论文标题
随机非线性抛物线方程的统计零可控性
Statistical null-controllability of stochastic nonlinear parabolic equations
论文作者
论文摘要
在本文中，我们考虑前向随机非线性抛物线方程，并在漂移期限内定位对照。在合适的假设下，我们证明了具有截短的非线性的小型全局无效控制性。我们还证明了真实系统的统计本地零控制性。证明依赖于对随机热方程无效控制性成本的精确估计以及对随机设置的源术语方法的适应。主要的难度来自于固定点参数中非线性的估计，这是由于缺乏定期（概率）的功能空间，在这些功能空间中，随机抛物线方程良好。这个主要问题是通过截断程序解决的。作为我们结果所涵盖的相关示例，让我们提及一维情况下的随机汉堡方程，并在三维环境中提及艾伦·卡恩方程。
In this paper, we consider forward stochastic nonlinear parabolic equations, with a control localized in the drift term. Under suitable assumptions, we prove the small-time global null-controllability, with a truncated nonlinearity. We also prove the statistical local null-controllability of the true system. The proof relies on a precise estimation of the cost of null-controllability of the stochastic heat equation and on an adaptation of the source term method to the stochastic setting. The main difficulty comes from the estimation of the nonlinearity in the fixed point argument due to the lack of regularity (in probability) of the functional spaces where stochastic parabolic equations are well-posed. This main issue is tackled through a truncation procedure. As relevant examples that are covered by our results, let us mention the stochastic Burgers equation in the one dimensional case and the Allen-Cahn equation up to the three-dimensional setting.
