On Recurrent Reachability for Continuous Linear Dynamical Systems
This work provides a decidability result for a fundamental problem in dynamical systems, but the restriction to low order and the hardness result for higher orders limit its immediate applicability.
The paper studies the recurrent reachability problem for continuous linear dynamical systems, asking whether a solution trajectory reaches a target halfspace infinitely often. They show decidability for systems of order at most 7, and prove that decidability at order 9 would require a major advance in Diophantine approximation.
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.