Geometric Jacobian linearization and LQR theory

The procedure of linearizing a control-affine system along a non-trivial reference trajectory is studied from a differential geometric perspective. A coordinate-invariant setting for linearization is presented. With the linearization in hand, the controllability of the geometric linearization is characterized using an alternative version of the usual controllability test for time-varying linear systems. The various types of stability are defined using a metric on the fibers along the reference trajectory and Lyapunov's second method is recast for linear vector fields on tangent bundles. With the necessary background stated in a geometric framework, linear quadratic regulator theory is understood from the perspective of the Maximum Principle. Finally, the resulting feedback from solving the infinite time optimal control problem is shown to uniformly asymptotically stabilize the linearization using Lyapunov's second method.