On the Decidability of Reachability in Continuous Time Linear Time-Invariant Systems

TL;DR

This paper investigates the decidability of reachability in continuous-time linear systems with bounded controls, providing new results for specific cases and conditions, and establishing complexity bounds for generalized problems.

Contribution

It introduces new decidability results for 2D systems under spectral conditions and rational matrices, and explores the complexity of generalized reachability problems.

Findings

Decidability in 2D systems with spectral conditions on matrix A.

Conditional decidability for matrices with rational or real eigenvalues.

Hardness results for generalized reachability problems with convex control sets.

Abstract

We consider the decidability of state-to-state reachability in linear time-invariant control systems over continuous time. We analyse this problem with respect to the allowable control sets, which are assumed to be the image under a linear map of the unit hypercube. This naturally models bounded (sometimes called saturated) controls. 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 decidability in two dimensions ($n=2$) if the matrix $A$ satisfies some spectral conditions, and conditional decidablility in general. If the transformation matrix $A$ is diagonal with rational entries (or rational multiples of the same algebraic number) then the reachability problem is decidable. If the transformation matrix $A$ only has real eigenvalues, the reachability problem is conditionally decidable. The time-bounded reachability problem is conditionally decidable, and unconditionally decidable in two dimensions. Some of our decidability results are conditional in that they rely on the decidability of certain mathematical theories, namely the theory of the reals with exponential ($\mathfrak{R}_{\exp}$) and with bounded sine ($\mathfrak{R}_{\exp,\sin}$). We also obtain a hardness result for a mild generalization of the problem where the target is simple set (hypercube of dimension $n-1$ or hyperplane) instead of a point, and the control set is a convex bounded polytope. In this case, we show that the problem is at least as hard as the \emph{Continuous Positivity problem} or the \emph{Nontangential Continuous Positivity problem}.