Incremental Versus Differential Approaches to Exponential Stability and Passivity

There are two main Lyapunov approaches to incremental stability analysis. One is to use incremental Lyapunov functions directly, and the other is based on so-called Finsler-Lyapunov functions via contraction analysis. A system is incrementally exponentially stable if it admits either an incremental or Finsler-Lyapunov function, and the converse is also true when the Jacobian of the drift vector field satisfies a boundedness assumption. However, the direct relation between these Lyapunov functions is not very clear yet. In this article, we show that if one type of Lyapunov function is found, the other can directly be constructed from it without the boundedness assumption. As an application of our approach, we also show that an open system is incrementally passive if and only if it is differentially passive under a mild technical assumption.