Stabilization of two-dimensional incompressible perfect fluids. (Sur la stabilisation des fluides parfaits incompressibles bidimensionnels.) (French) Zbl 1086.93511

Coron, Jean-Michel

Sémin. Équ. Dériv. Partielles, Éc. Polytech., Cent. Math., Palaiseau 1998-1999, Exp. No. VII, 17 p. (1999).

From the introduction: A natural question that arises is whether, if a control system is controllable, one can asymptotically stabilize its equilibrium points. It has long been known that the answer is affirmative for finite-dimensional linear control systems. This is also true for a great many infinite-dimensional linear systems; see, e.g., the works of Slemrod, of J.-L. Lions, of Lasiecka and Triggiani, and of Komornik. But the answer is negative for nonlinear systems, even finite-dimensional ones. The first counterexamples were due to Sussmann and to Sontag and Sussmann, and Brockett has given an elegant and powerful necessary condition for asymptotic stabilizability which is not satisfied in the case of many nonlinear systems.
It is therefore natural to consider the case of the Euler equation for incompressible fluids. We examine the simplest possible equilibrium, namely, zero velocity. Note that in the absence of control this equilibrium point is stable but not asymptotically stable. We show that for two-dimensional simply connected bounded domains this equilibrium point can be stabilized asymptotically using explicit controls.

MSC:

93D15	Stabilization of systems by feedback
34H05	Control problems involving ordinary differential equations
35Q35	PDEs in connection with fluid mechanics
76B75	Flow control and optimization for incompressible inviscid fluids
76E99	Hydrodynamic stability