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

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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.