LOCAL EXACT CONTROLLABILITY OF THE NAVIER-STOKES EQUATIONS

Let Omega subset of subset of R(n) and (u) over cap(t, x) be a given s olution of the equation partial derivative(t)u(t, x) + A(u) = f(t, x), t is an element of (0, T), x is an element of Omega. The equation is called exact controllable from boundary if for any initial condition u (0)(x) belonging to the E-neighbourhood of the point (u) over cap(0, . ) (epsilon = epsilon((u) over cap)) there exists such boundary control alpha that a solution u of the equation supplied with u\((0,T)x parti al derivative Omega) = alpha, u\(t=0) = u(0) satisfies the condition u (T, x) = (u) over cap(T, x). The local exact boundary controllability of the 2-D and 3-D Navier-Stokes as well as Boussinesq equations is es tablished in this paper. For 2-D Navier-Stokes equations the same prop erty is established also in the case when the control is defined on a boundary's arbitrary subset. For 2-D Euler (resp. Navier-Stokes) equat ions, global exact(resp. approximate) controllability has been shown ( with Navier Slip boundary conditions for Navier-Stokes) by Coron [5] ( resp. [7]).