Control in fluid mechanics and boundary layers – Jean-Michel Coron (Université Pierre et Marie Curie Paris)

January 7, 2022 @ 2:10 pm – 3:10 pm
Seminar Room 1
Newton Institute

A control system is a dynamical system on which one can act thanks to what is called the control. For example, in a car, one can turn the steering wheel, press the accelerator pedal etc. These are the control(s). One of the main problems in control theory is the controllability problem. One starts from a given situation and there is a given target. The controllability problem is to see if, by using some suitable controls depending on time, the given situation and target, one can move from the given situation to the target. We study this problem with a special emphasis on the case where the nonlinearities play a crucial role. In finite dimension in this case a key tool is the use of iterated Lie brackets as shown in particular by the Rashevski-Chow theorem. This key tool gives also important results for some control systems modelled by means of partial differential equations. However we do not know how to use it for many other control systems modelled by means partial differential equations. We present methods to avoid the use of iterated Lie brackets for the control in fluid mechanics (Euler and Navier-Stokes equations of incompressible fluids, shallow water equations). A special emphasis is put on the problem created by boundary layers.

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