Gradient in the Problem of Controlling Processes Described by Linear Pseudohyperbolic Equations

The paper considers the problem of controlling processes whose mathematical model is an initial–boundary value problem for a pseudohyperbolic linear differential equation of high order in the spatial variable and second order in the time variable. The pseudohyperbolic equation is a generalization of the ordinary hyperbolic equation typical in vibration theory. As examples, we consider models of vibrations of moving elastic materials. For the model problems, an energy identity is established and conditions for the uniqueness of a solution are formulated. As an optimization problem, we consider the problem of controlling the right-hand side so as to minimize a quadratic integral functional that evaluates the proximity of the solution to the objective function. From the original functional, a transition is made to a majorant functional, for which the corresponding upper bound is established. An explicit expression for the gradient of this functional is obtained, and adjoint initial–boundary value problems are derived.