Contraction theory: Hausdorff--Riemann Measures as Set-Based Lyapunov Functions
We offer a measure-theoretic extension of the concept and theory of $k$-contraction, including their generalization on fractional dimensions $d$. The respective contraction property is defined through the exponential decay of the $d$-dimensional volume of compact sets transported by a nonlinear flow. For autonomous systems on positively invariant compact sets, we derive comprehensive, i.e., necessary and sufficient, conditions for $d$-contractivity in two complementary forms. The first is expressed in terms of the finite-time Lyapunov characteristic exponents and is akin in spirit to the first Lyapunov method. The second one is consonant with the second Lyapunov method and relies on existence of a Riemannian metric ensuring exponential decay of the metric-induced $d$-dimensional Hausdorff measure. To acquire monotone measure-theoretic-based Lyapunov functions, we introduce a family of \emph{Hausdorff-Riemann measures}, which are elliptic, metric-dependent $d$-measures that strictly decrease along the trajectories and thus may serve as Lyapunov functions. These measures enable an anytime characterization of the rate of contraction and provide constructive tools for stability analysis and feedback design. To illustrate the applicability of the approach, we derive tractable criteria for orbital stability of periodic solutions of autonomous ODE's and employ several prototypical particular examples, including a rigid body with dissipation and constant torque, the R\"ossler system, and the Langford system.