AI 中文总结
本文采用Lebeau--Robbiano方法，克服Boussinesq系统耦合结构带来的困难，证明其在控制量减少时的局部零可控性，控制量小时间代价符合预期抛物阶。
AI 中文摘要
本文证明了二维和三维Boussinesq系统在控制量减少时的局部零可控性。更准确地说，控制量作用于温度方程，且仅作用于速度方程的N-2个分量。我们的证明基于Lebeau--Robbiano方法的谱方法思路。主要困难源于系统的耦合结构：速度方程由Stokes算子描述，温度方程由Dirichlet拉普拉斯算子描述，二者的自然谱分解不同，不存在自然适配耦合系统的共同谱局域化，因此无法直接应用常规的分量谱论证。我们证明线性化系统的级联结构可克服该困难，核心是混合可观性估计——仅对Stokes分量进行谱局域化，而对热分量不施加任何频率限制。结合该估计与Stokes高频的耗散，我们可对线性化系统开展Lebeau--Robbiano迭代。通过时间迭代论证，将得到的线性可控性估计推广到非线性Boussinesq系统。作为重要结论，控制量的小时间代价有界为Cexp(C/T)，既恢复了预期的抛物阶，又保留了控制量减少的特性。
英文摘要
In this paper, we prove the local null controllability of the Boussinesq system in dimensions two and three with a reduced number of localized controls. More precisely, the controls act on the temperature equation and on only $N-2$ components of the velocity equation. Our proof is based on a spectral approach in the spirit of the Lebeau--Robbiano method. The main difficulty comes from the coupled structure of the system. The velocity and temperature equations are governed by the Stokes operator and the Dirichlet Laplacian, respectively. Their natural spectral decompositions are different, and there is no common spectral localization naturally adapted to the coupled system. Therefore, the usual componentwise spectral argument cannot be applied directly. We show that the cascade structure of the linearized system makes it possible to overcome this difficulty. The main ingredient is a mixed observability estimate in which only the Stokes component is spectrally localized, while the heat component is treated without any frequency restriction. Combining this estimate with the dissipation of the high Stokes frequencies allows us to carry out a Lebeau--Robbiano iteration for the linearized system. The resulting linear controllability estimate is transferred to the nonlinear Boussinesq system through a time-iteration argument. As an important consequence, the controls have a small-time cost bounded by $C\exp(C/T)$, which recovers the expected parabolic order while preserving the reduced number of controls.
