Exact time-optimal control of the wave equation

In this paper the time-optimal control of a distributed parameter system is derived in ciosed form. The class of systems studied in this work are distributed parameter systems whose dynamics are governed by the wave equation. A frequency domain approach is utilized to arrive at the time-optimal solution which is bang-off-bang. To corroborate the optimality of the control profile derived for the distributed parameter system, the system is discretized in space and a series of time-optimal control problems are solved for the finite dimensional model, with increasing number of flexible modes. The limiting controller show the convergence of the first and last switch of the bang-bang controller of the finite dimensional system to the first and last switch of the bang-off-bang controller of the distributed parameter system, in addition to the convergence of the maneuver time. The number of switches in between the first and last switch is a function of the order of the finite dimensional system. The maneuver time of the distributed parameter system is compared to that of an equivalent rigid system and it is shown for certain maneuvers, that the bang-bang control profile of the rigid system is also the time-optimal control of the distributed system.