Local smooth stabilizability of 2-dimension nonlinear control systems in critical cases

This paper studies the local smooth stabilizability of the 2-dimension nonlinear control system, which possesses a pair of conjugated imaginary eigenvalues. The system is firstly transformed to a standard formula by the non-singularity linear coordinate transformation and the time scale transformation as well. Next, the approaches for determining smooth control law and Lyapunov function of the closed loop system are provided by the construction of several sets of linear equations based on the expanded canonical discriminant function and the formal progression method. Finally, a sufficient condition of the local smooth stabilizability to the system is obtained, the validity of which is shown by an example.