Optimization of the control by elastic boundary forces at two ends of a string in an arbitrarily large time interval

We further develop the method, devised earlier by the authors, which permits finding closed-form expressions for the optimal controls by elastic boundary forces applied at two ends, x = 0 and x = l, of a string. In a sufficiently large time T, the controls should take the string vibration process, described by a generalized solution u(x, t) of the wave equation

$$u_{tt} (x,t) - u_{tt} (x,t) = 0,$$

from an arbitrary initial state

$$\{ u(x,0) = \varphi (x), u_t (x,0) = \psi (x)$$

to an arbitrary terminal state

$$\{ u(x,T) = \hat \varphi (x), u_t (x,T) = \hat \psi (x).$$