In this paper, we are concerned with the internal control of a class of one-dimensional nonlinear parabolic systems with nonlocal and weakly degenerate diffusion coefficients. Our main theorem establishes a local null controllability result with only one internal control for a system of two equations. The proof, based on the ideias developed by Fursikov and Imanuvilov, is obtained from the global null controllability of the linearized system provided by Lyusternik's Inverse Mapping Theorem. This work extends the results previously treated by the authors for just one equation. For the system, the main issue is to obtain similar results with just one internal control, which requires a new Carleman estimate with the local term just depending on one of the state function.

中文翻译：

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具有非局部项和一个控制力的耦合退化系统的局部零可控性

在本文中，我们关注的是一类具有非局部和弱退化的扩散系数的一维非线性抛物系统的内部控制。我们的主要定理使用两个方程组的一个内部控制建立了局部零可控制性结果。该证明基于Fursikov和Imanuvilov提出的思想，是从Lyusternik的逆映射定理提供的线性化系统的全局零可控性中获得的。这项工作扩展了作者先前仅针对一个方程式处理的结果。对于该系统，主要问题是仅通过一个内部控制就可以获得类似的结果，这就需要使用局部项仅取决于状态函数之一的新的Carleman估计。