  Real-time nonlinear stabilization techniques are often limited by inefficient
or intractable online and/or offline computations, or a lack guarantee for
global stability. In this paper, we explore the use of Control Contraction
Metrics (CCM) for nonlinear stabilization because it offers tractable offline
computations that give formal guarantees for global stability. We provide a
method to solve the associated online computation for a CCM controller - a
pseudospectral method to find a geodesic. Through a case study of a stiff
nonlinear system, we highlight two key benefits: (i) using CCM for nonlinear
stabilization and (ii) rapid online computations amenable to real-time
implementation. We compare the performance of a CCM controller with other
popular feedback control techniques, namely the Linear Quadratic Regulator
(LQR) and Nonlinear Model Predictive Control (NMPC). We show that a CCM
controller using a pseudospectral approach for online computations is a middle
ground between the simplicity of LQR and stability guarantees for NMPC.
