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 is a continuous-time evolution law that allows a set of instantaneous velocities at each time and state instead of assigning one velocity. In its ordinary finite-dimensional form, $$ \dot x(t)\in F(t,x(t)), $$ where $F$ is set-valued. Under the standard solution convention used here, a trajectory $x$ is locally absolutely continuous and its derivative belongs to the indicated set for almost every time. Absolute continuity permits a derivative almost everywhere without requiring a classical derivative at every instant. The law constrains possible motions; it does not itself select one trajectory.^[1]

The set-valued rule can encode a real ambiguity or physical range at a discontinuity, as with static friction at zero slip. It can also collect admissible controlled velocities. The essential commitment is local derivative membership plus a declared solution convention, not convexity of $F$, uniqueness, or a nonempty reachable set. Those latter properties require additional hypotheses or further analysis. Bonnet and Frankowska preserve the same selection logic in a generalized setting where the state is a probability measure and selected velocity fields drive a continuity equation.^[1]^[2]^[3]

Structural Signature¶

Sig role-phrases: state and time carrier → set-valued velocity rule → absolutely continuous selection path → solution-convention and regularity scope.

State and time carrier. One specifies a state space and an interval on which a state curve can evolve. In the elementary formulation $x(t)\in\mathbb R^n$; in a sourced extension, a probability measure $\mu(t)$ is the state. A velocity set without a state and time domain is not yet an evolution law.^[1]^[3]
Set-valued velocity rule. $F(t,x)$ gives admissible derivatives. A singleton at every point reduces the form to an ordinary differential equation; genuine multiple values allow alternatives without asserting that every alternative yields a full solution.^[1]
Absolutely continuous selection path. A proposed $x(t)$ counts only if it is absolutely continuous and $\dot x(t)\in F(t,x(t))$ almost everywhere. In the Wasserstein formulation, a measurable selection of velocity fields must additionally satisfy the continuity equation distributionally; a bare curve or endpoint is insufficient.^[1]^[3]
Solution-convention and regularity scope. For discontinuous physical laws, the source's direct set-valued graph, a Filippov completion and other conventions can differ at the discontinuity. State which one is meant. Conditions such as upper semicontinuity, compactness, growth bounds or convexity may support a particular existence theorem, but are not universal membership roles of the bare inclusion.^[1]^[2]

What It Is Not¶

It is not simply an ODE with an unusual equals sign. An ODE $\dot x=f(t,x)$ has one prescribed local velocity; it is recoverable as the singleton case $F(t,x)=\{f(t,x)\}$. A genuinely multivalued $F$ can admit several locally possible velocities, but that does not prove multiple global solutions or any solution at all. A reachable set is constructed from whichever solutions actually exist; it is not part of the defining rule.^[1]^[2]

It is not a named existence theorem. Farkhi, Donchev and Baier prove a solution theorem for certain compact, upper-semicontinuous, potentially nonconvex multifunctions under growth and weak componentwise-monotonicity hypotheses. Those are conditions for their result, not a rule that all differential inclusions must have convex compact values. Equally, a formula for control-induced velocities needs a specified control class and selection assumptions before trajectory claims follow.^[2]^[3]

It is not a unique, convention-free cure for discontinuity. At zero slip, directly assigning a static-friction interval can represent physical forces not captured by a Filippov completion that ignores values on the discontinuity surface. The symbolic story “discontinuous friction” is too little to determine admissible trajectories until the graph and solution notion are fixed.^[1]

Scope of Application¶

In nonsmooth mechanics, Kiseleva, Kuznetsov and Leonov discuss dry-friction characteristics that take sign-dependent values away from zero slip but an interval at zero. A mechanical state law thereby permits a range of instantaneous forces and state derivatives on the sticking surface. Their analysis explicitly compares ways of defining solutions to discontinuous dynamics; the direct friction graph is a source-backed instance, not a claim that every friction system obeys the same completed graph.^[1]

In set-valued control and mean-field dynamics, Bonnet and Frankowska formulate a Wasserstein-space inclusion for curves of probability measures. A set $V(t,\mu)$ contains admissible velocity fields, and a measurable selected field must transport the measure through a continuity equation. This is a genuine generalized differential-inclusion setting; it must not be silently written as an ordinary finite-dimensional vector ODE or taken to prove that every selection corresponds to an admissible control without the paper's assumptions.^[3]

Clarity¶

The expression $\dot x\in F(t,x)$ answers a different question from “what is the solution?” It says which velocities are locally allowed. An initial condition and admissibility of a path are further matters. Neither the number of possible velocities at one point nor a plot of a reachable region tells us by itself how many full trajectories exist.^[1]^[2]

The clarity gain is especially sharp at discontinuities. A piecewise formula can be interpreted through several completions, and their behavior on the switching surface can differ. Writing a physical interval for static friction at zero slip preserves information that an almost-everywhere convexification of neighboring kinetic branches may discard. The researcher must identify the intended graph, not merely say “use a differential inclusion.”^[1]

Manages Complexity¶

A single multifunction $F$ compresses a family of admissible instantaneous motions into one governing object. Rather than writing a separate ODE for every friction regime or every selected control, one records the state-dependent Η set of choices and tests candidate paths against it. This representation is useful precisely because the choice may switch with state or time.^[1]^[3]

Compression does not confer well-posedness. Existence, uniqueness, compactness of solution sets and approximation by convexified laws are distinct mathematical questions. The nonconvex existence result of Farkhi and colleagues and the conditional relaxation results in Bonnet–Frankowska show that each conclusion has its own hypotheses. The inclusion notation organizes the problem; it does not solve all of these questions at once.^[2]^[3]

Abstract Reasoning¶

Given a proposed finite-dimensional model, first type the state and write $F(t,x)$ explicitly. Then ask whether a candidate absolutely continuous path satisfies derivative membership almost everywhere, including what happens when it passes through a discontinuity surface. Only afterward ask whether at least one such path exists from an initial state, whether there are several, and what can be reached. A solution theorem applies only after checking its regularity and growth assumptions against the selected $F$.^[1]^[2]

For a control interpretation, the set $F(t,x)=\{f(t,x,u):u\in U\}$ is meaningful only with a defined admissible-control class and a measurable selection that realizes the chosen velocity. Convexifying it can help establish approximations, but an averaged velocity in the convex hull is not automatically an instantaneous velocity of one original control. Bonnet and Frankowska prove a qualified relaxation result in their measure-valued context, not unconditional exact equivalence.^[3]

Knowledge Transfer¶

The set-valued evolution structure transfers literally from a dry-friction mechanical state to controlled measure dynamics at the level of state, allowed velocity family and selected trajectory. The mathematics of the carrier changes: an $\mathbb R^n$ derivative condition in the first case, a curve of probability measures driven by a selected field and continuity equation in the second. One cannot copy finite-dimensional existence assumptions into Wasserstein space without checking the source's separate theorem.^[1]^[3]

Outside continuous-time calculus, “a system has several options” is an analogy, not a differential inclusion. The named abstraction requires a time-indexed state, admissible instantaneous rates and a coherent path condition. Whether a more portable set-valued evolution-law skeleton deserves a prime is a future-prime question, not an automatic promotion of this typed mathematical node.

Examples¶

Dry friction at a zero-slip surface¶

Kiseleva, Kuznetsov and Leonov give a mechanical state form $\dot x=Ax+b\phi(\sigma)$ with a slip variable $\sigma=c^*x$. Their dry-friction characteristic takes a sign branch away from $\sigma=0$ but an interval such as $\phi(0)=[-1,1]$ at zero. At that surface the force, and thus the allowed state derivative, is set-valued. The authors explain why retaining the directly specified static-friction graph can matter: different discontinuous-equation solution conventions can ignore values assigned exactly on the surface and thereby change the modeled sticking behavior.^[1]

Mapped back: State and time carrier → mechanical $x(t)$ and slip $\sigma(t)$; set-valued velocity rule → $Ax+b\phi(\sigma)$ with interval-valued $\phi(0)$; absolutely continuous selection path → state trajectory whose derivative belongs to that set almost everywhere; solution-convention and regularity scope → direct static-friction graph is declared rather than silently replaced by a Filippov completion.

Mean-field control of a measure curve¶

Bonnet and Frankowska study a controlled distribution of states as a curve $\mu(t)$ of probability measures. For each time and measure, $V(t,\mu)$ gives a set of possible velocity fields. Their Definition 9 calls $\mu$ a solution when it is absolutely continuous in the Wasserstein space and some measurable selected field $v(t)\in V(t,\mu(t))$ satisfies $\partial_t\mu+\operatorname{div}(v\mu)=0$ in the distributional sense. The paper applies this framework to a mean-field optimal-control problem with closed-loop controls. This is not just a bigger vector $x$; the state and path test are typed differently.^[3]

Mapped back: State and time carrier → Wasserstein-space curve $\mu(t)$; set-valued velocity rule → the field family $V(t,\mu)$; absolutely continuous selection path → measurable $v(t)$ driving the distributional continuity equation; solution-convention and regularity scope → source Definition 9 fixes the generalized solution and subsequent existence/relaxation theorems introduce their own extra hypotheses.

Structural Tensions¶

T1: Preserve the intended alternatives versus obtain tractable existence guarantees. A genuinely nonconvex velocity family can represent distinct permissible motions. Convexifying or imposing strong regularity can simplify a proof, but may change the instantaneous alternatives being modeled. Conversely, keeping the raw alternatives may require a different theorem or leave existence unresolved. Farkhi and colleagues' nonconvex theorem shows that convexity is not constitutive; it is a choice tied to particular proof routes. Diagnostic: Was a condition included because the physical/control law warrants it, or only to fit a convenient existence theorem?^[2]

T2: Convexified analytic access versus exact admissible-control realization. A convexified inclusion can be easier to analyze, but a trajectory using an averaged velocity need not coincide at each instant with an original admissible-control trajectory. A relaxation theorem may establish approximation under explicit hypotheses, which is weaker than exact identity. Rejecting relaxation preserves literal choices but may forgo its useful compactness/approximation structure. Diagnostic: Does the conclusion require an exact original-control path, or a justified limit/approximation?^[3]

Structural–Framed Character¶

Differential Inclusion sits near the structural end inside a calculus-typed mathematical frame. Evaluative weight: inclusion is a formal membership relation, not a statement that branching is good or bad. Human-practice dependence: analysts choose models and solution conventions, but a path either meets the declared derivative-membership rule or it does not. Institutional origin: the theory has a research history, yet no institution's naming creates the set-valued evolution law. Vocabulary travel: the form moves from nonsmooth mechanics to measure-valued control because the state/velocity/selection relation survives, with carrier-specific qualifications. Import versus recognition: a new case requires an actual time-dependent state and admissible instantaneous-velocity relation, not simply multiple outcomes. Its character: a structural formalism with genuine cross-setting mathematical use, still domain-specific because derivatives, absolute continuity and typed velocity selections are indispensable.^[1]^[3]

Structural Core vs. Domain Accent¶

Skeletal relation. At each evolving state, a set of permissible local changes constrains whole paths. Whether that more portable “set-valued evolution” structure warrants a future prime is an open question; no live prime is assigned from this resemblance.

Domain-bound mechanism. In the standard form the local change is a derivative almost everywhere, the rule is $F(t,x)$, and the path is absolutely continuous. The generalized measure setting replaces a point-state derivative with a selected velocity field satisfying a continuity equation; that change is explicit, not a casual analogy. Static friction, a drilling system and mean-field controls are applications rather than essential names.^[1]^[3]

Why not a prime. Without time, derivative or an equivalent distributional evolution notion, and a precisely typed solution test, the remaining idea is merely “choose from a set.” The live Differential Equation node is not asserted as a strict parent because its current definition requires equality to a governing derivative expression; leaving this node provisionally unparented is more honest than a lexical edge.

Proposed DAG status: unparented. No checked live entry has a full necessary-genus signature for both the finite-dimensional set-membership law and its sourced generalized extension. Live Differential Equation is closely related and its singleton right-hand side is a special case, but its text presents an equality-governed rate law and cannot be made a strict parent without changing its stated boundary. Live Derivative names a mathematical component; derivative membership is not the entire inclusion identity.

The parentless status is provisional graph curation, not a claim that inclusions are unrelated to calculus. A future typed “set-valued evolution law” intermediary could connect them after separate admission and densification.

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

Not to Be Confused With¶

Ordinary differential equation. An equality $\dot x=f(t,x)$ selects one local velocity; it is the singleton-valued special case of the inclusion notation, not a synonym for a genuinely multivalued rule. Tell: Is $F(t,x)$ a set with multiple allowed derivatives?^[1]
Filippov completion. A particular way to assign values to a discontinuous single-valued field, not every directly specified set-valued physical graph. Tell: Which values on the discontinuity surface are actually retained?^[1]
Existence or uniqueness theorem. These establish whether and how solutions occur under extra hypotheses; neither is a component of the bare definition. Tell: Is a regularity/convexity assumption attached to a theorem or to the intended modeled law?^[2]
Reachable set or viability kernel. Objects derived by collecting or filtering trajectories; they presuppose a solution concept but do not replace the local rule. Tell: Is the subject an allowed local velocity or an aggregate of possible endpoints?
Convexified control inclusion. May admit averaged velocities not literally produced at an instant by a single original control; approximation needs a qualified relaxation result. Tell: Is the claim exact realization or convergence under assumptions?^[3]

References¶

[1] Maria Kiseleva, Nikolay Kuznetsov and Gennady Leonov, “Theory of Differential Inclusions and Its Application in Mechanics”, chapter 9 (2018), §9.2 Definitions 9.3–9.5 and Eqs.9.7–9.10, with §9.1 on frictional mechanics. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s

[2] Elza Farkhi, Tzanko Donchev and Robert Baier, “Existence of Solutions for Nonconvex Differential Inclusions of Monotone Type”, arXiv:1307.1871 (2013), pp.1–3, assumptions A1–A3, Theorem 1 and Example 2. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i

[3] Benoît Bonnet and Hélène Frankowska, “Differential Inclusions in Wasserstein Spaces: The Cauchy-Lipschitz Framework”, arXiv:2007.08906v2 (2020), §3.1 Definition 9 and Eq.20, §3.3 relaxation result, §4 mean-field control application. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n