Spectral decomposition method for controlled damped string reduction of control time

We consider the zero-controllability problem for the distributed parameter system governed by the damped wave equation where with boundary conditions The function f(t) is the control. This problem was studied in our recent (joint) work using the spectral decomposition method. The approach was based on our recent results on the spectral analysis of the nonselfadjoint dynamics generator of the above system. It was shown that the system is controllable in time To equals twice the time it takes for a wave to travel between the ends of the interval [0, a]. In this work, we investigate the conditions of the initial data, under which the control time can be reduced. We describe a class of initial data for which the Control time is T o where k is a positive integer. Based on this result, we show that the same effect can be achieved by changing the structure of the control (replacing by the sum without any restrictions on the class of initial data.