Periodic Reporting for period 1 - OUTCROSS (Operator semigroup techniques in the control of delay systems) Reporting period: 2016-11-01 to 2018-10-31 Summary of the context and overall objectives of the project This project addressed applications of operator theory to evolution equations in order to investigate fundamental properties of admissibility and controllability of state-delayed infinite-dimensional dynamical systems originating from such evolution equations.The importance for society of this project stems from the fact that concepts, methods and paradigms of dynamical systems significantly influence research in many sciences and have given rise to vast new areas of applications. A physical, chemical or engineering dynamical system, if assumed continuous in nature, is modelled by differential equations. A general class of models - evolution equations - comprises systems modelled by partial differential equations and by functional differential equations. Evolution equations class embraces models resulting from such diverse fields as aeronautics (e.g. wing vibrations), thermal engineering (e.g. heat conduction) and biology (e.g. cellular dynamics). Motivated by such a wide application area, evolution equations with boundary control and observation stand at the forefront of interdisciplinary subjects and thus are of particular interest to society.The overall objective of the project was to search for an answer to the question of necessary and sufficient conditions for admissibility and controllability of dynamical systems, accounting for weighted forms of admissibility and controllability. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far Initially the work was concentrated on methods used in the case of systems without state delay. The fellow learnt new techniques of analysis of infinite-dimensional systems, including the so-called Hilbert rigged spaces construction. This allowed him to formulate sufficient conditions for the problem of controllability of semilinear systems with a non-Lipschitz disturbance. This was done using the fixed point approach based on measures of noncompactness. Then the work concentrated on analysis of necessary and sufficient conditions of admissibility for contraction semigroups. In particular, the analysis covered the case when a delay is introduced to a contraction semigroup forming a retarded system. The approach selected was based on the Weiss conjecture reformulated for state-delayed systems. The achieved results showed that the assumption of contraction of the initial semigroup alone is not enough to draw conclusions on admissibility of a state-delayed system when described by the so-called delayed semigroup. To obtain more concrete results in the admissibility analysis the work focused on systems the structure of which is well known, namely diagonal systems. For diagonal state-delayed systems a set of sufficient conditions for admissibility was found in a form of a specific assumption on the eigenvalues of the generator of the undelayed semigroup. This allowed to draw conclusions on delayed diagonal semigroup and systems governed by it.While working with finite delay time the dependance between the Laplace transform of the delay function and a decomposition of Hardy spaces was taken in into account. This allowed us to formulate necessary and sufficient conditions for finite-time Laplace-Carleson embeddings and their relation to model spaces. This abstract analysis, due to Paley-Wiener theorem relating L2 spaces with Hardy spaces, allowed to draw conclusion on truncated Hankel and Toeplitz operators.The work currently concentrates on controllability analysis of diagonal systems. In particular, on applying various interpolation techniques to obtain results for diagonal state-delayed systems with square-integrable inputs.All results described above, that is generalisation of controllability analysis by means of fixed point theory to semilinear non-Lipschitz systems, analysis of retarded state-delayed systems, analysis of diagonal state-delayed systems and analysis of Laplace-Carleson embeddings for finite-time intervals together with applications to the Hankel/Toeplitz operators were disseminated in the form of scientific articles. All these results were also presented at various international conferences including CDPS Bordeaux 2017, WOTCA Guimarães 2018, INCPAA Erevan 2018; at seminar presentations at the University of Leeds, AGH University of Science and Technology in Cracow, Silesian University of Technology in Gliwice, University of Copenhangen; are scheduled to presented at seminars the University of Wuppertal and University of Oxford. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) 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.