Shape, Velocity, and Exact Controllability for the Wave Equation on a Graph with Cycle

Exact controllability is proven on a graph with cycle. The controls can be a mix of controls applied at the boundary and interior vertices. The method of proof first uses a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated to exact controllability. In the case of a single control, either boundary or interior, it is shown that exact controllability fails

Exact controllability for wave equation on general quantum graphs with non-smooth controls

In this paper we study the exact controllability problem for the wave equation on a finite metric graph with the Kirchhoff-Neumann matching conditions. Among all vertices and edges we choose certain active vertices and edges, and give a constructive proof that the wave equation on the graph is exactly controllable if $H^1(0,T)'$ Neumann controllers are placed at the active vertices and $L^2(0,T)$ Dirichlet controllers are placed at the active edges. The proofs for the shape and velocity controllability are purely dynamical, while the proof for the exact controllability utilizes both dynamical and moment method approaches. The control time for this construction is determined by the chosen orientation and path decomposition of the graph

Exact Controllability for the Wave Equation on a Graph with Cycle and Delta-Prime Vertex Conditions

Exact controllability for the wave equation on a metric graph consisting of a cycle and two attached edges is proven. One boundary and one internal control are used. At the internal vertices, delta-prime conditions are satisfied. As a second example, we examine a tripod controlled at the root and the junction, while the leaves are fixed. These examples are key to understanding controllability properties in general metric graphs

Shape, Velocity, and Exact Controllability for the Wave Equation

A new method to prove exact controllability for the wave equation is demonstrated and discussed on several examples. The method of proof first uses a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated to exact controllability

Source identification for the wave equation on graphs

International audienceWe consider source identification problems for the wave equation on graphs. The main advantage of our approach is its locality. Our algorithm reduces essentially to the resolution of linear integral Volterra equations of the second kind and is new even for an interval

Recovery of a potential on a quantum star graph from Weyl's matrix

The problem of recovery of a potential on a quantum star graph from Weyl's matrix given at a finite number of points is considered. A method for its approximate solution is proposed. It consists in reducing the problem to a two-spectra inverse Sturm-Liouville problem on each edge with its posterior solution. The overall approach is based on Neumann series of Bessel functions (NSBF) representations for solutions of Sturm-Liouville equations, and, in fact, the solution of the inverse problem on the quantum graph reduces to dealing with the NSBF coefficients. The NSBF representations admit estimates for the series remainders which are independent of the real part of the square root of the spectral parameter. This feature makes them especially useful for solving direct and inverse problems requiring calculation of solutions on large intervals in the spectral parameter. Moreover, the first coefficient of the NSBF representation alone is sufficient for the recovery of the potential. The knowledge of the Weyl matrix at a set of points allows one to calculate a number of the NSBF coefficients at the end point of each edge, which leads to approximation of characteristic functions of two Sturm-Liouville problems and allows one to compute the Dirichlet-Dirichlet and Neumann-Dirichlet spectra on each edge. In turn, for solving this two-spectra inverse Sturm-Liouville problem a system of linear algebraic equations is derived for computing the first NSBF coefficient and hence for recovering the potential. The proposed method leads to an efficient numerical algorithm that is illustrated by a number of numerical tests.Comment: arXiv admin note: substantial text overlap with arXiv:2210.1250

Ingham-type inequalities and Riesz bases of divided differences

We study linear combinations of exponentials e iλnt, λn ∈ Λ in the case where the distance between some points λn tends to zero. We suppose that the sequence Λ is a finite union of uniformly discrete sequences. In (Avdonin and Ivanov, 2001), necessary and sufficient conditions were given for the family of divided differences of exponentials to form a Riesz basis in space L 2 (0, T). Here we prove that if the upper uniform density of Λ is less than T/(2π), the family of divided differences can be extended to a Riesz basis in L 2 (0, T) by adjoining to {e iλnt} a suitable collection of exponentials. Likewise, if the lower uniform density is greater than T/(2π), the family of divided differences can be made into a Riesz basis by removing from {e iλnt} a suitable collection of functions e iλnt. Applications of these results to problems of simultaneous control of elastic strings and beams are given

An Inverse Problem for Quantum Trees with Delta-Prime Vertex Conditions

In this paper, we consider a non-standard dynamical inverse problem for the wave equation on a metric tree graph. We assume that the so-called delta-prime matching conditions are satisfied at 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 Neumann-to-Dirichlet map (acting from one boundary vertex to one boundary and all internal vertices) we recover the topology and geometry of the graph together with the coefficients of the equations