AI 中文总结
本文研究带状态和梯度观测的耦合线性随机抛物系统的不灵敏控制问题，通过将其转化为正-倒向随机抛物系统的零能控问题，利用Carleman估计证明了不同β取值下不灵敏控制的存在性。
AI 中文摘要
我们研究一类耦合线性随机抛物系统的不灵敏控制问题。我们建立了这样的控制的存在性：其中一个哨兵泛函（涉及状态变量及其空间梯度的局域观测）对零初始数据的小扰动不灵敏。我们首先将不灵敏控制问题重新表述为耦合正-倒向随机抛物系统的零能控问题，其中观测项诱导出零阶和二阶耦合项。通过对偶性，分析被简化为对应伴随系统的可观测性不等式。主要的分析贡献是，在控制和观测区域的合适几何假设下，推导了带零阶和二阶耦合项的耦合随机抛物系统的新全局Carleman估计。这些估计给出了所需的可观测性不等式，从而得到不灵敏控制的存在性。此外，根据权重参数β∈[0,1]的值（该参数决定两个状态分量对哨兵泛函的相对贡献），我们考虑两种情况：若β∈{0,1}，哨兵泛函仅依赖一个状态分量，作用于第一个方程漂移项的单个局域控制就足够；相反，若β∈(0,1)，两个状态分量均对哨兵泛函有贡献，作用于两个方程漂移项的两个局域控制就足够。而且，本文的控制策略还包括两个额外的、作用于整个扩散项的控制。
英文摘要
We study insensitizing control problems for a class of coupled linear stochastic parabolic systems. We establish the existence of controls such that a sentinel functional, involving localized observations of the state variables and their spatial gradients, is insensitive to small perturbations of the null initial data. We first reformulate the insensitizing control problem as a null controllability problem for a coupled forward--backward stochastic parabolic system, in which the observation terms induce both zeroth- and second-order coupling terms. By duality, the analysis is reduced to an observability inequality for the corresponding adjoint system. The main analytical contribution is the derivation of new global Carleman estimates for coupled stochastic parabolic systems with zeroth- and second-order coupling terms, under suitable geometric assumptions on the control and observation regions. These estimates yield the required observability inequalities and, consequently, the existence of insensitizing controls. Furthermore, depending on the value of a weighting parameter $β\in[0,1]$, which determines the relative contributions of the two state components to the sentinel functional, we consider two cases. If $β\in\{0,1\}$, the sentinel functional depends on only one state component, and a single localized control acting in the drift of the first equation is sufficient. In contrast, if $β\in(0,1)$, both state components contribute to the sentinel functional, and two localized controls acting in the drift terms of the two equations are sufficient. Moreover, the control strategy in this paper involves two additional controls acting throughout the diffusion terms.
