Ph.D. Thesis (Physical and Mathematical Sciences). Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk, 2000.

Zuyev A.L. Stabilization and stability of nonlinear dynamical systems with application to problems of rigid bodies mechanics (in Russian).
Key words:  stabilization, stability, local controllability, feedback law, solution in the sense of Filippov, Lyapunov function, partial stabilization, wind engine.
Abstract:  The dissertation is devoted to stabilization problems of nonlinear control systems in the critical cases. It has been proved that any locally controllable (reachable) system is nonasymptotically stabilizable by means of a discontinuous time-invariant feedback law provided that the solutions of closed-loop system are defined in the sense of A.F.Filippov. In addition, the set of discontinuity points of stabilizing feedback law is investigated for control affine systems. Such investigation is based on the selection of a continuous branch from the set-valued feedback control defining stable system of differential inclusions.

The asymptotic estimates of solutions of the model system are obtained for the critical case of two pairs of purely imaginary roots. An approach based on these estimates is proposed for solving the problem of optimal stabilization. The problem of partial stabilization of non-autonomous systems is investigated by means of Lyapunov functions having a negatively defined lower bound of derivatives. A theorem on stabilization of non-autonomous system in the sense of differential inclusions is proved. A constructive feedback design is proposed for control affine systems, provided that there exists a control Lyapunov function with respect to a part of the variables. The result obtained extends Artstein's theorem for the case of partial stabilization. For the case of autonomous system, the partial asymptotic stability is shown to be reached under a weaker condition on the Lyapunov function. With the help of these results, problems on partial stabilization of the orientation of a rigid body by means of a pair of control torques are solved. Two cases are considered. In the first case, the control is actuated by jet engines of orientation, while in the second one the control is implemented by means of flywheels.

A mathematical model of the wind engine is constructed in the dissertation. The proposed model consists of three rigid bodies interconnected by means of elastic joints. Conditions for stability of uniform rotations by linear approximation are obtained. These conditions are investigated for the case of large stiffness coefficients. The domains of stability are constructed in the planes of basic mechanical parameters.

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