On the decidability of reachability in linear time-invariant systems

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 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.

Set Propagation Techniques for Reachability Analysis

“…For instance, reachability is decidable for a discrete-time linear time-invariant system if the set of inputs is unconstrained. However, if the inputs are constrained to a finite union of affine subspaces the problem becomes undecidable [7]. In the case of hybrid automata, reachability is undecidable even for classes with very simple dynamics, such as systems with piecewise constant derivatives [8].…”

Answerable and Unanswerable Questions in Risk Analysis with Open‐World Novelty

“…This problem is undecidable even for some of the simplest and best-studied classes of deterministic dynamic systems (e.g., discrete-time linear time invariant [LTI] systems) if the feasible inputs are constrained (e.g., if the set of allowed inputs is nonconvex, consisting of a disjoint union of a finite number of convex polytopes) (Sousa-Pinto, 2017). It is also undecidable for many simple nonlinear systems, e.g., with piecewise linear dynamics that are linear, in each of multiple regions; or with saturated linear dynamics, where the dynamics are linear up to a maximum possible response rate and flat above it (Fijalkow, Ouaknine, Pouly, Sousa-Pinto, & Worrell, 1997;Sousa-Pinto, 2017). However, it is decidable for subsets of LTI systems with certain stability properties if the feasible inputs allow limited movement in any direction (more precisely, if they form a bounded convex polytope around the origin) (Fijalkow et al, 1997).…”

“…It is also undecidable for many simple nonlinear systems, e.g., with piecewise linear dynamics that are linear, in each of multiple regions; or with saturated linear dynamics, where the dynamics are linear up to a maximum possible response rate and flat above it (Fijalkow, Ouaknine, Pouly, Sousa-Pinto, & Worrell, 1997;Sousa-Pinto, 2017). However, it is decidable for subsets of LTI systems with certain stability properties if the feasible inputs allow limited movement in any direction (more precisely, if they form a bounded convex polytope around the origin) (Fijalkow et al, 1997). Detailed study shows that whether an arbitrary initial state can be driven to the origin is undecidable in piecewise linear systems with states having 22 or more dimensions, or, more generally, more than 21/(n -1) dimensions, where n is the number of different regions or "pieces"; n > 1 for any piecewise linear system (Blondel & Tsitsiklis, 2000).…”

PIRK: Scalable Interval Reachability Analysis for High-Dimensional Nonlinear Systems

“…Computing the exact reachable set is generally not possible. For example, even for the case of discrete-time LTI systems it is not known whether the exact reachable set is computable in many important applications [5]. Therefore, most practical methods resort to computing over-approximations or underapproximations of the reachable set, depending on the desired guarantee.…”