On the Decidability of Reachability in Linear Time-Invariant Systems

We consider the decidability of state-to-state reachability in linear time-invariant control systems, with control sets defined by boolean combinations of linear inequalities. Decidability of the sub-problem in which control sets are linear subspaces is a fundamental result in control theory. We first show that reachability is undecidable if the set of controls is a finite union of affine subspaces. We then consider two simple subclasses of control sets—unions of two affine subspaces and bounded convex polytopes respectively—and show that in these two cases the reachability problem for LTI systems is as hard as certain longstanding open decision problems concerning linear recurrence sequences. Finally we present some spectral assumptions on the transition matrix of an LTI system under which reachability becomes decidable with bounded convex polytopes as control sets. 1998 ACM Subject Classification F.2.m (Analysis of Algorithms and Problem Complexity)