Dr. Victoria Grushkovskaya

Institute of Mathematics, Julius Maximilian University of Würzburg
Emil-Fischer-Straße 40, 97074 Würzburg, Germany

Institute of Applied Mathematics and Mechanics,
National Academy of Sciences of Ukraine



Stabilization of Nonlinear Systems with Constraints on the Control

The thesis is devoted to stability and stabilization problems of the motion of nonlinear systems with constrained controls. Sufficient conditions for the attraction of trajectories of an abstract dynamical system with a weakly monotonic measure on the flow are obtained for almost all initial conditions. The limit set for solutions of nonlinear differential equations is described in terms of density functions. By using these results, the stabilization problem is solved for model equations describing the orientation of a rigid body in terms of quaternions with affine control. Asymptotic estimates of the solutions of a nonlinear system with critical and stable components are obtained for the case of q pairs of purely imaginary eigenvalues. By using these estimates, the decay rate of oscillations of a double pendulum with a partial dissipation is studied. The approach proposed in the thesis exploits the center manifold theory and the normal form method. In addition, the optimal stabilization problem is solved for nonlinear systems in a critical case of stability with a pair of purely imaginary eigenvalues. The optimality criterion is the maximization of the decay rate of solutions in a neighborhood of the origin. This criterion is formulated as a minimax problem with respect to non-integral functional. An explicit construction of a Lyapunov function is proposed to evaluate the optimal cost. As an example, a minimax optimal controller is obtained for a spring-pendulum system with partial measurements of the state vector. Necessary conditions for the stabilizability of control systems with three critical Hamiltonians are obtained. These conditions are used for the stabilization of the rotations of a rigid body in the case when the feedback values are in the set with three extremal points.