Optimal control in the mathematical model of internal waves

The paper presents the results of the study of the problem on the optimal control to solutions for a mathematical model of internal waves, which is based on a linear system of equations of hydrodynamics. This model describes the propagation of waves in a homogeneous incompressible stratified fluid. The mathematical model includes the Sobolev equation, the Cauchy and Dirichlet condition. We use a parallelepiped as a considered domain in the mathematical model. The paper shows existence and uniqueness of a strong solution to the Cauchy–Dirichlet problem for the Sobolev equation. Also, we obtain the sufficient conditions for existence and uniqueness of a solution to the problem on optimal control to such solutions in Hilbert spaces. Proof of existence and uniqueness of a strong solution is based on the theorem for an abstract incomplete inhomogeneous Sobolev type equation of the second order and the theory of relatively p-bounded operators. In this paper, we present the theorem on existence and uniqueness of the optimal control for the problem under study, which is based on the works of J.-L. Lyons.