Control Contraction Metrics and Universal Stabilizability
This work addresses the fundamental control theory problem of stabilizing all trajectories in nonlinear systems, which is incremental as it builds on existing concepts like control Lyapunov functions.
The paper tackles the problem of universal stabilizability for nonlinear systems, where every solution can be globally stabilized, by introducing control contraction metrics that can be found via pointwise linear matrix inequalities, with conditions shown to be necessary and sufficient for linear systems and certain nonlinear classes.
In this paper we introduce the concept of universal stabilizability: the condition that every solution of a nonlinear system can be globally stabilized. We give sufficient conditions in terms of the existence of a control contraction metric, which can be found by solving a pointwise linear matrix inequality. Extensions to approximate optimal control are straightforward. The conditions we give are necessary and sufficient for linear systems and certain classes of nonlinear systems, and have interesting connections to the theory of control Lyapunov functions.