Exponential Stabilisation of a Tree-Shaped Network of Strings With Variable Coefficients

Abstract

Abstract. In this paper, we deal with a tree-shaped network of strings with a fixed root node. By imposing velocity feedback controllers on all vertices except the root node, we show that the spectrum of the system operator consists of all isolated eigenvalues of finite multiplicity and is distributed in a strip parallel to the imaginary axis under certain conditions. Moreover, we prove that there exists a sequence of eigenvectors and generalised eigenvectors that forms a Riesz basis with parentheses, and that the imaginary axis is not an asymptote of the spectrum. Thereby, we deduce that the system is exponentially stable.2010 Mathematics Subject Classification. 93C20, 93D15, 35B35, 35P10, 47A70. Introduction.In past decades, there had been extensive literature devoted to research on controllability and stabilisation of one-dimensional (1D) networks. For instance, we refer to earlier works [5,8,9, 10], recent works [1, 2, 7, 18, 20, 22, 23] and the references therein. The main tools used in the literature are the Hilbert uniqueness method, the multiplier approach and the Riesz basis approach. We observe that the models discussed in the papers mentioned above are mainly of uniform structure; there are few results on non-homogeneous structure of 1D networks. Maybe part of the reason for the non-uniform networks is that there is no solution formula such as D'Alembert formula for 1D wave equation, and the investigation of controllability and observability of the networks tends to be very difficult. Therefore, Dager and Zuazua in [7] proposed that, as an open problem, the control problem of the network of 1D strings with variable coefficients is an important subject in future study. In this paper, we will study the stabilisation problem of a tree-shaped network of strings with variable coefficients using the spectral method. Since controllability and stabilisation of the system depend not only on the physical characteristics that support the graphs but also on the structures of the graphs, we suppose that the tree-shaped network is comprised of three strings with one fixed vertex called the root node, and three vertices

Networks of geometrically exact beams: Well-posedness and stabilization

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Energy decay for the damped wave equation on an unbounded network

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