Kawano & Bullo Unify Contraction Theory and Input-to-State Stability

Kawano and Bullo's latest work bridges two foundational concepts in nonlinear control theory: input-to-state stability (ISS) and contraction analysis. For systems defined on normed vector spaces, they characterize a specific incremental ISS property where the overshoot constant multiplies both initial-condition and input terms. By analyzing the associated variational system, they prove that an ISS-type bound on the variational dynamics is exactly equivalent to an incremental ISS bound on the original nonlinear system. This is a powerful theoretical unification that simplifies robustness analysis.

Going further, the authors establish an equivalence between an infinitesimal contraction condition—expressed through a Lyapunov-type function—and an incremental Lyapunov condition. Each of these yields both necessary and sufficient conditions for the ISS-type bounds, with the only gap being the input Lipschitz constant of the vector field. When the overshoot constant equals one, the infinitesimal contraction condition reduces to standard norm-based contraction conditions. The results are proven under only continuous differentiability of the vector field, making them widely applicable. They illustrate the findings with sensitivity matrices and Lyapunov characteristic exponents, offering concrete tools for practitioners in systems and control.