We consider the decidability of the membership problem for matrix-exponential semigroups: given k ∈ N and square matrices A 1 , . . . , A k , C, all of the same dimension and with real algebraic entries, decide whether C is contained in the semigroup generated by the matrix exponentials exp(A i t), where i ∈ {1, . . . , k} and t ≥ 0. This problem can be seen as a continuous analog of Babai et al.'s and Cai et al.'s problem of solving multiplicative matrix equations, and has applications to

more »

... ability analysis of linear hybrid automata and switching systems. Our main results are that the semigroup membership problem is undecidable in general, but decidable if we assume that A 1 , . . . , A k commute. The decidability proof is by reduction to a version of integer programming that has transcendental constants. We give a decision procedure for the latter using Baker's theorem on linear forms in logarithms of algebraic numbers, among other tools. The undecidability result is shown by reduction from Hilbert's Tenth Problem.