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

所提出的研究项目致力于研究热方程在有界和无界矩形区域上的零能控性，包括立方体、矩形、条形、平板、直角、半空间和R^d满区域。一个关键目标是得到控制代价的有效估计，或者等价地，可观测性常数。该项目的新特点是随机性进入。我们将研究三种情况。第一种情况是控制或观察集(描述传感器的放置)被随机扰动。这包括两种非常自然的情况：一方面，传感器随机缺省，因此丢失；另一方面，传感器随机偏离其原始位置的影响。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故意随机产生的。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点是可以用来减少优化问题中的自由度数。在球以Poisson点为中心的情况下，最优化问题中只剩下一个参数，即过程的强度。在项目的第三部分，我们考虑了由带随机系数的散度型算子建模的非均匀介质。在这里，我们想要分析这种随机不均匀如何影响控制成本。

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.