In this paper, we study the exact controllability and stabilization of a system of two wave equations coupled by velocities with an internal, local control acting on only one equation. We distinguish two cases. In the first one, when the waves propagate at the same speed: using a frequency domain approach combined with multiplier technique, we prove that the system is exponentially stable when the coupling region is a subset of the damping region and satises the geometric control condition GCC (see Definition 3.1 below). Following a result of Haraux [10], we establish the main indirect observability inequality. This results leads, by the HUM method, to prove that the total system is exactly controllable by means of locally distributed control. In the second case, when the waves propagate at different speed, we establish an exponential decay rate in the weak energy space under appropriate geometric conditions. Consequently, the system is exactly controllable using a result of [10].

中文翻译：

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局部耦合波动方程的精确可控制性和稳定性：理论结果

在本文中，我们研究由速度耦合的两个波动方程组的精确可控性和稳定性，而内部波动控制仅作用于一个方程。我们区分两种情况。在第一个中，当波以相同的速度传播时：使用频域方法与乘数技术相结合，我们证明了当耦合区域是阻尼区域的子集且满足几何控制条件时，系统是指数稳定的GCC（请参阅下面的定义3.1）。根据Haraux [10]的结果，我们建立了主要的间接可观测性不等式。结果表明，通过HUM方法可以证明整个系统可以通过本地分布式控制进行精确控制。在第二种情况下，当波以不同的速度传播时，我们在适当的几何条件下在弱能量空间中建立了指数衰减率。因此，使用结果[10]可以精确控制系统。