The progress beyond the state of the art was achieved in many places. In terms of controllability analysis of semilinear systems the application of rigged Hilbert spaces technique together with Schmidt existence theorem allowed us to obtain concrete results without the assumption of Lipschitz type of the nonlinearity disturbing the (otherwise linear) dynamical system. In terms of admissibility of state-delayed systems the obtained results show that the contraction assumption of the undelayed system is not enough, unlike in the undelayed case, to achieve admissibility in the presence of state delays. What is more, it was shown that the behaviour of eigenvalues of the generator of undelayed semigroup must resemble, in some sense, the behaviour of eigenvalues for such generator in a finite-dimensional case. In the case of finite-interval Laplace-Carleson embeddings the necessary and sufficient condition of their boundedness was found and applied to the analysis of truncated Toeplitz operators as well as to finite-time admissibility.
The current work is focused on searching for conditions of exact controllability in the case of state-delayed diagonal system with a scalar input. We expect that the use of Hardy space interpolation techniques and Carleson measures will give sufficient conditions for exact controllability.
The expected impact of this project is in three categories - the fellow, the subject and the wider society. By means of his fellowship in Leeds, the fellow acquired sound knowledge and new research skills in many aspects of modern analysis. This was obtained by working at a forefront research project, under the guidance of a renowned expert and by attending selected graduate level lectures including MAGIC courses (The MAGIC group runs a wide range of postgraduate-level lecture courses in mathematics, using IOCOM's Visimeet Video Conferencing technology). This equipped him with a unique knowledge, establishing him as an expert not only in the area of the project, but also in the field of applications of functional analysis through control theory to engineering problems and with an excellent set of potential collaborators from Europe and abroad. The approach and techniques used in this project as well as its findings will have deep and long-lasting consequences on many fundamental aspects of pure and applied operator theory and control theory. As such, they will also influence most branches of mathematical modelling of dynamical systems and their control-oriented analysis. Although research in this proposal was not primarily oriented towards industrial applications, its results will have a direct impact on the area of modelling and analysis of dynamical systems. As a consequence, in the long term this research project will enhance the ability of EU to become more competitive in such areas as aeronautics, astronautics, acoustics, fluid mechanics, control systems designs etc., where the project results have applications.
