LOCAL EXACT BOUNDARY CONTROLLABILITY OF THE BOUSSINESQ EQUATION

We study the local exact boundary controllability problem for the Bous sinesq equations that describe an incompressible fluid ow coupled to t hermal dynamics. The result that we get in this paper is as follows: s uppose that (y) over cap(t, x) is a given solution of the Boussinesq e quation where t is an element of (0, T), x is an element of Omega, Ome ga is a bounded domain with C-infinity-boundary partial derivative Ome ga. Let y(0)(x) be a given initial condition and \\(y) over cap 0, .) - y(0)\\ < epsilon where epsilon = epsilon((y) over cap) is small enou gh. Then there exists boundary control u such that the solution y(t; x ) of the Boussinesq equations satisfying y\((0, T) x partial derivativ e Omega) = u, y\(t=0) = y(0) coincides with (y) over cap(t, x) at the instant T : y(T;x) = (y) over cap(T, x).