Optimal active pointwise control of distributed parameter systems with time delays

A class of time-delayed optimal control systems governed by hyperbolic partial differential equations is considered. Problems of this type are of significant practical interest, for example, in controlling flexible or very large space structures. A control mechanism is proposed to suppress the undesirable vibrations in the structures by means of the simultaneous application of spatially distributed pointwise open-closed loop controllers. Two convex performance indices are introduced and minimized with respect to the open-loop control functions and the closed-loop control parameters. It is shown that the optimal pointwise open-loop control is related to the adjoint variable by a maximum principle. The maximum principle converts the solution of the original problem into the solution of a system of coupled initial-boundary-terminal-value problems with both delayed and advanced terms. The closed-loop control parameters are numerically determined from the minimization of the energy of the system subject to a constraint on the amount of closed-loop control force that can be applied. The proposed theory is demonstrated by applying it to a simply-supported beam subject to time-delayed feedback and open-loop controllers.