What does the inclusion symbol mean?

It says the derivative is an element of an allowed set of rates, rather than equal to one prescribed rate.

Does the model choose one trajectory?

Generally it admits a family. A selection criterion can choose a representative, but that is an additional modeling decision.

Is an alpha-cut itself fuzzy?

No. For a fixed alpha it is an ordinary set. The collection of cuts represents the graded fuzzy description.

AI Glossary / Graphs and symbolic AI / Advanced

Fuzzy differential inclusion

Definition

A fuzzy differential inclusion models change when the admissible rate of change is described by a fuzzy set. A common interpretation uses its alpha-cuts to form a family of ordinary differential inclusions, each allowing a set of possible rates at a chosen membership level.

What it means

An ordinary differential equation specifies a rate, such as x'(t) = 2. A differential inclusion allows a set of rates, such as x'(t) belonging to [1, 3]. Many trajectories can satisfy that condition.

A fuzzy version adds graded compatibility to the allowed rates. Values near two might fit the model more strongly than values near one or three. Selecting a membership level gives an ordinary set of rates that meet that level.

This is a description of compatible dynamics, not automatically a probability model over trajectories. Different formulations require reports to identify the solution interpretation and combination of information along paths.

How it works

Start with a fuzzy mapping that associates each time and state with a fuzzy set of rates. At a positive level alpha, take the set of rates whose memberships are at least alpha. An admissible trajectory must have its derivative in that set, typically almost everywhere over the time interval.

Solve or approximate the resulting inclusion, allowing all rate choices permitted by the model. If the initial state is also fuzzy, its corresponding cut supplies the initial-state set. Repeating the calculation across levels can describe nested sets of trajectories or reachable states.

Regularity assumptions are important for existence and numerical approximation. A calculation that samples only a few trajectories need not enclose every admissible one. Dependencies also matter: a parameter fixed for the whole experiment is different from a rate that can vary freely within the allowed set at each instant.

Three practical examples

A simulation explores possible positions when an object's velocity is described by graded ranges rather than one measured constant.
A process study represents uncertain net accumulation rates and examines which stock levels are reachable over a specified time horizon.
A mathematical modeling project compares families of trajectories compatible with imprecise dynamic rules, without inventing a probability distribution over those rules.

Think of it this way

Think of a map showing several allowed walking speeds, with some speeds fitting the description “comfortable pace” better than others. Choosing a compatibility level determines which speeds to include in a journey calculation.

The analogy helps explain families of possible paths. It does not say that a real walker changes pace arbitrarily or that the compatibility grades are observed frequencies of choosing each speed.

Worked example

Calculating reachable positions at two levels

Consider an invented scalar position model starting at x(0) = 0. Its velocity description is triangular, with closed support [1, 3] and membership one at 2. At positive level alpha, allowable velocities are [1 + alpha, 3 - alpha]. The set does not depend on position or time.

Allow any measurable velocity function that stays within that interval. After two time units, integration gives reachable positions [2 + 2alpha, 6 - 2alpha]. The bounds follow because integrating a velocity bounded below and above preserves those integral bounds. Every position between them is reachable here by selecting an appropriate constant velocity.

At alpha = 0.5, the interval is [3, 5]. At alpha = 0.8, it is [3.6, 4.4]. At alpha = 1, only position 4 remains. These nested intervals express compatibility levels, not 50%, 80% and 100% coverage probabilities.

This result assumes unrestricted scalar rates and constant bounds. State-dependent constraints, acceleration limits or linked uncertain parameters would require additional analysis.

Illustrative case study

The fictional Birch Trajectories team studies a simulated delivery cart with an imprecise speed description. Its first report displays a band of reachable positions and labels the inner band an 80% confidence region.

Review identifies that the underlying model uses a membership level of 0.8, without a probabilistic interpretation. The team changes the label and explains how the speed description was constructed. It also discovers that the model allows instantaneous speed changes that the cart cannot perform.

Birch adds the relevant motion constraints before comparing simulated paths with recorded runs. The revised documentation distinguishes compatibility, physical admissibility and empirical observation. The case illustrates model interpretation; it makes no claim that the simplified inclusion is sufficient for real vehicle operation.

Common misconceptions and mistakes

Replacing the inclusion by its midpoint equation discards admissible trajectories and can miss extremes.
Treating a high membership level as a high-confidence probability interval changes the mathematical meaning.
Assuming an uncertainty parameter can vary independently at every time can enlarge the trajectory family beyond the intended model.
Equating this formulation with every fuzzy differential equation overlooks alternative derivatives and solution concepts used in that literature.

Business use

For organizations, the practical question is whether the modeled trajectory family helps assess a decision under imprecise information. The approach may support sensitivity and reachability studies, but its sophistication does not make unsupported fuzzy inputs objective.

Ask which rates are allowed, why their grades were chosen and whether the same uncertainty must persist across time. Require clarity about exact reachable sets versus numerical approximations. If a probability of an adverse event is needed, membership grades alone do not answer that question; the decision requires an appropriately justified probabilistic or combined model.

Check your understanding

In the worked model at alpha = 0.5, can the position at time two equal 5.5?

Answer: No. The allowed velocity interval is [1.5, 2.5], so integration from zero permits at most position 5. A lower membership level could allow a wider interval, but that would be a different cut.

Related terms

Fuzzy differential equation
Fuzzy number
Reachable set
Trajectory
Possibility theory
Differential inclusion
alpha-cut
uncertainty propagation

Learn next

Use the worked velocity description to calculate the reachable interval at time three for alpha = 0.4. Then explain why adding a maximum acceleration would change the admissible trajectory calculation even if the speed membership function stayed unchanged.

Trajectory

Sources

Colombo and Krivan's original paper develops nonprobabilistic graded treatment of inclusion trajectories; indexed primary-paper text was available although direct PDF access failed: https://fix.prf.jcu.cz/~krivan/clanky/ColomboKrivan-Dynamic%20Systems%20and%20Applications-1992.pdf

An accessible research article discusses the inclusion interpretation alongside fuzzy derivatives: https://link.springer.com/article/10.1155/2009/395714

A primary research introduction explicitly displays the alpha-cut inclusion formulation: https://www.sciencedirect.com/science/article/pii/S0165011421003894

The scalar calculations and Birch case are original illustrations of a stated formulation, not a universal solution theorem.

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