Reachability Problems for Continuous Linear Dynamical Systems

James Worrell

This talk is about reachability problems for continuous linear dynamical systems. A central decision problem in this area is the Continuous Skolem Problem, which asks whether a real-valued function satisfying an ordinary linear differential equation has a zero. This can be seen as a continuous analog of the Skolem Problem for linear recurrence sequences, which asks whether the sequence satisfying a given recurrence has a zero term. For both the discrete and continuous versions of the Skolem Problem, decidability is open. We show that the Continuous Skolem Problem lies at the heart of many natural verification questions on linear dynamical systems, such as continuous-time Markov chains and linear hybrid automata. We describe some recent work, done in collaboration with Chonev and Ouaknine, that uses results in transcendence theory and real algebraic geometry to obtain decidability for certain variants of the problem. In particular, we consider a bounded version of the Continuous Skolem problem, corresponding to time-bounded reachability. We prove decidability of the bounded problem assuming Schanuel's conjecture, one of the main conjectures in transcendence theory. We describe some partial decidability results in the unbounded case and discuss mathematical obstacles to proving decidability of the Continuous Skolem Problem in full generality.