Dynamics for controlled 2-D Boussinesq systems with distributed controls (Q1856797)

The paper studies the long-time behavior of the solutions for optimal control problems of the Boussinesq systems in \(\Omega\times [0, \infty)\), where \(\Omega\) denotes a bounded domain in \(\mathbb{R}^2\). The Boussinesq system has the state \(({\mathbf u}(x,t), \theta(x,t))\) (the velocity field and its temperature) and the distribtited control \(h\) (the density of external heat source). Given the target \(({\mathbf U},\Theta)\) and the desired heat source \(H\), the problem is to minimize the cost functional \({\mathcal I}_\infty({\mathbf u}, \theta,h)\). First proven is the existence of the quasi-optimal solution \((\widetilde u,\widetilde\theta, \widetilde h)\) such that \(\|(\widetilde u(t),\widetilde \theta(t))-({\mathbf U}(t), \Theta(t)) \|_{L^2(\Omega)}\to 0\) as \(t\to\infty\), \((\widetilde u,\widetilde\theta, \widetilde h)\), of course, remaining in the admissible set. Moreover, an exponential convergence is ensured. Based on the property of the quasi-optimal solution, the long-time behavior of all solutions of Boussinesq system is derived. Then the existence of the optimal solution is derived, based on the similar problem with the cost functional \({\mathcal I}_T({\mathbf u},\theta,h)\) in each finite interval \((0,T)\). Finally, it is shown that the optimal solution asymptotically converges to 0 as \(t\to\infty\).