发表机构
School of Mathematics, Sichuan University; School of Mathematics, Southwest Jiaotong University（四川大学数学学院; 西南交通大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明具有空间解析噪声系数的随机抛物方程在平坦环面上可由单一漂移控制实现空虚可控性，并给出近似可控性，方法基于解析能量估计与小性传播。
AI 中文摘要
我们考虑在$n$维平坦环面$\nmathbb{T}^n=(\nmathbb{R}/(2\pi\mathbb{Z}))^n$（$n\in\mathbb{N}$）上的随机抛物方程，其唯一的控制作用于漂移项。乘性噪声系数是适应的，并且可能同时依赖于样本点、时间和空间。其空间轮廓被假定为实解析的，且具有一致的正解析半径，而相应的解析范数仅需在时间上平方可积，并沿样本路径一致有界。我们证明了一个仅涉及状态的可观测性不等式，其观测范数为时间上的$L^1$均方范数，并由此推导出从任意正测度的可测空间集通过一个适应的漂移控制实现空虚可控性。同一单次插值估计还给出了对任意平方可积随机终端目标的近似可控性。证明基于对伴随状态的直接解析能量估计，随后进行小性传播和 telescoping 论证。
英文摘要
We consider a stochastic parabolic equation on the $n$-dimensional flat torus $\mathbb{T}^n=(\mathbb{R}/(2π\mathbb Z))^n$, $n\in\mathbb N$, whose only control acts in the drift. The multiplicative-noise coefficient is adapted and may depend jointly on the sample point, time, and space. Its spatial profiles are assumed to be real analytic with a uniform positive radius of analyticity, while the corresponding analytic norm is only required to be square integrable in time, uniformly along sample paths. We prove a state-only observability inequality with an $L^1$-in-time mean-square observation norm and deduce null controllability from every measurable spatial set of positive measure by one adapted drift control. The same one-time interpolation estimate also yields approximate controllability to arbitrary square-integrable random terminal targets. The proof is based on a direct analytic energy estimate for the adjoint state, followed by propagation of smallness and a telescoping argument.
