The classical performance index optimization control algorithm is considered in order to check the real optimality of the control procedure; the basic steps for the optimal algorithm are reviewed, and the equation for the optimal control force derived. It is shown that the optimality conditions cannot be met with regard to the performance index, unless one is concerned with simple free oscillations. It is proved that in this case on one side the optimal control turns out to be of the linear closed-loop type, yielding explicit optimal control coe$cients, and on the other side that no solution can exist of the optimal problem for a generic forcing function. It is concluded that one is forced to calibrate the control force for free oscillations, and that the reliability of the index procedure mainly rests on some implicit expectation that linear control can be calibrated in the absence of the external disturbance and that it works under forced oscillations as well. Furthermore, the problem of delayed active control, with reference to a s.d.o.f. system controlled by a closed-loop linear algorithm and under the action of a dynamic forcing function is investigated.
In particular, the e!ects produced on the response of the structure by the introduction in the control law of assessed critical values of time delay are analysed and the comparison is proposed between the numerical results that one gets by adopting two di!erent procedures (on one hand the above-mentioned optimal linear control law and on the other hand the constrained minimization of the structural response norm) to compensate for time lag occurring in the actuation of the active control servomechanisms.