AI 中文总结
研究具有振荡系数的耦合双曲系统，通过划分频谱、利用不等式与谱隙分析证明投影状态的一致零可控性，分析ε→0时的极限，并用额外内部反馈控制策略实现全空间可控性。
AI 中文摘要
我们研究了一个耦合双曲系统的控制与均匀化性质，该系统的主项存在快速振荡系数。由于密度参数ε的高频微观结构变化，仅靠边界控制无法实现对所有初始数据的经典一致可控性。为克服这一限制，我们将频谱分为低频和高频分量。利用英厄姆 - 比尔林型不等式结合谱隙分析，证明了投影低频和高频状态的一致零可控性。此外，我们分析了ε趋于0时的结构极限，证明相关边界控制序列弱收敛到基础均匀化系统的零控制。最后，表明通过使用额外的内部反馈控制策略，可实现能量空间中所有初始数据的全空间可控性。
英文摘要
We study the control and homogenization properties of a coupled hyperbolic system characterised by a rapidly oscillating coefficient present in the principal term. Due to the high-frequency microstructural variations of the density parameter $\varepsilon$, classical uniform controllability results across all initial data fail under boundary controls alone. To overcome this limitation, we divide the spectrum into low and high-frequency components. Utilizing Ingham-Beurling-type inequalities combined with spectral gap analysis, we demonstrate the uniform null controllability of the projected low-frequency and high-frequency states. Furthermore, we analyse the structural limits as $\varepsilon \to 0$ and prove that the associated sequence of boundary controls converge weakly to a null control for the underlying homogenized system. Finally, we show that by using an additional interior feedback control strategy, full space controllability can be achieved for all initial data in the energy space.
Comments18 pages
