Construction of stabilizing feedback law by means of control Lyapunov function with respect to a part of the variables (Q2720926)

The problem of partial stabilization by means of a static time-invariant feedback law is investigated. The class of admissible feedback laws under consideration consists of locally bounded measurable functions, and the solutions of the closed-loop system are defined in the sense of A. F. Filippov. The notion of control Lyapunov function with respect to a part of the variables is introduced. It is proved that if the above function exists, then the control system is stabilizable with respect to a part of the variables. This result extends Artstein's theorem for the case of partial stabilization. The constructive feedback design is proposed for affine control system provided that there exists a control Lyapunov function with respect to a part of the variables. The obtained feedback law is proved to be continuous under some additional assumptions on the Lyapunov function.