the Decidability of Reachability in Linear Time-Invariant Systems

TL;DR

This paper investigates the decidability of reachability in linear time-invariant systems, revealing undecidability in some cases and establishing decidability under specific spectral conditions for convex polytopes.

Contribution

It demonstrates undecidability for finite unions of affine subspaces and proves decidability for convex polytopes with spectral assumptions, advancing understanding in control theory.

Findings

Undecidability for finite unions of affine subspaces.

Reachability as hard as Skolem's and Positivity Problems.

Decidability for convex polytopes under spectral conditions.

Abstract

We consider the decidability of state-to-state reachability in linear time-invariant control systems over discrete time. We analyse this problem with respect to the allowable control sets, which in general are assumed to be defined by boolean combinations of linear inequalities. Decidability of the version of the reachability problem in which control sets are affine subspaces of $\mathbb{R}^n$ is a fundamental result in control theory. Our first result is that reachability is undecidable if the set of controls is a finite union of affine subspaces. We also consider versions of the reachability problem in which (i)~the set of controls consists of a single affine subspace together with the origin and (ii)~the set of controls is a convex polytope. In these two cases we respectively show that the reachability problem is as hard as Skolem's Problem and the Positivity Problem for linear recurrence sequences (whose decidability has been open for several decades). Our main contribution is to show decidability of a version of the reachability problem in which control sets are convex polytopes, under certain spectral assumptions on the transition matrix.