**1. Introduction**

This paper concerns inverse problems for differential equations on quantum graphs. Under quantum graphs or differential equation networks (DENs) we understand differential operators on geometric graphs coupled by certain vertex matching conditions. Network-like structures play a fundamental role in many problems of science and engineering. The range for the applications of DENs is enormous. Here is a list of a few.

–*Structural Health Monitoring.* DENs, classically, arise in the study of stability, health, and oscillations of flexible structures that are made of strings, beams, cables, and struts. Analysis of these networks involve DENs associated with heat, wave, or beam equations whose parameters inform the state of the structure, see, e.g., [1].

–*Water, Electricity, Gas, and Traffic Networks.* An important example of DENs is the Saint-Venant system of equations, which model hydraulic networks for water supply and irrigation, see, e.g., [2]. Other important examples of DENs include the telegrapher equation for modeling electric networks, see, e.g., [3], the isothermal Euler equations for describing the gas flow through pipelines, see, e.g., [4], and the Aw-Rascle equations for describing road traffic dynamics, see e.g., [5].

–*Nanoelectronics and Quantum Computing.* Mesoscopic quasi-one-dimensional structures such as quantum, atomic, and molecular wires are the subject of extensive experimental and theoretical studies, see, e.g., [6], the collection of papers in [7–9]. The simplest model describing conduction in quantum wires is the Schrödinger operator on a planar graph. For similar models appear in nanoelectronics, high-temperature superconductors, quantum computing, and studies of quantum chaos, see, e.g., [10–12].

–*Material Science.* DENs arise in analyzing hierarchical materials like ceramic and metallic foams, percolation networks, carbon and graphene nano-tubes, and graphene ribbons, see, e.g., [13–15].

–*Biology.* Challenging problems involving ordinary and partial differential equations on graphs arise in signal propagation in dendritic trees, particle dispersal in respiratory systems, species persistence, and biochemical diffusion in delta river systems, see, e.g., [16–18].

Quantum graph theory gives rise to numerous challenging problems related to many areas of mathematics from combinatoric graph theory to PDE and spectral theories. A number of surveys and collections of papers on quantum graphs appeared in previous years; we refer to the monograph by Berkolaiko and Kuchment, [19], for a complete reference list. The inverse theory of network-like structures is an important part of a rapidly developing area of applied mathematics—analysis on graphs. It is tremendously important for all aforementioned applications. In this paper, we solve a non-standard dynamical inverse problem for the wave equation on a metric tree graph.

Let Ω = {*V*, *E*} be a finite compact and connected metric tree (i.e., graph without cycles), where *V* is a set of vertices and *E* is a set of edges. We recall that a graph is called a *metric graph* if every edge *ej* ∈ *E*, *j* = 1, ... , *N*, is identified with an interval (*a*2*j*−1, *a*2*j*) of the real line with a positive length *lj*. We denote the boundary vertices (i.e., vertices of degree one) by Γ = {*γ*0, ..., *γm*}, and interior vertices (whose degree is at least 2) by {*vm*+1, ...., *vN*}. The vertices can be regarded as equivalence classes of the edge end points *aj*. For each vertex *vk*, denote its degree by Υ*k*. We write *j* ∈ *J*(*v*) if *ej* ∈ *E*(*v*), where *E*(*v*) is the set of edges incident to *v*.

The graph <sup>Ω</sup> determines naturally the Hilbert space of square integrable functions H = *<sup>L</sup>*2(Ω). We define its subspace H<sup>1</sup> as the space of functions *<sup>y</sup>* on <sup>Ω</sup> such that *<sup>y</sup>*|*<sup>e</sup>* ∈ *<sup>H</sup>*1(*e*) for every *<sup>e</sup>* ∈ *<sup>E</sup>* and *<sup>y</sup>*|Γ\{*γ*0} <sup>=</sup> 0, and let <sup>H</sup>−<sup>1</sup> be the dual space to <sup>H</sup>1. When convenient, we denote the restriction of a function *w* on Ω to *ej* by *wj*. For any vertex *vk* and function *w*(*x*) on the graph, we denote by *∂wj*(*vk*) the derivative of *wj* at *vk* in the direction pointing away from the vertex.

Our system is described by the following initial boundary value problem (IBVP) with so-called delta-prime compatibility conditions at each internal vertex *vk*:

$$
\mu\_{tt} - \mu\_{xx} + q\mu \quad = \quad 0, \ (\mathbf{x}, t) \in (\Omega \backslash V) \times [0, T], \tag{1}
$$

$$u|\_{t=0} = u\_t|\_{t=0} = \begin{array}{ll} \end{array} \tag{2}$$

$$
\partial u\_i(v\_k, t) \quad = \quad \partial u\_j(v\_k, t), \; i, j \in J(v\_k), \; v\_k \in V \; \backslash \; \Gamma, \; t \in [0, T], \tag{3}
$$

$$\sum\_{j:j \in I(v\_k)} u\_j(v\_k, t) \quad = \quad 0, \ v\_k \in V \; \backslash \; \Gamma, \; t \in [0, T] , \tag{4}$$

$$\frac{\partial \mu}{\partial \mathbf{x}}(\gamma\_0, t) \quad = \quad f(t), \ t \in [0, T]. \tag{5}$$

$$\mu(\gamma\_k, t) \quad = \quad 0, \ k = 1, \ldots, m, \ t \in [0, T]. \tag{6}$$

Here, *<sup>T</sup>* is arbitrary positive number, *qj* <sup>∈</sup> *<sup>C</sup>*([*a*2*j*−1, *<sup>a</sup>*2*j*]) for all *<sup>j</sup>*, and *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(0, *<sup>T</sup>*). The physical interpretation of conditions (3) and (4), and some other matching conditions was discussed in [20].

The well-posedness of this system is discussed in Section 2; it will be proved that *<sup>u</sup>* ∈ *<sup>C</sup>*([0, *<sup>T</sup>*]; H1) ∩ *<sup>C</sup>*1([0, *<sup>T</sup>*]; H). In what follows, we refer to *<sup>γ</sup>*<sup>0</sup> as the root of <sup>Ω</sup> and *<sup>f</sup>* as the control.

We now pose our inverse problem. Assume an observer knows the boundary condition (5), and that (6) holds at the other boundary vertices, and that the graph is a tree. The unknowns are the number of boundary vertices and interior vertices, the adjacency relations for this tree, i.e., for each pair of vertices, whether or not there is an edge joining them, the lengths {*<sup>j</sup>*}, and the function *q*. We wish to determine these quantities with a set of measurements that we describe now. We can suppose *vN* is the interior vertex adjacent to *γ*<sup>0</sup> with *e*<sup>1</sup> the edge joining the two, see Figure 1. Our first measurement is then the following measurement at *γ*0:

$$(\mathcal{R}\_{0,1}f)(t) := u\_1^f(\gamma\_{0\prime}t). \tag{7}$$

We show that from operator *R*0,1 one can recover -<sup>1</sup> and the degree Υ*<sup>N</sup>* of *vN*. Then by a well known argument, see [21], one can then determine *q*1.

**Figure 1.** A metric tree.

Having established these quantities, in our second step, we propose to place sensors on the edges incident to *vN*, and using these measurements together with *R*0,1 to determine the data associated to these edges. Note that the one control remains at *γ*0. The goal is to repeat these steps until all data associated to the graph have been determined. To define the interior measurements we require more notation. For each interior vertex *vk* we list the incident edges by {*ek*,*<sup>j</sup>* : *j* = 1, ..., Υ*k*}. Here *ek*,1 is chosen to be the edge lying on the unique path from *γ*<sup>0</sup> to *vk*, and the remaining edges are labeled randomly, see Figure 2. Then the sensors measure

$$\left( \left( R\_{k,j} f \right) (t) := u\_j^f \left( v\_k, t \right), \ k = m+1, \ldots, N, \ j = 2, \ldots, \mathbb{Y}\_k - 1. \tag{8}$$

We show that we do not need sensors at *ek*,1,*ek*,Υ*<sup>k</sup>* . Thus the total number of sensors is 1 + ∑*<sup>N</sup> <sup>j</sup>*=*m*+1(Υ*<sup>j</sup>* − <sup>2</sup>). It is easy to check that this number is equal to |Γ| − 1. We denote by *<sup>R</sup><sup>T</sup>* the (|Γ| − 1)-tuple (*R*0,1, *RN*,2, *RN*,3, ....) acting on *L*2(0, *T*).

**Figure 2.** Sensors at vertex *vk* marked by arrows.

#### **2. Results**

Let be equal to the maximum distance between *γ*<sup>0</sup> and any other boundary vertex. Our main result is the following

**Theorem 1.** *Assume qj* ∈ *C*([*a*2*j*−1, *a*2*j*]) *for all j*. *Suppose T* > 2-*. Then from R<sup>T</sup> one can determine the number of interior and boundary vertices, the adjacency relations of the tree, q, and the lengths of the edges.*

#### **3. Discussion**

We now compare this result to others in the literature. We are unaware of any works treating the inverse problem on general tree graphs with delta-prime conditions on the internal vertices. The most common conditions for internal vertices are continuity together with Kirchhoff–Neumann condition: <sup>∑</sup>*j*∈*J*(*vk* ) *<sup>∂</sup>uj*(*vk*, *<sup>t</sup>*) = 0 and all references in this paragraph assume these conditions. In [21], the authors assume that controls and measurements take place at all boundary vertices but one. The authors use an iterative method called "leaf peeling", where the response operator on Ω is used first to determine the data on the edges adjacent to the boundary, and then to determine the response operator associated to a proper subgraph. In [21], the leaf peeling argument includes spectral methods that require knowing *R<sup>T</sup>* for all *T*. In [22], the methods of [21] are extended to the case where masses are placed at internal vertices, see also [23]; however these methods still require knowledge of *R<sup>T</sup>* for all *T*. Also in [22], it is

proven that that for a single string of length with *N* attached masses and *T* > 2-, *R<sup>T</sup>* 0,1 is sufficient to solve the inverse problem. In particular, [23] uses a spectral variant of the boundary control method, together with the relationship between the response operator and the connecting operator. In [24,25], a dynamical leaf peeling argument is developed for a tree with no masses and with response operators at all but one boundary points, allowing for the solution of the inverse problem for finite *T* sufficiently large. An important ingredient in their leaf peeling is determining the response operators associated with subtrees, called "reduced response operators", from the response operator associated to the original tree. In all of these papers, it is assumed that there are no interior measurements. In [26], the iterative methods from [24,25,27] are adapted to a tree with masses placed at internal vertices, with a single control at the roof and measurements there and at internal vertices. For other works on quantum graphs, see [1,16,19,28–31].

A special feature of the present paper is that we use only one control together internal observations. This may be useful in some physical settings where some or most boundary points are inaccessible. Another potential advantage of the method presented here is that we recover all parameters of the graphs, including its topology, from the (|Γ| − <sup>1</sup>)-tuple response operator acting on *<sup>L</sup>*2(0, *<sup>T</sup>*). In previous papers, the authors recovered the graph topology from a larger number of measurements: the (|Γ| − 1) × (|Γ| − 1) matrix (boundary) response operator or, equivalently, from (|Γ| − 1) × (|Γ| − 1) Titchmarsh–Weyl matrix function. In [32], the inverse problems on a star graph for the wave equation with general self-adjoint matching conditions was solved by the (|Γ| − 1) × (|Γ| − 1) matrix boundary response operator.

## **4. Materials and Methods**
