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Abstract In this paper, we explore the inverse dynamic problem for the Dirac system on finite metric graphs, including trees and graphs with a cycle. Our primary objective is to reconstruct the graph’s topology (connectivity), determine the lengths of its edges, and identify the matrix potential function on each edge. By using only the dynamic matrix response operator as our inverse data, we adapt the leaf peeling method to recover the unknown data on a tree graph. We then introduce a new approach to reconstruct the unknown data on a graph with a cycle. Additionally, we present a novel dynamic algorithm to address the forward problem for the Dirac system on finite metric graphs. 

ABSTRACT The question of what conditions should be set at the nodes of a discrete graph for the wave equation with discrete time is investigated. The variational method for the derivation of these conditions is used. A parallel with the continuous case is also drawn. As an example, the problem of shape controllability from the boundary is studied. 

Abstract A method for successive synthesis of a Weyl matrix (or Dirichlet-to-Neumann map) of an arbitrary quantum tree is proposed. It allows one, starting from one boundary edge, to compute the Weyl matrix of a whole quantum graph by adding on new edges and solving elementary systems of linear algebraic equations in each step. 

The inverse problem of recovery of a potential on a quantum tree graph from the Weyl matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm–Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two‐spectrum inverse problems on separate edges and reduces them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests. 

In this paper, we consider initial boundary value problems and control problems for the wave equation on finite metric graphs with Dirichlet boundary controls. We propose new constructive algorithms for solving initial boundary value problems on general graphs and boundary control problems on tree graphs. We demonstrate that the wave equation on a tree is exactly controllable if and only if controls are applied at all or all but one of the boundary vertices. We find the minimal controllability time and prove that our result is optimal in the general case. The proofs for the shape and velocity controllability are purely dynamical, while the proof for the full controllability utilizes both dynamical and moment method approaches.