Perspectives on Contractivity in Control, Optimization, and Learning

Contraction theory is a mathematical framework for studying the convergence, robustness, and modularity properties of dynamical systems and algorithms. In this opinion paper, we provide five main opinions on the virtues of contraction theory. These opinions are (i) contraction theory is a unifying framework emerging from classical and modern works, (ii) contractivity is computationally-friendly, robust, and modular stability, (iii) numerous dynamical systems are contracting, (iv) contraction theory is relevant to modern applications, and (v) contraction theory can be vastly extended in numerous directions. We survey recent theoretical and applied research in each of these five directions.