Exact boundary controllability of Galerkin’s approximations of Navier-Stokes equations. (English) Zbl 1053.93009

Lions, Jacques-Louis; Zuazua, Enrique

Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 26, No. 4, 605-621 (1998).

Summary: We consider the 2D and 3D Navier-Stokes equations in a bounded smooth domain with a boundary control acting on the system through the Navier slip boundary conditions. We introduce a finite-dimensional Galerkin approximation of this system. Under suitable assumptions on the Galerkin basis we prove that this Galerkin approximation is exactly controllable. Moreover, we prove that the cost of control is independent of the presence of a nonlinearity on the system. Our assumptions on the Galerkin basis are related to the linear independence of suitable traces of its elements over the boundary. In this respect, the one-dimensional Burgers equation provides a particularly degenerate example that we study in detail. In this case we prove local controllability results.

MSC:

93B05	Controllability
35Q30	Navier-Stokes equations
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76D05	Navier-Stokes equations for incompressible viscous fluids

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