Control, Observation and Identification Problems for the Wave Equation on Metric Graphs

Avdonin, Sergei

doi:10.1016/j.ifacol.2019.08.010

Cited by 11 publications

(6 citation statements)

References 20 publications

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“…In future work, we will study inverse problems of graphs with cycles, in which case both boundary and internal observations appear to be necessary. For a discussion on inverse problems for graphs with cycles see [4] and references therein.…”

Section: Figure 3 ω and Subtree ω Kjmentioning

confidence: 99%

An inverse problem for quantum trees with observations at interior vertices

Avdonin¹,

Edward²

2021

NHM

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In this paper we consider a non-standard dynamical inverse problem for the wave equation on a metric tree graph. We assume that positive masses may be attached to the internal vertices of the graph. Another specific feature of our investigation is that we use only one boundary actuator and one boundary sensor, all other observations being internal. Using the Dirichlet-to-Neumann map (acting from one boundary vertex to one boundary and all internal vertices) we recover the topology and geometry of the graph, the coefficients of the equations and the masses at the vertices.

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Section: Figure 3 ω and Subtree ω Kjmentioning

confidence: 99%

An inverse problem for quantum trees with observations at interior vertices

Avdonin¹,

Edward²

2021

NHM

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“…We especially mention the works of S. Avdonin. A large number of works are devoted to control, observation, identification, and inverse problems of the wave equation on metric graphs (also on star graphs) (see [26][27][28][29]). The boundary control method, the leaf-peeling method, and the distributed-parameter system have been used to investigate problems of controllability, observability, and stability for the wave equation, but the method of generalized functions has not been used.…”

Section: Introductionmentioning

confidence: 99%

Solution to the Dirichlet Problem of the Wave Equation on a Star Graph

Arepova,

Alexeyeva,

Arepova

2023

Mathematics

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In this paper, the solution to the Dirichlet problem for the wave equation on the star graph is constructed. To begin, we solve the boundary value problem on the interval (on one edge of the graph). We use the generalized functions method to obtain the wave equation with a singular right-hand side. The solution to the Dirichlet problem is determined through the convolution of the fundamental solution with the singular right-hand side of the wave equation. Thus, the solution found on the interval is determined by the initial functions, boundary functions, and their derivatives (the unknown boundary functions). A resolving system of two linear algebraic equations in the space of the Fourier transform in time is constructed to determine the unknown boundary functions. Following inverse Fourier transforms, the solution to the Dirichlet problem of the wave equation on the interval is constructed. After determining all the solutions on all edges and taking the continuity condition and Kirchhoff joint condition into account, we obtain the solution to the wave equation on the star graph.

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“…To reach the exact controllability of systems on graphs with cycles we need to use not only boundary but also interior controls, as was proposed in [3], for the graph, denoted Ω, consisting of a ring with two attached edges, see Figure 1.1. First we prove the shape and velocity controllability using the dynamical method -we reduce these problems to the Volterra integral equations of the second kind.…”

Section: Introductionmentioning

confidence: 99%