A simple extension of contraction theory to study incremental stability properties

Contraction theory is a recent tool enabling to study the stabil-ity of nonlinear systems trajectories with respect to one another,and therefore belongs to the class of incremental stability meth-ods. In this paper, we extend the original definition of contrac-tion theory to incorporate in an explicit manner the control in-put of the considered system. Such an extension, calleduniver-sal contraction, is quite analogous in spirit to the well-knownInput-to-State Stability (ISS). It serves as a simple formulationof incremental ISS, external stability, and detectability in adif-ferentialsetting. The hierarchical combination result of con-traction theory is restated in this framework, and a differentialsmall-gain theorem is derived from results already available inLyapunov theory.