Demise of Roger Brocket, trailblazer in intertwining engineering and mathematics

On March 19, he passed away. Roger Brockett, a professor at Harvard University, whose contributions have left a lasting imprint on the world of engineering and mathematics. Brockett carried out fundamental work in applied areas such as robotics, astrodynamics, artificial intelligence, computer vision or the control of systems at the quantum level. To do this, he promoted areas of mathematics that are now also recognized as one of his reference figures: geometric control theory, dynamic systems or stochastic calculus.

Roger Brockett was born in 1938, on a farm in the small city of Seville in Ohio (USA), within a large family – he was the youngest of seven siblings. As a child, while the other children played with the animals, he was interested in agricultural machines. This premature vocation led him to study engineering because, according to him, it was a subject with which he could cover everything and his obsession in life has always been to understand and comprehend what was around him. In 1964 he received his doctorate from the Case Institute of Technology (Ohio) with a thesis on dynamical systems and their applications to control. One day, after attending a conference, he stayed for a while talking with the speaker, Peter Elias, head of the Department of Electrical Engineering at the Massachusetts Institute of Technology (MIT). He must have been so impressed that, a few days later, he received a letter offering him a contract at MIT.

At MIT, surrounded by some of the best engineers in the world, Brockett began to excel in both theoretical and applied problem solving. He studied, for example, the control of satellites, in which the non-linearity of the system determines its temporal evolution. These systems are described with equations that are not linear. Let’s imagine that we launched a paper airplane into the air several times; if we launch it with twice the speed, the trajectory can be very different, that is, the second does not follow the first with twice the speed. The complex effect exerted by the airflow on the wings of the plane makes it very difficult to predict its trajectory. To simplify these complex problems, Roger Brockett analyzed when it is possible to transform a nonlinear system to a linear one, much easier to solve. His ideas were the beginning of a line of theoretical and applied research, which is now being worked on all over the world.

Brockett decided to transfer to Harvard, looking for a new environment in which to develop his mathematical vision of engineering.

Later, Brockett decided to transfer to Harvard University, looking for a new environment in which to develop his mathematical vision of engineering and, in his words, “force me to think more deeply about fundamental concepts of science, rather than ingenious techniques to solve problems”. In the 80s of the last century he founded the Harvard Robotics Laboratory, where ambitious projects in robotic manipulation or control of quantum systems are currently being led. To do this, they make use of innovative mathematical tools from differential geometry, topology or dynamic systems.

A crucial area in current technological development, pioneered by Roger Brockett, is the so-called geometric control theory. It was designed to deal with problems such as those raised by the aerospace industry, or those that arise in the planning of robot trajectories.

To treat the inherent nonlinearity of these systems, Brockett used one of the most powerful branches of mathematics, differential geometry. Specifically, it allows the use of differential calculus techniques, started by Isaac Newton and Gottfried Leibniz, for non-linear spaces, such as a sphere or other curved surface, in order to study their intrinsic properties.

His mathematical vision helped him to see, within the complexity of different technological processes, the substantial, leaving aside the superfluous, and finding the ideal geometric framework for its treatment. In this way he was able to devise effective techniques to control technological systems to do various tasks, or at least know if they are capable of doing them.

Brockett was an exceptional person, kind and charming to deal with. When it came to scientific issues with him, he always looked for the fundamental, leaving aside the technicalities of theory, in his eagerness to understand and comprehend, which has always accompanied him throughout his life.

Their silent contribution has served and will serve to improve our society and our way of life, since their contributions have been essential in topics that we now recognize as important: robotics, satellite control, artificial intelligence… For example, the techniques developed by Brockett and his Collaborators offer the ideal way to deal with problems such as reorientation or stabilization of a satellite or a robot in a desired position.

His legacy will live on in the work of hundreds of scientists around the world who follow in his footsteps. One of the most important facets of Brockett was, without a doubt, the training of future generations. He directed more than sixty doctoral theses and some of his students are today scientific leaders in different fields. Among them are Daniel Liberzon, Jan Willems, Anthony Bloch, John Baras, John Bailleul, David Dobkin, Peter Crouch, PS Krishnaprasad. With all of them and with his scientific legacy, Roger Brockett, mathematical engineer, will always be with us.

David Martin de Diego He is a researcher at the Higher Council for Scientific Research at the Institute of Mathematical Sciences

Agate Rudder Garcia-Longoria is coordinator of the ICMAT Mathematical Culture Unit

Coffee and Theorems is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”

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