science

Ross–Fahroo lemma

Ross–Fahroo lemma is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Ross–Fahroo lemma rather than just read about it. In short: Named after I. Michael Ross and F.

Key takeaways

Ross–Fahroo lemma belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Ross–Fahroo lemma to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Ross–Fahroo lemma from memory before moving on to harder problems.

Reference excerpt

Named after I. Michael Ross and F. Fahroo, the Ross–Fahroo lemma is a fundamental result in optimal control theory. It states that dualization and discretization are, in general, non-commutative operations. The operations can be made commutative by an application of the covector mapping principle.

Description of the theory A continuous-time optimal control problem is information rich. A number of interesting properties of a given problem can be derived by applying the Pontryagin's minimum principle or the Hamilton–Jacobi–Bellman equations. These theories implicitly use the continuity of time in their derivation. When an optimal control problem is discretized, the Ross–Fahroo lemma asserts that there is a fundamental loss of information. This loss of information can be in the primal variables as in the value of the control at one or both of the boundary points or in the dual variables as in the value of the Hamiltonian over the time horizon. To address the information loss, Ross and Fahroo introduced the concept of closure conditions which allow the known information loss to be put back in. This is done by an application of the covector mapping principle.

Applications to pseudospectral optimal control When pseudospectral methods are applied to discretize optimal control problems, the implications of the Ross–Fahroo lemma appear in the form of the discrete covectors seemingly being discretized by the transpose of the differentiation matrix. When the covector mapping principle is applied, it reveals the proper transformation for the adjoints. Application of the transformation generates the Ross–Fahroo pseudospectral methods.

See also Ross' π lemma Ross–Fahroo pseudospectral methods

References

Worked examples

Example 1 — a first encounter with Ross–Fahroo lemma

Start with the simplest possible case. Write down what Ross–Fahroo lemma claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Ross–Fahroo lemma before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Ross–Fahroo lemma ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Ross–Fahroo lemma

In research: Ross–Fahroo lemma appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Ross–Fahroo lemma in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Ross–Fahroo lemma is common in secondary-school and first-year university syllabi. It links to neighbouring topics Control theory, Numerical analysis, Optimal control, so understanding it makes those chapters shorter.
In everyday life: Look for Ross–Fahroo lemma outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Ross–Fahroo lemma in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Ross–Fahroo lemma means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Ross–Fahroo lemma out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ross–Fahroo lemma in simple terms?

Named after I. Michael Ross and F.

Why does Ross–Fahroo lemma matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Ross–Fahroo lemma?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Ross–Fahroo lemma.