Boundary control problem for the one-dimensional Klein-Gordon-Fock equation with a variable coefficient. The case of control by displacement at one endpoint with the other endpoint being fixed

We consider the problem of boundary control by displacement at one boundary point x = 0 for a process described by the Klein-Gordon-Fock equation with a variable coefficient on a finite interval 0 ≤ x ≤ l with the Dirichlet condition u(l, t) = 0 at the other boundary point. For the critical time interval T = 2l, we show that there exists a unique boundary function u(0, t) = µ(t) bringing the system from an arbitrary initial state into an arbitrary terminal state.