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

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.

At present we use all the standard means for disseminating the results of our research, such as journal articles, conference and seminar presentations, and the World Wide Web (see http://www.maths.leeds.ac.uk/~pmt6jrp/pubsjrp.html). I have had extensive experience of the opportunities for publishing research work in all its forms; in particular, I am a former editor of the Journal of the London Mathematical Society and member of several editorial boards (currently Mathematics of Control, Signals and Systems), and the analysis adviser to the LMS Student Texts series (C.U.P.). My experience of conference organization is also relevant; for example I ran the LMS-funded meeting Modelling and Control of Infinite-dimensional Systems in Leeds in 1998; this was the beginning of a series of IFAC workshops, now running every two years, under the title Control of Distributed Parameter Systems . Both Prof. Jacob and I are on the International Programme Committee for these meetings. A recent meeting very relevant to the themes of this project was the workshop on Operator Theory and its Applications , which was held in Edinburgh (ICMS) in September 2009: I co-organized this with Prof. Pott. With Jacob, Pott and Prof. Zwart of Twente, we are also running an Oberwolfach mini-workshop Well-posedness and controllability of evolution equations in December 2010. Some other meetings which I participated in organizing were the annual meetings in the programme of the European network in Analysis and Operators. A future meeting in this area, for which Jacob is the main organizer, is the 7th workshop on the control of distributed parameter systems, scheduled for Wuppertal in July 2011. We can thus be confident that we will find opportunities of presenting the results of this project to an interested audience, on this and on later occasions. The School of Mathematics at Leeds has recently opened a Research Visitors' Centre, as part of a new initiative to attract visitors and encourage collaborative research. It is expected that this will also enable us to broaden the impact of our work outside the limits of our own institutions. Although this project has applied mathematical components, we cannot claim an immediate impact in industrial applications; however, the University of Leeds is always on the lookout for consultancy opportunities through Consulting Leeds , a company that manages consultancy projects on behalf of its academics and external partners. In addition, having spent four years as a postdoc in the Cambridge University Engineering Department, I have many contacts among academic engineers in this country, for example Profs. K. Glover and M. C. Smith in Cambridge, Prof. Q.-C. Zhong (moving soon to Loughborough), and Prof. D. Limebeer (now in Oxford). Thus if the work turns out to have more direct industrial applications, there are opportunities for exploiting these.