When planning trajectories, a reasonable demand on the generated curves is that they interpolate through given points, or given intervals, at given specified times, or that they go through these points or intervals at times lying in specified time windows. This problem is addressed here as an optimal control problem, minimizing the infinity norm of the inputs of a discretized linear control system since the infinity norm minimization problem arises naturally when doing trajectory planning for both controlled switching systems and non-holonomic robotic systems.