Global asymptotic stabilization of periodic nonlinear systems with stable free dynamics

We consider a nonlinear control system with periodic coefficients. We study the problem of asymptotic stabilization of the equilibrium x=0x=0 of the closed-loop system by state feedback. We assume that the free dynamic system possesses a periodic Lyapunov function ensuring Lyapunov stability of the equilibrium x=0x=0. We have developed a method for constructing a damping control for affine systems with periodic coefficients based on a generalization of weak Jurdjevic–Quinn conditions, using an extension of the notion of the commutator to non-stationary vector fields. Using this approach, we obtain sufficient conditions for uniform local and global asymptotic stabilization of general nonlinear systems, in particular, affine control systems, with periodic coefficients. Stabilization results generalize known results for time-invariant systems to time-varying periodic systems.