Bebiia M. Stabilization and bounded control synthesis for nonlinear systems with uncontrollable unstable first approximation

The stabilization and bounded control synthesis problems for some classes of nonlinear systems with uncontrollable unstable first approximation are solved. The approach for stabilizing (finite-time stabilizing) control design is based on the study of a nonlinear approximation to the initial system. We find an explicit change of coordinates and control that transforms several classes of singular triangular systems to nonlinear systems of a special form; it is used to construct classes of stabilizing and finite-time stabilizing controls. We obtain necessary and sufficient conditions for the solvability of the singular Lyapunov matrix equation, which arises in the feedback construction process. The class of positive definite solutions of this equation is described.