Nonlinear stabilization by adding integrators

International audienceThe global stabilization problem of a nonlinear control system with an integrator is considered when the initial system (subsystem without integrator) is stabilizable. It is supposed that the stabilizing feedback and the Lyapunov function for the initial system satisfy the Barbashin-Krasovski\u ı asymptotic stability theorem. Explicit formulae for stabilizing feedback of a nonlinear system with an integrator are derived. The use of Lyapunov functions with derivatives of constant but not fixed signs significantly simplifies the computing of stabilizing feedback. This is confirmed by examples. The global stabilization of a nonlinear control system of the form \dot x= f(x, y), \qquad \qquad \dot y=u \tag 1 is studied, where x∈ \bbfRⁿ, y∈ \bbfR^p, u∈ \bbfR^p and f is a smooth vector field such that f (0, 0) =0. It is proved that to find a feedback stabilizer for this system we do not need to have a strict Lyapunov function for the subsystem (2) \dot x= f(x, v), where v is the input. Moreover, it is proved how to asymptotically stabilize system (1) without stabilizing system (2)