Gradient in the problem of controlling processes described by linear pseudohyperbolic equations

The paper considers the problem of controlling processes, the mathematical model of which 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, which is typical in vibration theory. As examples, models of vibrations of moving elastic materials were considered. For model problems, an energy identity is established, and conditions for the uniqueness of a solution are formulated. As an optimization problem, we considered the problem of controlling the right side in order to minimize the quadratic integral functional, which evaluates the proximity of the solution to the objective function. From the original functional a transition was made to the majorant functional, for which the corresponding upper bound was established. An explicit expression for the gradient of this functional is obtained, and conjugate initial-boundary value problems are derived.