On boundary control problems for the Klein–Gordon–Fock equation with an integrable coefficient

We consider the process described in the rectangle Q T = [0 ≤ x ≤ l ] × [0 ≤ t ≤ T ] by the equation u tt - u xx - q ( x, t ) u = 0 with the condition u ( l, t ) = 0, where the coefficient q ( x, t ) is only square integrable on Q T . We show that for T = 2 l the problem of boundary control of this process by the condition u (0, t ) = µ( t ) has exactly one solution in the class W 2 1 ( Q T ) under minimum requirements on the smoothness of the initial and terminal functions and under natural matching conditions at x = l .