mathematics

Jan Camiel Willems

Jan Camiel Willems is a mathematics 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 Jan Camiel Willems rather than just read about it. In short: Jan Camiel Willems (18 September 1939 – 31 August 2013) was a Belgian mathematical system theorist who has done most of his scientific work while residing in the Netherlands and the United States. He is most noted for the introduction of the notion of a dissipative system and for the development of the behavioral approach to systems theory.

Key takeaways

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

Reference excerpt

Jan Camiel Willems (18 September 1939 – 31 August 2013) was a Belgian mathematical system theorist who has done most of his scientific work while residing in the Netherlands and the United States. He is most noted for the introduction of the notion of a dissipative system and for the development of the behavioral approach to systems theory.

Biography Jan Willems was born in Bruges in 1939. He studied engineering at the University of Ghent, obtained the M.Sc. degree from the University of Rhode Island, and the Ph.D. degree from the Massachusetts Institute of Technology in electrical engineering in 1968. He was an assistant professor in the department of electrical engineering at MIT from 1968 to 1973. On 1 February 1973 he was appointed professor of systems and control in the mathematics department of the University of Groningen. In 2003 he became emeritus professor. Afterwards, he became guest professor at the KU Leuven. He served terms as chairperson of the European Union Control Association and of the Dutch Mathematical Society (Wiskundig Genootschap). He was managing editor of the SIAM Journal of Control and Optimization and as founding and managing editor of Systems & Control Letters.

Research contributions In his Ph.D. dissertation, Willems worked on input/output stability. In an often-cited 1972 paper he introduced the notion of a dissipative system. This notion is a generalization of Lyapunov function to input/state/output systems. The construction of the storage function, as the analogue of a Lyapunov function is called, led to the study of the linear matrix inequality (LMI) in control theory. Applied to linear-quadratic-Gaussian control, the construction of the storage function leads to the Kalman–Yakubovich–Popov lemma. In the 1980s Willems worked on the geometric theory of linear systems, where he introduced the notion of almost invariant subspace. Since the 1990s, he has devoted his interest to the development of the behavioral approach to systems theory and control. In the behavioral approach a dynamical system is simply viewed as a family of trajectories. This approach avoids having to separate the system variables into inputs and outputs.

Awards and honors Willems was a fellow of the IEEE, the Society for Industrial and Applied Mathematics, the American Mathematical Society and International Federation of Automatic Control. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. In 1998, he received the IEEE Control Systems Award "for seminal contributions to control theory and leadership in systems research". In 2010, he became doctor honoris causa of the University of Liège.

Selected publications

References

External links Jan Willems' homepage S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (book in pdf)

Worked examples

Example 1 — a first encounter with Jan Camiel Willems

Start with the simplest possible case. Write down what Jan Camiel Willems claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Jan Camiel Willems 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 Jan Camiel Willems 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 Jan Camiel Willems

In research: Jan Camiel Willems appears in mathematics 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 Jan Camiel Willems 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: Jan Camiel Willems is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1939 births, 2013 deaths, Academic staff of the University of Groningen, so understanding it makes those chapters shorter.
In everyday life: Look for Jan Camiel Willems 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 Jan Camiel Willems in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Jan Camiel Willems 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 Jan Camiel Willems out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Jan Camiel Willems in simple terms?

Why does Jan Camiel Willems matter?

Because it connects several mathematics 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 Jan Camiel Willems?

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 Jan Camiel Willems.