  We introduce the concept of a control contraction metric, extending
contraction analysis to constructive nonlinear control design. We derive
sufficient conditions for exponential stabilizability of all trajectories of a
nonlinear control system. The conditions have a simple geometrical
interpretation, can be written as a convex feasibility problem, and are
invariant under coordinate changes. We show that these conditions are necessary
and sufficient for feedback linearizable systems, and also derive novel convex
criteria for exponential stabilization of a nonlinear submanifold of state
space. We illustrate the benefits of convexity by constructing a controller for
an unstable polynomial system that combines local optimality and global
stability, using a metric found via sum-of-squares programming.
