Title:

Decidability boundaries in linear dynamical systems

The object of this thesis is the study of the decidability properties of linear dynamical systems, which have fundamental ties to theoretical computer science, software verification, linear hybrid systems, and control theory. In particular, we describe a method for deciding the termination of simple linear loops, partly solving a 10yearold open problem of Tiwari (2004) and Braverman (2006). We also study the membership problem for semigroups of matrix exponentials, which we show to be undecidable in general by reduction from Hilbert's Tenth Problem, and decidable for all instances where the matrices defining the semigroup commute. In turn, this entails the undecidability of the generalised versions of the Continuous Orbit and Skolem Problems to a multimatrix setting. We also study pointtopoint controllability for linear timeinvariant systems, which is a central problem in control theory. For discretetime systems, we show that this problem is undecidable when the set of controls is nonconvex, and at least as hard as the Skolem Problem even when it is a convex polytope; for continuoustime systems, we show that this problem reduces to the Continuous Orbit Problem when the set of controls is a linear subspace, which entails decidability. Finally, we show how to decide whether all solutions of a given linear ordinary differential equation starting in a given convex polytope eventually leave it; this problem, which we call the "Polytope Escape Problem'', relates to the liveness of states in linear hybrid automata. Our results rely on a number of theorems from number theory, logic, and algebra, which we introduce in a selfcontained way in the preamble to this thesis, together with a few new mathematical results of independent interest.
