The problem of maximising the time during which an open-loop system can operate without exceeding a specified error bound is considered for linear systems that are subject to uncertainties about their parameters and their initial conditions, and whose operation is hampered by disturbance signals. The objective is to characterise an optimal input signal that keeps performance errors within specified bounds for the longest time. It is shown that such an input signal exists, and that it can be approximated by a bang-bang input signal without significantly affecting performance.