论文标题
稳定的库拉莫托 - 苏瓦辛斯基系统的控制不敏感的控制问题
Insensitizing control problems for the stabilized Kuramoto-Sivashinsky system
论文作者
论文摘要
在这项工作中，我们解决了对四阶和二阶抛物线方程的非线性耦合系统的不敏感控制的存在，称为稳定的库拉莫托 - sivashinsky模型。主要思想是寻找控件，使状态的某些功能（所谓的前哨）对初始数据的扰动局部不敏感。由于基础模型是耦合的，因此我们将考虑一个哨兵，在该哨兵中，我们可以在局部观察集中观察系统的一个或两个组成部分。通过某些经典参数，对于级联系统的级联系统，可以将不敏感的问题简化为零控制性。一旦线性化，通过卡尔曼估计值研究了该新系统的无效控制性，但是与标量模型的其他不敏感问题，Carleman工具的选举和整体控制策略取决于哨兵的初始选择，这是由于扩展系统中（缺乏）耦合而导致的。最后，通过应用局部反转定理获得了扩展（非线性）系统的局部零可控性（因此，不敏感的属性）。
In this work, we address the existence of insensitizing controls for a nonlinear coupled system of fourth- and second-order parabolic equations known as the stabilized Kuramoto-Sivashinsky model. The main idea is to look for controls such that some functional of the state (the so-called sentinel) is locally insensitive to the perturbations of the initial data. Since the underlying model is coupled, we shall consider a sentinel in which we may observe one or two components of the system in a localized observation set. By some classical arguments, the insensitizing problem can be reduced to a null-controllability one for a cascade system where the number of equations is doubled. Upon linearization, the null-controllability for this new system is studied by means of Carleman estimates but unlike other insensitizing problems for scalar models, the election of the Carleman tools and the overall control strategy depends on the initial choice of the sentinel due to the (lack of) couplings arising in the extended system. Finally, the local null-controllability of the extended (nonlinear) system (and thus the insensitizing property) is obtained by applying a local inversion theorem.
