Robust stabilization to non-linear delayed systems via delayed state feedback: The averaging method

The problem of robust stabilization to dynamical systems via delayed feedback is important in applications. Due to the fact that the characteristic quasi-polynomial of a delayed system is difficult to analyze when uncertain parameters are involved, this problem has been most frequently solved on the basis of the method of Lyapunov functional, by solving Riccati equations, or by solving linear matrix inequalities. By applying the averaging method that reduces the delay differential equation of infinite dimensional to an ordinary differential equation, this paper presents a simple method to stabilize the trivial solution or periodic solutions of a type of non-linear delayed vibration systems via delayed state feedback. In particular, this method is applied to the robust stabilization of those systems when the system parameters are uncertain, but fall into given intervals, respectively. In addition, an extension is made to this problem for a general class of delayed systems that result from a small perturbation of a linear delay system with characteristic roots of non-positive real parts only. This can serve as a straightforward application to the Hopf bifurcation control of delayed systems with weak non-linearity.