*4.1. Preliminaries*

In what follows, we use the notations

$$\mathcal{F}^n = \{ f \in \mathcal{H}^n(\mathbb{R}) \, : \, f(t) = 0 \text{ if } t \le 0 \}, \tag{9}$$

where <sup>H</sup>*n*(R) are the standard Sobolev spaces. We define the Heaviside function by *<sup>H</sup>*(*t*) = 1 for *<sup>t</sup>* <sup>&</sup>gt; 0, and *<sup>H</sup>*(*t*) = 0 for *<sup>t</sup>* < 0. Then, we define *Hn* ∈ F*<sup>n</sup>* as the unique solution to

$$\frac{d^n}{dt^n} H\_n = H\_{\tilde{r}}$$

at times we use *H*−1(*t*), resp. *H*−2(*t*) for *δ*(*t*), resp. *δ* (*t*). Here *δ*(*t*) denotes the Dirac delta function supported at *t* = 0. In this section and those that follow, we drop the superscript *T* from *R<sup>T</sup>* when convenient.

Consider a star shaped graph with edges *e*1, ...,*eN*. For each *j*, we identify *ej* with the interval (0, *<sup>j</sup>*) and the central vertex with *x* = 0, see Figure 3.

**Figure 3.** Star with coordinate system: *ej* identified with [0, *j*].

Recall the notation *qj* = *q*|*ej* , and *uj*(∗, *t*) = *u*(∗, *t*)|*ej* . Thus, we consider the system

$$\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} + qu \quad = \quad 0, \; x \in \mathfrak{e}\_{j\prime} \; j = 1, \dots, N, \; t \in \times [0, T]. \tag{10}$$

$$
\mu|\_{t=0} = \mu\_t|\_{t=0} = \quad 0,\tag{11}
$$

$$
\partial u\_i(0, t) \quad = \quad \partial u\_j(0, t), \; i \neq j, \; t \in [0, T]. \tag{12}
$$

$$\sum\_{j=1}^{N} \mu\_j(0, t) \quad = \quad 0, \ t \in [0, T], \tag{13}$$

$$\exists \theta u\_1(\ell\_1, t) \; \; \; \; \; \; f(t), \; t \in [0, T] ; \; \; \tag{14}$$

$$u\_j(\ell\_j, t) \quad = \ 0, \ j = 2, \ldots, N, \ t \in [0, T]. \tag{15}$$

Let *u<sup>f</sup>* solve (10)–(15), and set

$$\mathcal{G}\_{\hat{j}}(t) = u\_{\hat{j}}^{f}(0, t), \; j = 1, \dots, N. \tag{16}$$

For (11), it is standard that the waves have unit speed of propagation on the interval, so *gj*(*t*) = 0 for *t* < -<sup>1</sup> and all *j*. It will be useful first to consider the vibrating string on an interval.

#### *4.2. Representation of Solution on an Interval and Reduced Response Operator*

We adapt a representation of *uf*(*x*, *t*) developed in [27], where only Dirichlet control and boundary conditions were considered. Fix *j* ∈ {1, ..., *N*}. We extend *qj* to (0, ∞) as follows: first evenly with respect to *x* = *<sup>j</sup>*, and then periodically. Thus *qj*(2*k<sup>j</sup>* ± *x*) = *qj*(*x*) for all positive integers *k*.

Define *wj* to be the solution to the Goursat-type problem

$$\begin{cases} \frac{\partial w^2}{\partial s^2}(\mathbf{x}, \mathbf{s}) - \frac{\partial w^2}{\partial \mathbf{x}^2}(\mathbf{x}, \mathbf{s}) + q\_j(\mathbf{x})w(\mathbf{x}, \mathbf{s}) = 0, & 0 < \mathbf{x} < \mathbf{s} < \mathbf{s}, \\\ w\_x(0, s) = 0, \ w(\mathbf{x}, \mathbf{x}) = -\frac{1}{2} \int\_0^\mathbf{x} q\_j(\eta) d\eta, \ \mathbf{x} > 0. \end{cases}$$

A proof of solvability of this problem can be found in [33]. Consider the IBVP on the interval (0, *j*):

$$
\tilde{u}\_{tt} - \tilde{u}\_{xx} + q\_j(\mathbf{x})\tilde{u}\_{\;\;\;\;t} = \; \; 0, \; 0 < \mathbf{x} < \ell\_j, \; t \in (0, T), \tag{17}
$$

$$
\overline{u}(\mathbf{x},0) = \overline{u}\_l(\mathbf{x},0) \ = \ \ 0 \ \ 0 < \mathbf{x} < \ell\_{j\prime} \tag{18}
$$

$$\begin{array}{rcl}\partial\vec{u}(0,t)&=&p(t),\end{array} \tag{19}$$

$$\|\ell(\ell\_j, t)\| \quad = \quad 0, \ t > 0. \tag{20}$$

Let *<sup>P</sup>*(*t*) = <sup>−</sup> *<sup>t</sup>* <sup>0</sup> *p*(*s*)*ds*. Then, the solution to (17)–(20) on *ej* can be written as

$$\Pi\_{j}^{p}(\mathbf{x},t) = \sum\_{\substack{n \geq 0 \ 0 \leq 2n\ell\_{j} + x \leq t}} (-1)^{n} \left( P(t - 2n\ell\_{j} - \mathbf{x}) + \int\_{2n\ell\_{j} + \mathbf{x}}^{t} w\_{j}(2n\ell\_{j} + \mathbf{x}, \mathbf{s}) P(t - s) d\mathbf{s} \right)$$

$$+ \sum\_{\substack{n \geq 1 \ 0 \leq 2n\ell\_{j} - x \leq t}} (-1)^{n} \left( P(t - 2n\ell\_{j} + \mathbf{x}) + \int\_{2n\ell\_{j} - \mathbf{x}}^{t} w\_{j}(2n\ell\_{j} - \mathbf{x}, \mathbf{s}) P(t - s) d\mathbf{s} \right). \tag{21}$$

In what follows, we only consider *t* ≤ *T* for some finite *T*, so all sums will be finite.

Let us now change the condition (20) to *u*˜*x*(*<sup>j</sup>*, *t*) = 0. In this case, the solution becomes

$$\begin{split} \|\boldsymbol{p}\_{j}^{p}(\mathbf{x},t)\| &= \sum\_{\substack{n\geq 0:\ 0\leq 2n\ell\_{j}+x\leq t \\ n\geq 1:\ 0\leq 2n\ell\_{j}-x\leq t}} \left( P(t-2n\ell\_{j}-\mathbf{x}) + \int\_{2n\ell\_{j}+\mathbf{x}}^{t} w\_{j}(2n\ell\_{j}+\mathbf{x},\mathbf{s})P(t-s)ds \right) \\ &+ \sum\_{\substack{n\geq 1:\ 0\leq 2n\ell\_{j}-x\leq t \\ n\geq 1:\ 0\leq 2n\ell\_{j}-x\leq t}} \left( P(t-2n\ell\_{j}+\mathbf{x}) + \int\_{2n\ell\_{j}-\mathbf{x}}^{t} w\_{j}(2n\ell\_{j}-\mathbf{x},\mathbf{s})P(t-s)ds \right). \end{split}$$

To represent the solution of the wave equation on the edge *e*<sup>1</sup> in a star graph, we must account for the control at *x* = -1. Thus it will also be useful to represent the solution of a wave equation on an interval when the control is on the right end. Consider the IBVP:

$$v\_{tt} - v\_{xx} + q\_1(\mathbf{x})v = 0, \ 0 < \mathbf{x} < \ell\_1, \ t > 0,$$

$$v(\mathbf{x}, 0) = v\_t(\mathbf{x}, 0) \ = \ 0, \ 0 < \mathbf{x} < \ell\_1.$$

$$\begin{aligned} \partial v(0, t) &= 0, \\ \partial v(\ell\_1, t) &= \ f(t), \ t > 0. \end{aligned} \tag{22}$$

Set *q*˜1(*x*) = *q*1(-<sup>1</sup> − *x*), and extend *q*˜1 to [0, ∞) by *q*˜1(2*k*-<sup>1</sup> ± *x*) = *q*˜1(*x*). Define *k*<sup>1</sup> to be the solution to the Goursat-type problem

$$\begin{cases} & \frac{\partial k^2}{\partial s^2}(\mathbf{x}, \mathbf{s}) - \frac{\partial k^2}{\partial \mathbf{x}^2}(\mathbf{x}, \mathbf{s}) + \tilde{q}\_1(\mathbf{x})k(\mathbf{x}, \mathbf{s}) = \mathbf{0}, & 0 < \mathbf{x} < \mathbf{s}, \\\ k\_\mathbf{x}(\mathbf{0}, \mathbf{s}) = \mathbf{0}, & k(\mathbf{x}, \mathbf{x}) = -\frac{1}{2} \int\_0^\mathbf{x} \tilde{q}\_1(\eta) d\eta, \; \mathbf{x} < \ell\_j. \end{cases}$$

Let *<sup>F</sup>*(*t*) = <sup>−</sup> *<sup>t</sup>* <sup>0</sup> *f*(*s*)*ds*. One can then verify that

$$\begin{aligned} \mathbf{w}^f(\mathbf{x}, t) &= \quad F(t - \ell\_1 + \mathbf{x}) + \int\_{\ell\_1 - \mathbf{x}}^t k\_1(\ell\_1 - \mathbf{x}, \mathbf{s}) F(t - \mathbf{s}) \, \mathrm{d}s \\ &+ F(t - \ell\_1 - \mathbf{x}) + \int\_{\ell\_1 + \mathbf{x}}^t k\_1(\ell\_1 + \mathbf{x}, \mathbf{s}) F(t - \mathbf{s}) \, \mathrm{d}s \\ &+ F(t - 3\ell\_1 + \mathbf{x}) + \int\_{3\ell\_1 - \mathbf{x}}^t k\_1(3\ell\_1 - \mathbf{x}, \mathbf{s}) F(t - \mathbf{s}) \, \mathrm{d}s \\ &+ F(t - 3\ell\_1 - \mathbf{x}) + \int\_{3\ell\_1 + \mathbf{x}}^t k\_1(3\ell\_1 + \mathbf{x}, \mathbf{s}) F(t - \mathbf{s}) \, \mathrm{d}s \\ &\dots \end{aligned} \tag{23}$$

Thus

$$\mathbb{P}\left(\mathbf{0},t\right) = 2\sum\_{n=1}^{\infty} \left(\mathbb{P}(t-(2n-1)\ell\_1) + \int\_{(2n-1)\ell\_1}^{t} k\_1((2n-1)\ell\_1,s)\mathbb{P}(t-s)\right). \tag{24}$$

We now show that the system (10)–(15) is well-posed. Recall that F<sup>1</sup> was defined in (9), and *gj*(*t*) = *<sup>u</sup><sup>f</sup> <sup>j</sup>* (0, *t*).
