On Recurrent Reachability for Continuous Linear Dynamical Systems

TL;DR

This paper investigates the problem of determining whether solutions to linear differential equations reach a target repeatedly, establishing decidability for equations up to order 7 and linking higher orders to deep number theory challenges.

Contribution

It proves decidability of the recurrent reachability problem for linear systems of order at most 7 and connects higher-order cases to major open problems in Diophantine approximation.

Findings

Decidability established for systems up to order 7.

Higher orders (9 and above) relate to computing Lagrange constants.

Decision problem remains open for order 8.

Abstract

The continuous evolution of a wide variety of systems, including continuous-time Markov chains and linear hybrid automata, can be described in terms of linear differential equations. In this paper we study the decision problem of whether the solution $\boldsymbol{x}(t)$ of a system of linear differential equations $d\boldsymbol{x}/dt=A\boldsymbol{x}$ reaches a target halfspace infinitely often. This recurrent reachability problem can equivalently be formulated as the following Infinite Zeros Problem: does a real-valued function $f:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$ satisfying a given linear differential equation have infinitely many zeros? Our main decidability result is that if the differential equation has order at most $7$, then the Infinite Zeros Problem is decidable. On the other hand, we show that a decision procedure for the Infinite Zeros Problem at order $9$ (and above) would entail a major breakthrough in Diophantine Approximation, specifically an algorithm for computing the Lagrange constants of arbitrary real algebraic numbers to arbitrary precision.