Inverse problem for a differential operator on a star-shaped graph with nonlocal matching condition

Abstract

In this paper, we develop two approaches to investigation of inverse spectral problems for a new class of nonlocal operators on metric graphs. The Laplace differential operator is considered on a star-shaped graph with nonlocal integral matching condition. This operator is adjoint to the functional-differential operator with frozen argument at the central vertex of the graph. We study the inverse problem that consists in the recovery of the integral condition coefficients from the eigenvalues. We obtain the spectrum characterization, reconstruction algorithms, and prove the uniqueness of the inverse problem solution.

Appendix

In the Appendix, we describe some physical processes modeled by the boundary value problems L and $L^*$. The majority of the known mathematical models with integral BCs are constructed for differential operators on intervals. Here, we illustrate the idea of generalizing such models to the case of the star-shaped graph.

Firstly, we describe the model of a thermostat, following the ideas of [12, 58]. Although in [12, 58] nonlinear equations were studied, we consider the simplest linear heat equation

$$\begin{aligned} \frac{\partial u}{\partial t} = a^2 \frac{\partial ^2 u}{\partial x^2} , \quad t > 0, \quad x \in (0, 1), \end{aligned}$$

where $u = u(x, t)$ is a temperature of a heated bar at the point x at the time t, $a > 0$ is a given constant. The BCs

$$\begin{aligned} \frac{\partial u(x, t)}{\partial x}\bigg |_{x = 0} - h u(0, t) = 0, \quad \frac{\partial u(x, t)}{\partial x} \bigg |_{x = 1} + p u(\eta , t) = 0, \quad h, p \in {\mathbb {R}}, \quad \eta \in (0, 1], \end{aligned}$$

correspond to the heat exchange with the zero temperature environment at $x = 0$ and a controller at $x = 1$ adding or removing heat dependent on the temperature detected by a sensor at $x = \eta$. If there are several sensors at the points $\eta _1$, $\eta _2$, ..., $\eta _k$, then the right-hand BC takes the form

$$\begin{aligned} \frac{\partial u(x, t)}{\partial x}\bigg |_{x = 1} + \sum _{j = 1}^k p_j u(\eta _j, t) = 0. \end{aligned}$$

Furthermore, if the sensors collect information on the temperature from the whole bar, then we obtain the integral condition

$$\begin{aligned} \frac{\partial u(x, t)}{\partial x}\bigg |_{x = 1} + \int _0^1 p(x) u(x, t) \, dx = 0. \end{aligned}$$

Now consider m bars of equal length $\pi$ connected all together as a star-shaped graph. The boundary value problem

$$\begin{aligned}&\frac{\partial u_j}{\partial t} = a^2 \frac{\partial ^2 u_j}{\partial x_j^2} , \quad t > 0, \quad x_j \in (0, \pi ), \quad j = \overline{1, m}, \nonumber \\&\frac{\partial u_j(x_j, t)}{\partial x_j}\bigg |_{x_j = 0} - h_j u_j(0, t) = 0, \quad j = \overline{1, m}, \end{aligned}$$

(A.1)

$$\begin{aligned}&u_1(\pi , t) = u_j(\pi , t), \quad j = \overline{2, m}, \end{aligned}$$

(A.2)

$$\begin{aligned}&\sum _{j = 1}^m \left( \frac{\partial u_j(x_j, t)}{\partial x_j}\bigg |_{x_j = \pi } + \int _0^{\pi } p_j(x_j) u_j(x_j, t) \, dx_j \right) = 0 \end{aligned}$$

(A.3)

models the heating process in this structure with a controller at the central vertex, corresponding to $x_j = \pi$. The controller adds or removes heat depending on the temperature on the whole graph.

Similarly, a star-shaped graph of vibrating strings connected all together at the central vertex can be considered. Suppose that a controller is located at the central vertex. If the influence of the controller depends on the displacements of the whole graph, then such model is described by the equations

$$\begin{aligned} \frac{\partial ^2 u_j}{\partial t^2} = a^2 \frac{\partial ^2 u_j}{\partial x_j^2} , \quad t > 0, \quad x_j \in (0, \pi ), \quad j = \overline{1, m}, \end{aligned}$$

with the conditions (A.1)–(A.3). Here $u_j(x_j, t)$ is the lateral displacement of the j-th string at the point $x_j$ at the time t, (A.2) is the continuity condition at the central vertex, and the condition (A.3) describes the balance of forces at this vertex.

Obviously, the separation of variables in the both boundary value problems implies the eigenvalue problem L of form (1)–(4).

On the other hand, some physical models with frozen argument can be constructed by using the ideas of [7, 38]. The eigenvalue problem $L^*$ can be obtained by separating the variables in the boundary value problem for the following parabolic (hyperbolic) equations:

$$\begin{aligned} \frac{\partial ^{\nu } u_j}{\partial t^{\nu }} = a^2 \frac{\partial ^2 u_j}{\partial x_j^2} - \overline{p_j(x_j)} u_j(\pi , t) \quad \nu = 1 \, (\nu = 2), \quad t > 0, \quad x_j \in (0, \pi ), \quad j = \overline{1, m}, \end{aligned}$$

(A.4)

with the conditions (A.1)–(A.2) and

$$\begin{aligned} \sum _{j = 1}^m \frac{\partial u_j(x_j, t)}{\partial x_j}\bigg |_{x_j = \pi } = 0. \end{aligned}$$

(A.5)

In the parabolic case, the system (A.4), (A.1), (A.2), (A.5) models the heat conduction in a star-shaped structure of rods possessing a sensor at the central vertex and an external distributed heat source. This source is described by the functions

$$\begin{aligned} f_j(x_j, t) = -\overline{p_j(x_j)} u_j(\pi , t), \quad j = \overline{1, m}, \end{aligned}$$

(A.6)

on the edges of the graph, that is, the source power is proportional to the temperature at the central vertex measured by the sensor. In is shown in [38] that such model can be implemented by electric conductive rods of a constant thermal conductivity but possessing the variable electrical resistance independent of the temperature.

In the hyperbolic case, the system (A.4), (A.1), (A.2), (A.5) corresponds to an oscillatory process under a damping external force $f_j(x, t)$ defined by (A.6). This force is proportional to the displacement at the internal point measured by a sensor. In particular, this model is relevant to vibrating wires affected by a magnetic field (see [38] for details). It is worth mentioning that the MC (A.5) corresponds to the law of energy conservation in the parabolic case and to the balance of forces at the central vertex in the hyperbolic case.

Thus, from the physical point of view, there are two principal situations:

• 1. A controller located at one point is influenced by sensors that collect information from the whole structure. Then, the process is modeled by a boundary value problem with integral MCs.

• 2. A system is influenced by a distributed force depending on measurements of a sensor located at one point. Such processes are described by functional-differential equations with frozen argument.

From the mathematical point of view, these two situations correspond to mutually adjoint operators. The inverse spectral problem, which consists in the recovery of the unknown coefficients from the spectrum, corresponds to the construction of such physical systems with desired properties.