Controlled heat equation with random control set and/or stochastic inhomogeneous diffusivity

具有随机控制集和/或随机非均匀扩散率的受控热方程

• 批准号: 471212562
• 负责人: Professor Dr. Ivan Veselic
• 金额: --
• 依托单位: Lehrstuhl IX: Analysis, Mathematische Physik und Dynamische Systeme
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/471212562?language=en
• 关键词: Controlled; heat; equation; random; control

The proposed research project it devoted to studying the null-controllability of the heat equation on bounded and unbounded rectangular domains, including cubes, rectangles, strips, slabs, orthants, halfspaces, and the full of R^d. A key goal is to obtain efficient estimates on the control cost or, equivalently, the observability constant. The new feature of the project is that randomness enters. Three cases how this can happen will be studied.The first one is that the control or observation set (describing the placement of sensors) is randomly perturbed. This includes two very natural scenarios: On one hand the situation that sensors default randomly and hence are missing, on the other, the effect that sensors are randomly displaced from their original positions. The question is how much the random perturbation affects the control cost.The second case is that the control set is by purpose randomly generated. This can be modelled, for instance, by a union of unit balls centered around points of a Poisson process. Here the goal is to describe the resulting probability distribution of the (random) control cost. A key feature of this approach is that it can be used to reduce the number of degrees of freedom in the optimization problem. In the case of balls centered at Poissonian points only one parameter in the optimization problem is left, namely the intensity of the process.In the third part of the project we consider inhomogeneous media which are modelled by divergence type operators with stochastic coefficients. Here we want to analyze how this stochastic inhomogeneity affects the control cost.

本文研究了有界和无界矩形区域上热方程的零能控性，包括立方体、矩形、条、板、正交体、半空间和满R^d区域。一个关键的目标是获得有效的估计控制成本，或等价地，可观性常数。该项目的新特点是随机性进入。我们将研究三种情况：第一种是控制或观测集（描述传感器的位置）被随机扰动。这包括两个非常自然的场景：一方面是传感器随机默认并因此丢失的情况，另一方面是传感器从其原始位置随机移位的效果。问题是随机扰动对控制成本的影响有多大。第二种情况是控制集是有意随机生成的。例如，这可以通过以泊松过程的点为中心的单位球的并集来建模。这里的目标是描述（随机）控制成本的结果概率分布。这种方法的一个关键特征是它可以用来减少优化问题中的自由度。在球的泊松点为中心的情况下，只有一个参数的优化问题是左，即强度的process.In该项目的第三部分，我们考虑非均匀介质的随机系数的发散型运营商建模。在这里，我们要分析这种随机不均匀性如何影响控制成本。