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Abstract

In this paper, we consider the control and identification problem for systems described by partial differential equations on graphs. Our aim is to recover the potentials and identification source for a parabolic equation on graphs. The method of boundary control which was proposed by St.-Petersburg mathematicians in the late 80-ies of the twentieth century is the basic method. The
method is based on the connection between inverse problems (identification) and controllability of dynamical systems. A special role is played by the Leaf Peeling Method developed by S.A. Avdonin. The implementation of this research is carried out using methods and results from different areas of mathematics: mathematical analysis, functional analysis, special sections of the theory of differential equations, optimal control theory, theory of trigonometric Fourier series, computational mathematics, numerical analysis, and the spectral theory of differential operators including operators on metric graphs. A new approach to solving control and inverse problems for differential equations on arbitrary graphs is proposed. The following results were obtained during the research: a new approach to solving control problems and inverse problems for differential equations with memory on graphs is developed; new results on controllability and identification of potentials and sources for differential equations on graphs are obtained; algorithms for solving control problems and inverse problems for differential equations with memory on an interval, star graph and trees have been developed; numerical tests to solve dynamic problems on graphs are performed. The results of this study are applicable both theoretically - the development of research in the theory of differential equations with memory on graphs, and in biological processes, in particular neurobiology, nanotechnology, in the chemical and petroleum industries.