Local exact controllability of Boussinesq equations (Q2719469)

The authors propose a symbolic form of notation for the Boussinesq equations NEWLINE\[NEWLINE \partial_ty(t,x) +A(y) = f(t,x),\quad t=(0,T),\quad x=\Omega,\tag{1} NEWLINE\]NEWLINE where \(\Omega\subset \mathbb{R}^3\), \(y(t,x)\) is the vector field of velocity and temperature, and \(f(t,x)\) is the vector field of the external effects. It is assumed that the solution \(\widehat y(t,x)\) of the Boussinesq equations is known and the initial condition \(y_0(x)\) is close to \(\widehat y(t,x)\) with respect to the corresponding norm. It is proved that there exists the control \(u\) specified on the lateral surface \(\Sigma = (0,T)\times \partial\Omega\) of the cylinder \((0,T)\times\Omega\) such that the solution \(y(t,x)\) of the mixed boundary value problem for the Boussinesq equations with the initial condition \(y|_{t=0} = y_0\) coincides with the given solution \(\widehat y\) at \(t=T\).