Regular Pairings for Nonquadratic Lyapunov Functions and Contraction Analysis

Recent studies on stability and contractivity have highlighted the importance ofsemi-inner products, which we refer to as ``pairings,"" associated with general norms. A pairing isa binary operation that relates the derivative of a curve's norm to the radius vector of the curveand its tangent. This relationship, known as the curve norm derivative formula, is crucial whenusing the norm as a Lyapunov function. Another important property of the pairing, used in stabilityand contraction criteria, is the so-called Lumer inequality, which relates the pairing to the inducedlogarithmic norm. We prove that the curve norm derivative formula and Lumer's inequality are, infact, equivalent to each other and to several simpler properties. We then introduce and characterizeregular pairings that satisfy all of these properties. Our results unify several independent theoriesof pairings (semi-inner products) developed in previous work on functional analysis and controltheory. Additionally, we introduce the polyhedral max pairing and develop computational tools forpolyhedral norms, advancing contraction theory in non-Euclidean spaces