On solution of a optimal control problem governed by a linear wave equation

This paper studies the minimization problem governed by a wave equation with homogeneous Neumann boundary condition and where the control function is a initial velocity of the system. We give necessary conditions for the existence and uniqueness of the optimal solution. We get the Frechet derivation of the cost functional via the solution of the corresponding adjoint problem. We construct a minimizing sequence and show that the limit of the minimizing sequence is the solution of the optimal control problem.