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

本课题主要研究有界和无界矩形域上热方程的零可控性，包括立方体、矩形、条状、平板、正交体、半空间和满R^d。一个关键的目标是获得控制成本的有效估计，或者等价地，可观测常数。这个项目的新特点是随机性的加入。将研究三种情况下这种情况是如何发生的。第一个是控制或观察集（描述传感器的位置）是随机扰动的。这包括两种非常自然的情况：一方面是传感器随机默认，因此丢失的情况，另一方面是传感器从原始位置随机移动的效果。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点是它可以用来减少优化问题中的自由度。在以泊松点为中心的球的情况下，优化问题只剩下一个参数，即过程的强度。在项目的第三部分，我们考虑非均匀介质，它是由散度型算子与随机系数建模。这里我们想分析这种随机非均匀性是如何影响控制成本的。

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.