Spaces of analytic functions, applied to well-posedness and controllability of linear systems

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The theory of systems and control is a key component of modern engineering and applied mathematics, and includes, for example, problems related to electrical networks, the heat equation, and the evolution of dynamical systems. This project develops tools from pure mathematics, mostly the areas of complex analysis and functional analysis, in order to analyse three fundamental notions:(i) admissibility, needed for the equations governing the system's behaviour to be well-posed in a certain sense;(ii) controllability, a way of making precise the idea that a system can be steered into any desired behaviour by a suitable choice of control; and(iii) observability, the property that measurements of the system yield full information about the state of the system.These properties are studied in a general framework, which enables one to analyse many types of evolving behaviour. In addition to the techniques in complex analysis and functional analysis, which it is planned to develop in directions motivated by the applications above, it is also planned to pursue the idea of using recent results in number theory to obtain further information in this area.