We study the geodesic connectedness of a globally hyperbolic spacetime
(M, g) admitting a complete smooth Cauchy hypersurface S and endowed with
a complete causal Killing vector field K. The main assumptions are that the
kernel distribution D of the one-form induced by K on S is non-integrable and
that the gradient of g(K, K) is orthogonal to D. We approximate the metric g
by metrics gε smoothly depending on a real parameter ε and admitting K as a
timelike Killing vector field. A known existence result for geodesics of such type
of metrics provides a sequence of approximating solutions, joining two given
points, of the geodesic equations of (M, g) and whose Lorentzian energy turns
out to be bounded thanks to an argument involving trajectories of some affine
control systems related with D.