Identity of Optimality Conditions for Elastic Vibrations in Different Auxiliary Interpretations of Wave Problem

In this paper an optimal control problem is considered, where the controlled process is described by a non-linear wave equation. The state vector consists of the solution for the non-linear wave equation and its first-order partial derivatives with respect to the independent variables. The vector of controls includes distributed and boundary controls. The statement of the problem allows any combinations of boundary conditions of the first, second, and third kinds on both left and right borders of the domain. For the initial optimal control problem two types of equivalent auxiliary optimal control problems are constructed. They differ by the ways of describing the controlled process. The first approach describes the controlled process by a hyperbolic system of four differential equations of the first order. The second approach uses one differential equation of the second order and two DE of the first order, identical to the ones used at the first equivalent problem. It is necessary to reduce the initial wave equation to the corresponding equivalent systems in order not only to construct a convenient concept of a generalized solution, but to obtain the necessary optimality conditions as well. In spite of formally different Pontryagin's functions used for the two equivalent control problems, it was proved that the corresponding conjugate problems take such forms that for the same set of controls the values of the Pontryagin's functions coincide throughout the domain. This property justifies identity of optimality conditions in forms of variational and Pontryagin's maximum principles, obtained for the both equivalent optimal control problems.