Differential Inclusion¶

A continuous-time evolution law that permits a state-dependent set of instantaneous velocities rather than prescribing just one.

Version: v1 · 2026-10-03 · History
Domain-specific #: 13141
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Dynamical Systems → Mathematics

Core Idea¶

A differential inclusion permits a set of instantaneous velocities at each state and time. In the usual finite-dimensional form, \(\dot x(t)\in F(t,x(t))\); a solution is an absolutely continuous path whose derivative belongs to that set almost everywhere. The law constrains possible paths but does not guarantee any solution or prescribe a unique one. A singleton-valued \(F\) recovers an ordinary differential equation as a special case.[^{ref-13e3272d9e9a][}ref-1853d37c99e3]

Scope of Application¶

In nonsmooth mechanics, a dry-friction force can be interval-valued at zero slip even when kinetic friction takes sign-dependent values away from zero. In a distinct mean-field control setting, Bonnet and Frankowska let a set \(V(t,\mu)\) contain admissible velocity fields for a probability-measure state; a measurable selection drives an absolutely continuous measure curve through a continuity equation. That generalized form is not simply a larger finite-dimensional vector ODE.[^{ref-13e3272d9e9a][}ref-8311008cb6de]

Clarity¶

The inclusion is the local rule, not the reachable set derived from all its solutions. A discontinuous single-valued formula also needs a declared completion: a direct static-friction graph and a Filippov construction may represent different at-zero behavior. Compactness, upper semicontinuity, convexity and growth assumptions belong to specific existence arguments, not to the bare identity. Farkhi and colleagues even prove existence for a structured class with nonconvex allowed values.[^{ref-13e3272d9e9a][}ref-1853d37c99e3]

Manages Complexity¶

One set-valued map records alternative admissible local motions without writing a separate equation for every regime or control. The compression leaves existence, uniqueness and trajectory realization as separate questions. In particular, convexifying allowed velocities can aid analysis but need not make every averaged-velocity path exactly realizable by a single original control; relaxation claims require their theorem hypotheses.[^{ref-1853d37c99e3][}ref-8311008cb6de]

Abstract Reasoning¶

To assess a model, specify the state carrier and \(F\), choose a solution convention, and test candidate paths by derivative membership almost everywhere. At a discontinuity surface, ask which graph values are retained. Only then apply an existence or relaxation theorem after checking its assumptions. Bonnet and Frankowska's measure-valued version uses a selected velocity field satisfying a distributional continuity equation, so its own definition must be used rather than silently substituting the Euclidean one.[^{ref-13e3272d9e9a][}ref-1853d37c99e3][^ref-8311008cb6de]

Knowledge Transfer¶

The state–allowed-velocity–selected-path structure transfers from frictional mechanics to controlled measure dynamics, with distinct mathematical carriers and regularity conditions. Live Differential Equation is related but currently equality-based, so no strict DAG parent is proposed. A more portable set-valued evolution skeleton is a future-prime question; without continuous-time derivative or distributional path semantics, “several options” is only an analogy.[^{ref-13e3272d9e9a][}ref-8311008cb6de]

[^ref-13e3272d9e9a]: Maria Kiseleva, Nikolay Kuznetsov and Gennady Leonov, “Theory of Differential Inclusions and Its Application in Mechanics”, chapter 9 (2018), §9.2. [^ref-1853d37c99e3]: Elza Farkhi, Tzanko Donchev and Robert Baier, “Existence of Solutions for Nonconvex Differential Inclusions of Monotone Type”, arXiv:1307.1871 (2013), pp.1–3. [^ref-8311008cb6de]: Benoît Bonnet and Hélène Frankowska, “Differential Inclusions in Wasserstein Spaces: The Cauchy-Lipschitz Framework”, arXiv:2007.08906v2 (2020), §3.1, §3.3 and §4.

Neighborhood in Abstraction Space¶

Differential Inclusion sits in a moderately populated region (57^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Continuum Mechanics & Field Models (42 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08