Time varying feedback synthesis for a class of non-homogeneous systems

This paper demonstrates that a previously introduced differential geometric approach to the synthesis of time varying stabilizing feedback controls for homogeneous systems (systems without drift) also applies to a wide class of non-homogeneous systems (systems with drift). The approach is universal in the sense that it is independent of the vector fields determining the motion of the system, or of the choice of a Lyapunov function. The proposed feedback law is a composition of a standard stabilizing feedback control for a Lie bracket extension of the original system and a periodic continuation of a specific solution to an open loop control problem stated for an abstract equation on a Lie group, an equation which describes the evolution of flows of both the original and extended systems. The open loop problem is solved as a trajectory interception problem in logarithmic coordinates of flows.