A simple extension of contraction theory to study incremental stability properties

Contraction theory is a recent tool enabling to study the stability of nonlinear systems trajectories with respect to one another, and therefore belongs to the class of incremental stability methods. In this paper, we extend the original definition of contraction theory to incorporate in an explicit manner the control input of the considered system. Such an extension, called universal contraction, is quite analogous in spirit to the well-known Input-to-State Stability (ISS). It serves as a simple formulation of incremental ISS, external stability, and detectability in a differential setting. The hierarchical combination result of contraction theory is restated in this framework, and a differential small-gain theorem is derived from results already available in Lyapunov theory.