Continuous feedback stabilization of nonlinear control systems by composition operators

Abstract

The ability to asymptotically stabilize control systems through the use of continuous feedbacks is an important topic of control theory and applications.  In this paper, we provide a complete characterization of continuous feedback stabilizability using a new approach that does not involve control Lyapunov functions.  To do so, we first develop a slight generalization of feedback stabilization using composition operators and characterize continuous stabilizability in this expanded setting.  Employing the obtained characterizations in the more general context, we establish relationships between continuous stabilizability in the conventional sense and in the generalized composition operator sense.  This connection allows us to show that the continuous stabilizability of a control system is equivalent to the stability of an associated system formed from a local section of the vector field inducing the control system.  That is, we reduce the question of continuous stabilizability to that of stability.  Moreover, we provide a universal formula describing all possible continuous stabilizing feedbacks for a given system.