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