Motivated by an energy efficient building application, we want to optimize a quadratic cost functional subject to the Boussinesq approximation of the Navier-Stokes equations and to bilateral state and control constraints. Since the computation of such an optimal solution is numerically costly, we design an efficient strategy to compute a sub-optimal (but applicationally acceptable) solution with significantly reduced computational effort. We employ an economic Model Predictive Control (MPC) strategy to obtain a feedback control. The MPC sub-problems are based on a linear-quadratic optimal control problem subjected to mixed control and state constraints and a convection-diffusion equation, reduced with proper orthogonal decomposition. To solve each sub-problem, we apply a primal-dual active set strategy. The method can be fully parallelized, which enables the solution of large problems with real-world parameters.

受节能建筑应用的启发，我们希望根据 Navier-Stokes 方程的 Boussinesq 近似以及双边状态和控制约束来优化二次成本函数。由于这种最优解的计算在数值上是昂贵的，我们设计了一种有效的策略来计算次优（但在应用上可接受）的解，并显着减少计算量。我们采用经济模型预测控制 (MPC) 策略来获得反馈控制。MPC 子问题基于受混合控制和状态约束以及对流扩散方程的线性二次最优控制问题，并通过适当的正交分解进行简化。为了解决每个子问题，我们应用原始对偶活动集策略。该方法可以完全并行化，