Optimal Control of the Behavior of Solutions to an Initial-Boundary Value Problem Arising in the Mechanics of Discrete-Continuum Systems.

We consider an initial-boundary value problem for a system of two differential equations, one of which is ordinary and the other a partial differential equation, where the connection between the equations is carried out through an integral functional. In this case, the boundary conditions contain higher-order time derivatives of the unknown functions. The initial-boundary value problem models the rotation of a mechanical system consisting of two rigid bodies connected by an elastic rod. The rotation is considered around the center of mass of one of the rigid bodies and carried out by a moment of external forces (control torque) applied to the axis of rotation of the rigid body. For the initial-boundary value problem, we introduce the concept of a generalized solution and prove theorems on the existence and uniqueness of a generalized solution and on the well-posedness of the problem formulation. We solve problems of optimal control of the rotation of a mechanical system from the initial state to the final state at a given point in time, minimizing the value of the control torque and minimizing the energy functional of the control torque. In this formulation, we also solve the performance problems under restrictions on the value of the control torque and on the value of the energy integral of the control torque. [ABSTRACT FROM AUTHOR]