Solvability of Matrix-Exponential Equations

Abstract: We consider a continuous analogue of Babai et al.'s and Cai et al.'s problem of solving multiplicative matrix equations. Given $k+1$ square matrices $A_{1}, \ldots, A_{k}, C$, all of the same dimension, whose entries are real algebraic, we examine the problem of deciding whether there exist non-negative reals $t_{1}, \ldots, t_{k}$ such that \begin{align*} \prod \limits_{i=1}^{k} \exp(A_{i} t_{i}) = C . \end{align*} We show that this problem is undecidable in general, but decidable under the assumption that the matrices $A_{1}, \ldots, A_{k}$ commute. Our results have applications to reachability problems for linear hybrid automata. Our decidability proof relies on a number of theorems from algebraic and transcendental number theory, most notably those of Baker, Kronecker, Lindemann, and Masser, as well as some useful geometric and linear-algebraic results, including the Minkowski-Weyl theorem and a new (to the best of our knowledge) result about the uniqueness of strictly upper triangular matrix logarithms of upper unitriangular matrices. On the other hand, our undecidability result is shown by reduction from Hilbert's Tenth Problem.