Habilitation Thesis (Doctor of Science Thesis). Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk, 2008.

Zuyev A.L. Motion control and stabilization of infinite dimensional mechanical systems with elastic elements (in Russian, 288 pages).
Key words:  stabilization, stability, feedback control, Lyapunov functional, flexible link manipulator, Euler - Bernoulli beam, Timoshenko beam, Galerkins method, degree of a map, semigroup of operators.
Abstract:  In the thesis, stabilization and control problems are solved for models of mechanical systems with elastic elements having infinite number of degrees of freedom. A class of generalized dynamical systems with the multi-valued solutions flow is introduced, and conditions of partial asymptotic stability are investigated for these systems in Banach and metric spaces. These results are applied for the synthesis of control functionals of a mechanical system consisting of a rigid body and an arbitrary number of the Euler - Bernoulli beams. The reduction scheme of a finite dimensional model of a system with elastic beam to the standard Brunovsky canonical form is proposed. An optimal control problem is solved, and the approximate controllability of the infinite dimensional system is established. The motion equations are derived for a manipulating robot with links represented by the Euler - Bernoulli and Timoshenko beams. For a single-link manipulator, a feedback control is found that ensures strong stabilization in an infinite dimensional state space. For the partial differential equations of motion, approximate Galerkin systems are constructed, and the problems of controllability, stabilization, observability, and observer-based stabilization are solved. Numerical results are presented to illustrate the efficiency of the control laws proposed. A new approach for the stabilization of nonlinear control systems is proposed, based on the application of the critical Hamiltonians.

Click here for the detailed abstract in Ukrainian (33 pages).

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