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arXiv 2302.06237math.APmath.OC
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右端 Dirichlet 控制下 Korteweg-de Vries 方程的局部能控性

Local controllability of the Korteweg-de Vries equation with the right Dirichlet control

Hoai-Minh Nguyen

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arXiv admin note: text overlap with arXiv:2010.04478
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AI总结：

本文研究右端 Dirichlet 控制 KdV 方程在临界长度下的局部能控性，刻画临界长度与一维不可达空间，并给出有限时间精确能控或任意正时间非零能控的判据。

摘要：

带有右端 Dirichlet 控制的 Korteweg-de Vries（KdV）方程最早在二十多年前得到研究。已有结果表明，该系统在所有非临界长度下均具有小时间局部精确能控性，而其线性化系统在所有临界长度下都不能控。尽管过去二十年间 KdV 系统的能控性已得到广泛研究，但该系统在临界长度下的局部能控性仍是一个开放问题。本文对这一问题给出了确定性回答。首先，我们刻画了线性化系统的所有临界长度及相应的不可达空间；特别地，证明了不可达空间的维数始终为 1。其次，我们证明了带有右端 Dirichlet 控制的 KdV 系统在小时间内不是局部零能控的。第三，针对所有临界长度，我们给出了一个判据，用以判定系统究竟在有限时间内局部精确能控，还是在任意正时间内都不是局部零能控的。由此，我们证明存在某些临界长度，使系统在小时间内不是局部零能控的，但在有限时间内局部精确能控。这些事实令人惊讶，并且与相关已知结果明显不同。第一，已知相应的右端零 Dirichlet KdV 系统借助内部控制可在小时间内局部精确能控。第二，相应的右端 Neumann 控制 KdV 系统，其线性化系统的不可达空间可能具有任意维数。第三，如果相应的不可达空间维数为 1，则右端 Neumann 控制的 KdV 系统在小时间内局部精确能控。

英文摘要：

The Korteweg-de Vries (KdV) equation with the right Dirichlet control was initially investigated more than twenty years ago. It was shown that this system is small time, locally, exactly controllable for all non-critical lengths and its linearized system is not controllable for {\it all} critical lengths. Even though the controllability of the KdV system has been studied extensively in the last two decades, the local controllability of this system for critical lengths remains an open question. In this paper, we give a definitive answer to this question. First, we characterize all critical lengths and the corresponding unreachable space for the linearized system. In particular, we show that the unreachable space is always of dimension 1. Second, we prove that the KdV system with the right Dirichlet control is not locally null controllable in small time. Third, we give a criterion to determine whether the system is locally exactly controllable in finite time or {\it not} locally null controllable in any positive time for {\it all} critical lengths. Consequently, we show that there exist critical lengths such that the system is not locally null controllable in small time but is locally exactly controllable in finite time. These facts are surprising and distinct in comparison with related known results. First, it is known that the corresponding KdV system with the right zero Dirichlet is locally exactly controllable in small time using internal controls. Second, the unreachable space of the linearized system of the corresponding KdV system with the right Neumann control might be of arbitrary dimension. Third, the KdV system with the right Neumann control is locally exactly controllable in small time if the corresponding unreachable space is of dimension 1.

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