The ’ theory of moments ‘ originated in the late nineteenth century and is associated with the work of Chebyshev, Markov and Stieltjes. This paper applies some moment theorems developed by Akhiezcr and Krein in the 1930s to a number of time optimal control problems : acceleration control of a mass, control of the harmonic oscillator, boundary control of a vibrating string and a discrete time version of the accelerated mass problem.