Work

The scientific work is devoted to problems of motion control and stabilization of the mechanical system consisting of a rigid body and a thin elastic plate. A reduction scheme that allows transforming the equations of motion with partial derivatives to an infinite system of ordinary differential equations is proposed. Controllability conditions are obtained for a model in a finite dimensional state space. Conditions of spectral controllability are studied as well. A mathematical model of the Kirchhoff plate with the rotational inertia of its cross section is considered. For such a model, a system of ordinary differential equations with finite numbers of modal coordinates is derived, and the optimal control problem with a quadratic cost is solved. We consider a dynamical system with distributed parameters in order to describe controlled vibrations of the Kirchhoff plate. A class of optimal controls corresponding to its finite-dimensional approximations is used to study the reachable set. Analytic estimates of the norm of these control functions are obtained depending on the boundary conditions. These estimates are used to study the reachable set for the infinite-dimensional system. For a model with incommensurable frequencies, an estimate of the reachable set is obtained under the condition of power decay of the amplitudes of generalized coordinates. Feedback control functionals, depending on the generalized velocities, are con-structed for the system considered. A theorem on the partial asymptotic stability of the equilibrium of the closed-loop system is proved. A mechanical system consisting of a rigid body and an elastic Kirchhoff plate is considered under the action of three independent controls. The equations of motion for a nonlinear model are derived in the form of a system of ordinary and partial differential equations. The operator form of this problem is presented as an abstract differential equation in a Hilbert space. A feedback control law is constructed such that the corresponding infinitesimal generator of the closed-loop system is dissipative. Key words: Kirchhoff plate, controllability, optimal control problem, reachable set, feedback control, asymptotic stability, infinitesimal generator.