The time optimal control of transmission lines with amplitude constraints on the control is considered as a typical problem involving systems governed by hyperbolic partial differential equations. Using a Laplace transformation formulation to yield a time ‘optimal’ solution, it is shown how this sub-optimal control which is bang-bang develops into an optimal control which is not always at its limiting values—demonstrating the effect which the nature of the differential equation has on the form of the optimal control. A simple physical interpretation of the results is given.