一类拟线性系统的局部零能控性及相关数值实验
Local null controllability of a quasi-linear system and related numerical experiments
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中文总结 AI 辅助
本文针对扩散系数依赖状态梯度的拟线性抛物系统，通过线性化系统估计与局部逆定理证明其局部零能控性，提出拟牛顿型迭代算法并经数值实验验证。
中文摘要 AI 辅助
本文研究扩散系数依赖于状态变量梯度的拟线性抛物型系统的零控制问题。在主要理论结果中，针对扩散系数的正则性与增长性及正则初始数据给出若干假设，证明局部零能控性成立。为此，考虑线性化系统的零能控问题，推导控制量与状态的新估计，再应用局部逆定理。还构造了一类拟牛顿型迭代算法以计算零控制及对应状态，将该方法应用于问题的若干数值近似，通过多项实验展示结果。
英文摘要
This paper concerns the null control of quasi-linear parabolic systems where the diffusion coefficient depends on the gradient of the state variable. In our main theoretical result, with some assumptions on the regularity and growth of the diffusion coefficient and regular initial data, we prove that local null controllability holds. To this purpose, we consider the null controllability problem for the linearized system, we deduce new estimates on the control and the state and, then, we apply a Local Inversion Theorem. We also formulate an iterative algorithm of the quasi-Newton kind for the computation of a null control and an associated state. We apply this method to some numerical approximations of the problem and illustrate the results with several experiments.
