The aim of this part of the project is to investigate the question of the cost of controllability (and hence also observability) of hyperbolic PDEs in a multiscale setting. Multiscale setting means that the domain of PDE is a region which has the same structure at different scales. A simple example is given in Figure 3. Our motivation is the fact that the complex structures composed of simpler ones are ubiquitous in modern engineering systems, and obtaining new insights in this area of research would have considerable impact on the study of systems appearing in practical applications.
Natural laws are to a large part described by evolution equations. These often take the form of a partial differential equation. Often it is desirable, and indeed possible, to influence the evolution of a system in time. However, for the purpose of bringing the system into one specific, desired state, only limited means of influence are at disposal. Here the relevant question is: Is the partial influence on the system available sufficient to drive the system into a desired final state? This is so-called controllability problem.
Another, related problem is the following one. To understand the laws of nature, scientists rely on observations. However, oftentimes it is only possible to observe merely a part of the system. The question is: to what extent it is possible to identify the true state of the system based on this partial information. This is the so-called observability problem.
These questions are the topic of the control theory of partial differential equations (sometimes called distributed parameter systems), a multidisciplinary discipline with intensive research and large literature in the last 40 years. We refer to the survey paper by Russell and to the SIAM Review article and book by J.-L. Lions for an introduction to the case of systems modelled by means of PDE.
One issue, which is still an important topic of investigation, is the question of the cost of the control. That is the question of how large the size has to be (measured in some appropriate way) of the control needed to reach (a suitably small neighbourhood of) the final state. This problem was analysed by Lebeau in the context of wave equations with analytic coefficients. The important ingredient of these results was the use of Carleman inequalities. Since there is a well-known duality connection between controllability and observability problems, an answer to the former also implies an answer to the later.
Our goal will be to find estimates for the cost of null-controllability for hyperbolic PDEs on multiscale domains and investigate the question of efficient regions of control for a suitable class of hyperbolic PDEs.