The system is described by the Navier-Stokes equations \[ \partial_t v(t, x) + (v(t, x) \cdot \nabla)v(t, x) = \Delta v(t, x) - \nabla p(t, x) , \quad \nabla \cdot v(t, x) = 0 \] in a bounded simply connected domain \(\Omega \subset R^3\) with \(C^\infty\) boundary \(\Gamma.\) The velocity \(v(t, x)\) is controlled at the boundary through the Dirichlet boundary condition \[ v(t, x) = u(t, x) \quad (t, x) \in (0, T) \times \Gamma , \] where the control \(u(t, x)\) belongs to a boundary space \(W^s((0, T) \times \Gamma)\) (see definition in the paper). The instantaneous values \(v(t, \cdot )\) of the velocity belong to the subspace \(H^3(\Omega)\) of solenoidal vectors of the Sobolev space \(W^3_2(\Omega)^3.\)
The main result in this paper is: if \(T > 0\) and \(v_0(x)\) is an initial velocity in \(H^3(\Omega)\) satisfying \(\|v_0\|_{H^3(\Omega)} \leq \varepsilon\) for sufficiently small \(\varepsilon,\) then there exists a control \(u \in W^{(3/2)}((0, T) \times \Gamma)\) such that \(v(T, x) = 0\) \((x \in \Omega).\) Moreover, the control is tangential to the boundary (that is, \(u(t, x) \cdot n(x) = 0\) for \(x \in \Gamma)\) and \[ \|v(t, \cdot) \|_{H^3(\Omega)} \leq C e^{- k/(T - t)^2} \quad \text{as} \;t \to T \] for suitable \(C, k > 0.\)