where $\tau\geq\tau^{0}$ such that if $\chi\in S(1,g_{\mathsf{T}})$,
homogeneous of degree $0$ in $(\sigma,\xi^{\prime},\tau)$ with
$\operatorname{supp}(\chi)\subset\mathscr{U}$, there exist $C>0$ and
$\tau_{0}>0$ such that

${|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)v)|}_{3,1/2,\tilde{\tau}}\leq
C\Big{(}\sum\limits_{j=1}^{2}{|B_{j,\varphi}v_{|x_{N}=0^{+}}|}_{7/2-k_{j},\tilde{\tau}}+{\|Q_{\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tilde{\tau}}+{|\operatorname{tr}(v)|}_{3,-1/2,\tilde{\tau}}\Big{)},$

for $\tau\geq\tau_{0}$ and $v\in\overline{}\mathscr{C}^{\infty}_{c}(W_{+})$.

###### Proof.

We choose a conic open neighborhood $\mathscr{U}_{0}$ of $\varrho^{0\prime}$
according to Proposition 3.24 and such that $\mathscr{U}_{0}\subset
W\times\mathbb{R}\times\mathbb{R}^{d-1}\times(0,+\infty)\times[1,+\infty)\times(0,1]$
. Assume moreover that $\tau\geq\tau_{0}$ in $\mathscr{U}_{0}$.

In Section 3.10 we introduced the symbols
$e_{j,\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})$,
$j=1,\dots,m^{\prime}=m^{-}+2=6-m^{+}$. For a conic set $\mathscr{W}$ denote
$\mathbb{S}_{\mathscr{W}}=\\{\varrho^{\prime}=(s,x,\sigma,\xi^{\prime},\tau,\gamma,\varepsilon)\in\mathscr{W};\
|(\sigma,\xi^{\prime},\tilde{\tau})|=1\\}$.

Consequence of the Lopatinskiĭ-Šapiro condition holding at
$\varrho^{0\prime}$, by Lemma 3.26 for all
$\varrho^{\prime}\in\mathbb{S}_{\mathscr{U}_{0}}$ there exists $C>0$ such that

$\displaystyle\sum_{j=1}^{m^{\prime}}\big{|}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})\big{|}^{2}\geq
C|\mathbf{z}|_{\mathbb{C}^{4}}^{2},\qquad\mathbf{z}=(z_{0},\dots,z_{3})\in\mathbb{C}^{4}.$

Let $\mathscr{U}_{1}$ be a second conic open neighborhood of
$\varrho^{0\prime}$ such that
$\overline{\mathscr{U}_{1}}\subset\mathscr{U}_{0}$. Since
$\mathbb{S}_{\overline{\mathscr{U}_{1}}}$ is compact (recall that $W$ is
bounded), there exists $C_{0}>0$ such that

$\displaystyle\sum_{j=1}^{m^{\prime}}\big{|}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})\big{|}^{2}\geq
C_{0}|\mathbf{z}|_{\mathbb{C}^{4}}^{2},\qquad\mathbf{z}=(z_{0},\dots,z_{3})\in\mathbb{C}^{4},\
\varrho^{\prime}\in\mathbb{S}_{\overline{\mathscr{U}_{1}}}.$

Introducing the map
$N_{t}\varrho^{\prime}=(s,x,t\sigma,t\xi^{\prime},t\tilde{\tau})$, for
$\varrho^{\prime}=(s,x,\xi^{\prime},\sigma,\tau,\gamma,\varepsilon)$ with
$t=\lambda_{\mbox{\tiny${\mathsf{T}}$},\tilde{\tau}}^{-1}$ one has

$\displaystyle\sum_{j=1}^{m^{\prime}}\big{|}\Sigma_{e_{j,\varrho^{0\prime}}}(N_{t}\varrho^{\prime},\mathbf{z})\big{|}^{2}\geq
C_{0}|\mathbf{z}|_{\mathbb{C}^{4}}^{2},\qquad\mathbf{z}=(z_{0},\dots,z_{3})\in\mathbb{C}^{4},\
\varrho^{\prime}\in\mathscr{U}_{1},$ (4.1)

since $N_{t}\varrho^{\prime}\in\mathbb{S}_{\overline{\mathscr{U}_{1}}}$. Now,
for $j=1,2$ one has

$\displaystyle\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})=\sum_{\ell=0}^{k_{j}}e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})z_{\ell},$

with $e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})$ homogeneous of degree
$k_{j}-\ell$, and for $3\leq j\leq m^{\prime}$ one has

$\displaystyle\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})=\sum_{\ell=0}^{3}e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})z_{\ell},$

with $e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})$ homogeneous of degree
$m^{+}+j-\ell-3$. We define $\mathbf{z}^{\prime}\in\mathbb{C}^{4}$ by
$z^{\prime}_{\ell}=t^{\ell-7/2}z_{\ell}$, $\ell=0,\dots,3$. One has

$\displaystyle\Sigma_{e_{j,\varrho^{0\prime}}}(N_{t}\varrho^{\prime},\mathbf{z}^{\prime})=t^{k_{j}-7/2}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z}),\qquad
j=1,2,$

and

$\displaystyle\Sigma_{e_{j,\varrho^{0\prime}}}(N_{t}\varrho^{\prime},\mathbf{z}^{\prime})=t^{m^{+}+j-13/2}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z}),\qquad
j=3,\dots,m^{\prime}.$

Thus, from (4.1) we deduce

$\displaystyle\sum_{j=1}^{2}\lambda_{\mbox{\tiny${\mathsf{T}}$},\tilde{\tau}}^{2(7/2-k_{j})}\big{|}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})\big{|}^{2}+\sum_{j=3}^{m^{\prime}}\lambda_{\mbox{\tiny${\mathsf{T}}$},\tilde{\tau}}^{2(13/2-m^{+}-j)}\big{|}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})\big{|}^{2}$
$\displaystyle\geq
C_{0}\sum_{\ell=0}^{3}\lambda_{\mbox{\tiny${\mathsf{T}}$},\tilde{\tau}}^{2(7/2-\ell)}|z_{\ell}|^{2},$
(4.2)

for $\mathbf{z}=(z_{0},\dots,z_{3})\in\mathbb{C}^{4}$, and
$\varrho^{\prime}\in\mathscr{U}_{1}$, since
$t=\lambda_{\mbox{\tiny${\mathsf{T}}$},\tilde{\tau}}^{-1}.$

We now choose $\mathscr{U}$ a conic open neighborhood of $\varrho^{0\prime}$,
such that $\overline{\mathscr{U}}\subset\mathscr{U}_{1}$. Let $\chi\in
S(1,g_{\mathsf{T}})$ be as in the statement and let $\tilde{\chi}\in
S(1,g_{\mathsf{T}})$ be homogeneous of degree $0$, with
$\operatorname{supp}(\tilde{\chi})\subset\mathscr{U}_{1}$ and
$\tilde{\chi}\equiv 1$ in a neighborhood of $\overline{\mathscr{U}}$, and thus
in a neighborhood of $\operatorname{supp}(\chi)$.

For $j=3,\dots,m^{\prime}$ one has
$e_{j,\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})=\kappa^{+}_{\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})\xi_{d}^{j-3}\in
S_{\tilde{\tau}}^{m^{+}+j-3,0}$. Set
$E_{j}=\mathrm{Op}(\tilde{\chi}e_{j,\varrho^{0\prime}})$. The introduction of
$\tilde{\chi}$ is made such that $\tilde{\chi}e_{j,\varrho^{0\prime}}$ is
defined on the whole tangential phase-space. Observe that

$\displaystyle\mathscr{B}(w)$
$\displaystyle=\sum\limits_{j=1}^{2}{|B_{j,\varphi}w_{|x_{N}=0^{+}}|}_{7/2-k_{j},\tilde{\tau}}^{2}+\sum\limits_{j=3}^{m^{\prime}}{|E_{j}w_{|x_{N}=0^{+}}|}_{13/2-m^{+}-j,\tilde{\tau}}^{2}$
$\displaystyle=\sum\limits_{j=1}^{2}{|\Lambda_{\mbox{\tiny${\mathsf{T}}$},\tilde{\tau}}^{7/2-k_{j}}B_{j,\varphi}w_{|x_{N}=0^{+}}|}_{\partial}^{2}+\sum\limits_{j=3}^{m^{\prime}}{|\Lambda_{\mbox{\tiny${\mathsf{T}}$},\tilde{\tau}}^{13/2-m^{+}-j}E_{j}w_{|x_{N}=0^{+}}|}_{\partial}^{2}$

is a boundary quadratic form of type $(3,1/2)$ as in Definition 2.6. From
Proposition 2.8 (Gårding inequality for a boundary differential form of type
$(m-1,r)$) and inequality (4.1) we have

$\displaystyle{|\operatorname{tr}(u)|}_{3,1/2,\tilde{\tau}}^{2}$
$\displaystyle\lesssim\sum\limits_{j=1}^{2}{|B_{j,\varphi}u_{|x_{N}=0^{+}}|}_{7/2-k_{j},\tilde{\tau}}^{2}+\sum\limits_{j=3}^{m^{\prime}}{|E_{j}u_{|x_{N}=0^{+}}|}_{13/2-m^{+}-j,\tilde{\tau}}^{2}+{|\operatorname{tr}(v)|}_{3,-M,\tilde{\tau}}^{2}.$
(4.3)

for $u=\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)v$ and $\tau\geq\tau_{0}$
chosen sufficiently large.

In $\mathscr{U}_{1}$ one can write

$\displaystyle
q_{\varphi}=q_{\varphi}^{+}q_{\varphi}^{-}=\kappa^{+}_{\varrho^{0\prime}}\kappa^{-}_{\varrho^{0\prime}},$

with $\kappa^{+}_{\varrho^{0\prime}}$ of degree $m^{+}$ and
$\kappa^{-}_{\varrho^{0\prime}}$ of degree $m^{-}$. In fact we set

$\displaystyle\tilde{\kappa}^{+}_{\varrho^{0\prime}}(\varrho^{\prime})=\prod_{j\in\mathscr{E}^{+}}\big{(}\xi_{d}-\tilde{\chi}\pi_{j}(\varrho^{\prime})\big{)},\qquad\tilde{\kappa}^{-}_{\varrho^{0\prime}}(\varrho^{\prime})=\prod_{j\in\mathscr{E}^{-}}\big{(}\xi_{d}-\tilde{\chi}\pi_{j}(\varrho^{\prime})\big{)},$

with the notation of Section 3.9, thus making the two symbols defined on the
whole tangential phase-space. In $\mathscr{U}$, one has also

$\displaystyle
q_{\varphi}=\tilde{\kappa}^{+}_{\varrho^{0\prime}}\tilde{\kappa}^{-}_{\varrho^{0\prime}}.$

The factor $\tilde{\kappa}^{-}_{\varrho^{0\prime}}$ is associated with roots
with negative imaginary part. With Lemma 2.9 given in Section 2.1 one has the
following microlocal elliptic estimate

${\|\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)w\|}_{m^{-},\tilde{\tau}}+{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)w)|}_{m^{-}-1,1/2,\tilde{\tau}}\lesssim{\|\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{-}_{\varrho^{0\prime}})\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)w\|}_{+}+{\|w\|}_{m^{-},-M,\tilde{\tau}},$

for $w\in\overline{\mathscr{S}}(\mathbb{R}^{N}_{+})$ and $\tau\geq\tau_{0}$
chosen sufficiently large. We apply this inequality to
$w=\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})v$.
Since

$\displaystyle\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{-}_{\varrho^{0\prime}})\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})=\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)Q_{\varphi}\mod\Psi_{\tilde{\tau}}^{4,-1},$

one obtains

$\displaystyle{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})v)|}_{m^{-}-1,1/2,\tilde{\tau}}\lesssim{\|Q_{\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tilde{\tau}}.$

With
$[\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi),\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})]\in\Psi_{\tilde{\tau}}^{m^{+},-1}$
one then has

$\displaystyle{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})u)|}_{m^{-}-1,1/2,\tilde{\tau}}\lesssim{\|Q_{\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tilde{\tau}}+{|\operatorname{tr}(v)|}_{3,-1/2,\tilde{\tau}},$

with $u=\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)v$ as above, using that
$m^{+}+m^{-}=4$. Note that

$\displaystyle{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})u)|}_{m^{-}-1,1/2,\tilde{\tau}}$
$\displaystyle\qquad\asymp\sum_{j=0}^{m^{-}-1}{|D_{d}^{j}\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})u_{|x_{N}=0^{+}}|}_{m^{-}-j-1/2,\tilde{\tau}}$
$\displaystyle\qquad\gtrsim\sum_{j=3}^{m^{\prime}}{|E_{j}u_{|x_{N}=0^{+}}|}_{5/2+m^{-}-j,\tilde{\tau}}-{|\operatorname{tr}(v)_{|x_{N}=0^{+}}|}_{3,-1/2,\tilde{\tau}},$

using that
$\xi_{d}^{j}\tilde{\kappa}^{+}_{\varrho^{0\prime}}=\tilde{\chi}e_{j+3,\varrho^{0\prime}}$
in a conic neighborhood of $\operatorname{supp}(\chi)$ and using that
$m^{-}=m^{\prime}-2$. We thus obtain

$\displaystyle\sum_{j=3}^{m^{\prime}}{|E_{j}u_{|x_{N}=0^{+}}|}_{13/2-m^{+}-j,\tilde{\tau}}\lesssim{\|Q_{\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tilde{\tau}}+{|\operatorname{tr}(v)|}_{3,-1/2,\tilde{\tau}},$

since $13/2-m^{+}=5/2+m^{-}$. With (4.3) then one finds

$\displaystyle{|\operatorname{tr}(u)|}_{3,1/2,\tilde{\tau}}$
$\displaystyle\lesssim\sum\limits_{j=1}^{2}{|B_{j,\varphi}u_{|x_{N}=0^{+}}|}_{7/2-k_{j},\tilde{\tau}}^{2}+{\|Q_{\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tilde{\tau}}+{|\operatorname{tr}(v)|}_{3,-1/2,\tilde{\tau}}.$

In addition, observing that

$\displaystyle{|B_{j,\varphi}u_{|x_{N}=0^{+}}|}_{7/2-k_{j},\tilde{\tau}}\lesssim{|B_{j,\varphi}v_{|x_{N}=0^{+}}|}_{7/2-k_{j},\tilde{\tau}}+{|\operatorname{tr}(v)|}_{3,-1/2,\tilde{\tau}},$

the result of Proposition 4.2 follows. ∎

###  4.2 Proof of Proposition 4.1

As mentioned above the proof relies on a patching procedure of microlocal
estimates given by Proposition 4.2.

We set

$\displaystyle\Gamma^{d}=\\{(\sigma,\xi^{\prime},\tau,\gamma,\varepsilon)\in\mathbb{R}\times\mathbb{R}^{d-1}\times(0,+\infty)\times(0,1];\
\tau\geq\tau_{0},\gamma\geq\gamma_{0}\\},$

and

$\displaystyle\mathbb{S}^{d}=\\{(\sigma,\xi^{\prime},\tau,\gamma,\varepsilon)\in\Gamma^{d};\
|(\sigma,\xi^{\prime},\tilde{\tau})|=1\\}.$

Consider
$(\sigma^{0},\xi^{0\prime},\tau^{0},\gamma^{0},\varepsilon^{0})\in\mathbb{S}^{d}$.
Since the Lopatinskiĭ-Šapiro condition holds at
$\varrho^{0\prime}=(\sigma^{0},x^{0},\xi^{0\prime},\tau^{0},\gamma^{0},\varepsilon^{0})$,
we can invoke Proposition 4.2:

  1. 1.

There exists a conic open neighborhood $\mathscr{U}_{\varrho^{0\prime}}$ of
$\varrho^{0\prime}$ in
$W\times\mathbb{R}^{d-1}\times(0,+\infty)\times[1,+\infty)\times(0,1]$ where
$\tau\geq\tau_{0},\gamma\geq\gamma_{0}$;

  2. 2.

For any $\chi_{\varrho^{0\prime}}\in S(1,g_{\mathsf{T}})$ homogeneous of
degree $0$ supported in $\mathscr{U}_{\varrho^{0\prime}}$ the estimate of
Proposition 4.2 applies to
$\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi_{\varrho^{0\prime}})v$ for
$\tau\geq\tau_{0},\gamma\geq\gamma_{0}$.

Without any loss of generality we may choose $\mathscr{U}_{\varrho^{0\prime}}$
of the form
$\mathscr{U}_{\varrho^{0\prime}}=\mathscr{O}_{\varrho^{0\prime}}\times\Gamma_{\varrho^{0\prime}}$,
with
$\mathscr{O}_{\varrho^{0\prime}}\subset W$ an open neighborhood of
$(s^{0},x^{0})$ and $\Gamma_{\varrho^{0\prime}}$ a conic open neighborhood of
$(\sigma^{0},\xi^{0\prime},\tau^{0},\gamma^{0},\varepsilon^{0})$ in
$\mathbb{R}\times\mathbb{R}^{d-1}\times(0,+\infty)\times[1,+\infty)\times(0,1]$
where $\tau\geq\tau^{0},\gamma\geq\gamma_{0}$.

Since $\\{(s^{0},x^{0})\\}\times\mathbb{S}^{d}$ is compact we can extract a
finite covering of it by open sets of the form of
$\mathscr{U}_{\varrho^{0\prime}}$. We denote by $\tilde{\mathscr{U}}_{i}$,
$i\in I$ with $|I|<\infty$, such a finite covering. This is also a finite
covering of $\\{(s^{0},x^{0})\\}\times\Gamma^{d}$.

Each $\tilde{\mathscr{U}}_{i}$ has the form
$\tilde{\mathscr{U}}_{i}=\mathscr{O}_{i}\times\Gamma_{i}$, with
$\mathscr{O}_{i}$ an open neighborhood of $(s^{0},x^{0})$ and $\Gamma_{i}$ a
conic open set in $\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$ where
$\tau\geq\tau^{0}$. We set $\mathscr{O}=\cap_{i\in I}\mathscr{O}_{i}$ and
$\mathscr{U}_{i}=\mathscr{O}\times\Gamma_{i}$, $i\in I$.

Let $W^{0}$ be an open neighborhood of $(s^{0},x^{0})$ such that
$W^{0}\Subset\mathscr{O}$. The open sets $\mathscr{U}_{i}$ give also an open
covering of $\overline{W^{0}}\times\mathbb{S}^{d}$ and
$\overline{W^{0}}\times\Gamma^{d}$. With this second covering we associate a
partition of unity $\chi_{i}$, $i\in I$, of
$\overline{W^{0}}\times\mathbb{S}^{d}$, where each $\chi_{i}$ is chosen smooth
and homogeneous of degree one for $|(\sigma,\xi^{\prime},\tilde{\tau},)|\geq
1$, that is:

$\displaystyle\sum\limits_{i\in I}\chi_{i}(\varrho^{\prime})=1$

for $\varrho^{\prime}=(s,x,\sigma,\xi^{\prime},\tau,\gamma,\varepsilon)$ in a
neighborhood of $\overline{W^{0}}\times\Gamma^{d}$, and
$|(\sigma,\xi^{\prime},\tilde{\tau})|\geq 1$.

Let $u\in\overline{}\mathscr{C}^{\infty}_{c}(W^{0}_{+})$. Since each
$\chi_{i}$ is in $S(1,g_{\mathsf{T}})$ and supported in $\mathscr{U}_{i}$,
Proposition 4.2 applies:

${|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi_{i})v)|}_{3,1/2,\tilde{\tau}}\leq
C_{i}\Big{(}\sum\limits_{j=1}^{2}{|B_{j,\varphi}v_{|x_{N}=0^{+}}|}_{7/2-k_{j},\tilde{\tau}}+{\|Q_{\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tilde{\tau}}+{|\operatorname{tr}(v)|}_{3,-1/2,\tilde{\tau}}\Big{)},$
(4.4)

for some $C_{i}>0$, for $\tau\geq\max\limits_{i}\tau_{i}$ for some
$\tau_{i}>0,\gamma\geq 1$.

We set $\tilde{\chi}=1-\sum\limits_{i\in I}\chi_{i}$. One has
$\tilde{\chi}\in\cap_{M\in\mathbb{N}}S(\lambda_{\mbox{\tiny${\mathsf{T}}$},\tilde{\tau}}^{-M},g_{\mathsf{T}})$
microlocally in a neighborhood of $\overline{W^{0}}\times\Gamma^{d}$. Thus,
considering the definition of $\Gamma^{d}$, we then have
$\tilde{\chi}\in\cap_{M\in\mathbb{N}}S(\lambda_{\mbox{\tiny${\mathsf{T}}$},\tilde{\tau}}^{-M},g_{\mathsf{T}})$
locally in a neighborhood of $\overline{W^{0}}$.

For any $M\in\mathbb{N}$ using that $\operatorname{supp}(v)\subset W^{0}$ one
has

$\displaystyle{|\operatorname{tr}(v)|}_{3,1/2,\tilde{\tau}}$
$\displaystyle\leq\sum\limits_{i\in
I}{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi_{i})v)|}_{3,1/2,\tilde{\tau}}+{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\chi})v)|}_{3,1/2,\tilde{\tau}}$
$\displaystyle\lesssim\sum\limits_{i\in
I}{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi_{i})v)|}_{3,1/2,\tilde{\tau}}+{|\operatorname{tr}(v)|}_{3,-M,\tilde{\tau}}$
$\displaystyle\lesssim\sum\limits_{i\in
I}{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi_{i})v)|}_{3,1/2,\tilde{\tau}}+{\|v\|}_{4,-M,\tilde{\tau}}.$

Summing estimates (4.4) together for $i\in I$ we thus obtain

$\displaystyle{|\operatorname{tr}(v)|}_{3,1/2,\tilde{\tau}}\lesssim\sum\limits_{j=1}^{2}{|B_{j,\varphi}v_{|x_{N}=0^{+}}|}_{7/2-k_{j},\tilde{\tau}}+{\|Q_{\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tilde{\tau}}+{|\operatorname{tr}(v)|}_{3,-1/2,\tilde{\tau}},$

for $\tau\geq\max\limits_{i\in I}\tau_{i}$ for some $\tau_{i}>0$. Therefore,
by choosing $\tau\geq\tau_{0}$ sufficiently large and $\gamma\geq 1$ one
obtains the result of Proposition 4.1. ∎

##  5 Local Carleman estimate for the augmented conjugated operator

We recall that $Q=D^{4}_{s}+\Delta^{2}=Q_{1}Q_{2}$ with
$Q_{k}=(-1)^{k}iD^{2}_{s}+A$ and $A=-\Delta.$ Setting
$Q_{\varphi}=e^{\tau\varphi}Qe^{-\tau\varphi},$

$Q_{\varphi}=P_{1}P_{2}~{}~{}~{}\text{with}~{}~{}~{}P_{k}=e^{\tau\varphi}Q_{k}e^{-\tau\varphi}=(-1)^{k}i(D_{s}+i\tau\partial_{s}\varphi(z))^{2}+A_{\varphi},$
(5.1)

with, in the selected normal geodesic coordinates,

$A_{\varphi}=e^{\tau\varphi}Ae^{-\tau\varphi}=(D_{d}+i\tau\partial_{d}\varphi(z))^{2}+R(x,D_{x}^{\prime}+i\tau
d_{x^{\prime}}\varphi(z)),~{}~{}~{}z=(s,x).$

In [9, Section 4.5], a local Carleman estimate for the augmented conjugated
operator $\mathbb{Q}_{\varphi}$ is obtained by combining microlocal estimates
in three different microlocal regions.

Let $V$ be an open bounded neighborhood of $z_{0}=(s_{0},x_{0})$ in
$\mathbb{R}^{N}.$ We use the following result due to J. Le Rousseau and L.
Robbiano.

######  Proposition 5.1 ([9]).

Let $\varphi(z)=\varphi_{\gamma,\varepsilon}(z)$ be define as in section 2.1.
There exists an open neighborhood $W$ of $z_{0}$ in
$(0,S_{0})\times\mathbb{R}^{d}$, $W\subset V$, and there exist
$\tau_{0}\geq\tau_{*},\gamma\geq 1,\varepsilon_{0}\in(0,1]$ such that

$\gamma{\|\tilde{\tau}^{-1}v\|}_{4,0,\tilde{\tau}}+{|\operatorname{tr}(v)|}_{3,1/2,\tilde{\tau}}\lesssim{\|Q_{\varphi}v\|}_{+}+{|\operatorname{tr}(v)|}_{1,5/2,\tilde{\tau}},$

for $\tau\geq\tau_{0},$ $\gamma\geq\gamma_{0},$
$\varepsilon\geq\varepsilon_{0},$ and $v=w_{|\overline{\mathbb{R}^{N}_{+}}}$
with $w\in\mathscr{C}^{\infty}_{c}(\mathbb{R}^{N})$ and
$\operatorname{supp}(w)\subset W.$

Putting together the estimates of Propositions 4.1 and 5.1, we produce the
following final local estimate for the augmented operator $Q.$

######  Theorem 5.2.

Let $Q=\Delta^{2}+D_{s}^{4}$ and
$(s^{0},x^{0})\in(0,S_{0})\times\partial\Omega$, with $\Omega$ locally given
by $\\{x_{d}>0\\}$ and $S_{0}>0$. Assume that $(Q,B_{1},B_{2},\varphi)$
satisfies the Lopatinskiĭ-Šapiro condition of Definition 3.21 at
$\varrho^{\prime}=(s,x,\sigma,\xi^{\prime},\tau,\gamma,\varepsilon)$ for all
$(\sigma,\xi^{\prime},\tau,\gamma,\varepsilon)\in\mathbb{R}\times\mathbb{R}^{d-1}\times(0,+\infty)\times[1,+\infty)\times(0,1].$
Let $\varphi(z)=\varphi_{\gamma,\varepsilon}(z)$ be define as in section 2.1.
There exists an open neighborhood $W$ of $z_{0}$ in
$(0,S_{0})\times\mathbb{R}^{d}$, $W\subset V$, and there exist
$\tau_{0}\geq\tau_{*},\gamma\geq 1,\varepsilon_{0}\in(0,1]$ such that

$\gamma{\|\tilde{\tau}^{-1}e^{\tau\varphi}u\|}_{4,0,\tilde{\tau}}+{|\operatorname{tr}(e^{\tau\varphi}u)|}_{3,1/2,\tilde{\tau}}\lesssim{\|e^{\tau\varphi}Qu\|}_{+}+\sum\limits_{j=1}^{2}{|e^{\tau\varphi}B_{j}u_{|x_{N}=0^{+}}|}_{7/2-k_{j},\tilde{\tau}},$

for $\tau\geq\tau_{0},$ $\gamma\geq\gamma_{0},$
$\varepsilon\geq\varepsilon_{0},$ and
$u\in\overline{}\mathscr{C}^{\infty}_{c}(W_{+}).$

We recall that the notion of the function space is introduced in (1.4).

Using the notations of Section 2 for semi-classical norms, we can write the
Carleman inequality of Theorem 5.2 as follows:

$\gamma\sum_{|\alpha|\leq
4}\|\tilde{\tau}^{3-|\alpha|}e^{\tau\varphi}D^{\alpha}_{s,x}u\|_{+}+\sum\limits_{r=0}^{3}|e^{\tau\varphi}D^{r}_{x_{N}}u_{|\partial
Z}|_{7/2-r,\tilde{\tau}}\lesssim{\|e^{\tau\varphi}Qu\|}_{+}+\sum\limits_{j=1}^{2}{|e^{\tau\varphi}B_{j}u_{|x_{N}=0^{+}}|}_{7/2-k_{j},\tilde{\tau}}.$

##  6 Spectral inequality and an application

Before we state and prove the spectral inequality, we start by stating an
interpolation-type inequality which follows from the Carleman inequality of
Theorem 5.2. Finally, as an application of such a spectral inequality, we
deduce a null-controllability result of the fourth order parabolic equation
(1.1).

###  6.1 An interpolation inequality

Let $S_{0}>0$ and $\alpha\in(0,S_{0}/2)$. We introduce the cylinder
$Z=(0,S_{0})\times\Omega$ and we define $Y=(\alpha,S_{0}-\alpha)\times\Omega$
for some $\alpha>0.$ We recall that $Q$ denotes the augmented elliptic
operator $Q:=D^{4}_{s}+\Delta^{2}.$
Following the proof the the interpolation inequality of Theorem 5.1 in [9,
Section 5], one can derive the following interpolation inequality.

######  Theorem 6.1.

(interpolation inequality). Let $\mathscr{O}$ be nomempty subset of $\Omega.$
There exist $C>0$ and $\delta\in(0,1)$ such that for $u\in H^{4}(Z)$ that
satisfies

$B_{1}u_{|{x\in\partial\Omega}}=0,~{}~{}~{}~{}B_{2}u_{|{x\in\partial\Omega}}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}s\in(0,S_{0}),$

we have

$\|u\|_{H^{3}(Y)}\leq
C\|u\|^{1-\delta}_{H^{3}(Z)}\left(\|Qu\|_{L^{2}(Z)}+\sum\limits_{j=1}^{2}|B_{j}u_{|s=0}|_{H^{3-k_{j}}(\mathscr{O})}\right)^{\delta},$
(6.1)

where $H^{m}(\Omega)$ denote the classical Sobolev spaces in $\Omega.$

###  6.2 Spectral inequality

Let $\phi_{j}$ and $\mu_{j}$ be eigenfunctions and associated eigenvalues of
the bi-Laplace operator $P=\Delta^{2}$ with the boundary operators $B_{1}$ and
$B_{2}$, which form a Hilbert basis for $L^{2}(\Omega)$ (see section 3.3 ),
with

$\displaystyle 0<\mu_{0}\leq\mu_{1}\leq\cdots\leq\mu_{k}\leq\cdots.$

We now prove the spectral inequality of Theorem 1.3, viz., for some $K>0,$

$\|y\|_{L^{2}(\Omega)}\leq
Ke^{K\mu^{1/4}}\|y\|_{L^{2}(\omega)},~{}~{}\mu>0,~{}~{}y\in\operatorname{span}\\{\phi_{j};~{}\mu_{j}\leq\mu\\}.$
(6.2)

We use the following corollary, whose proof can be adapted to the proof of
Corollary 3.3 in [11, Section 3.1].

######  Corollary 6.2.

Assume that $(P,B_{1},B_{2})$ fulfills the Lopatinskiĭ-Šapiro condition on
$\partial\Omega.$ Then there exists $C>0$ such that

$\|u\|_{H^{4}(\Omega)}\leq
C\left(\|Pu\|_{L^{2}(\Omega)}+|B_{1}u_{|\partial\Omega}|_{H^{7/2-k_{1}}(\partial\Omega)}+|B_{1}u_{|\partial\Omega}|_{H^{7/2-k_{2}}(\partial\Omega)}\right)$

for all $u\in H^{4}(\Omega).$

We recall that $k_{1}$ and $k_{2}$ denote respectively the orders of $B_{1}$
and $B_{2}.$

###### Proof.

Let $\mu>0$ and $y\in\operatorname{span}\\{\phi_{j};~{}\mu_{j}\leq\mu\\},$
meaning that $y=\sum\limits_{\mu_{j}\leq\mu}\alpha_{j}\phi_{j}.$ We set

$u(s,x)=\sum\limits_{\mu_{j}\leq\mu}\alpha_{j}\mu_{j}^{-3/4}f(s\mu_{j}^{1/4})\phi_{j}(x).$

Here $\alpha_{j}\in\mathbb{C}$ for $j=0,\dots,k$ with $k\in\mathbb{N},$ where
$f(s)=\beta\sin(\beta s)\cosh(\beta s)-\beta\cos(\beta s)\sinh(\beta s)$ with
$\beta=\sqrt{2}/2.$ Straightforward computation shows that
$D_{s}^{4}f=\partial^{4}_{s}f=-f.$ Therefore, we have $Qu=0$ with
$Q=D^{4}_{s}+\Delta^{2}$. Moreover, we note that

$f(0)=f^{\prime}(0)=f^{(2)}(0)=0,f^{(3)}(0)=1$

and

$f(s)=h(\beta
s)=\frac{1}{2}\left(e^{-s}\cos(s-\frac{\pi}{4})-e^{s}\cos(s+\frac{\pi}{4})\right).$

Applying inequality (6.1) of Theorem 6.1 to $u$ reads

$\|u\|_{H^{3}(Y)}\leq
C\|u\|^{1-\delta}_{H^{3}(Z)}\left(\sum\limits_{j=1}^{2}|B_{j}u_{|s=0}|_{H^{3-k_{j}}(\mathscr{O})}\right)^{\delta}.$

As
$\partial_{s}^{3}u_{|s=0}=\sum\limits_{\mu_{j}\leq\mu}\alpha_{j}\phi_{j}(x)=y,$
we have
$\sum\limits_{j=1}^{2}|B_{j}u_{|s=0}|_{H^{3-k_{j}}(\mathscr{O})}\asymp\|y\|_{L^{2}(\mathscr{O})}.$
Then

$\|u\|_{H^{3}(Y)}\leq
C\|u\|^{1-\delta}_{H^{3}(Z)}\|y\|_{L^{2}(\mathscr{O})}^{\delta}.$

We also have $\|u\|_{L^{2}(Y)}\lesssim\|u\|_{H^{3}(Y)}$ with

$\displaystyle\|u\|_{L^{2}(Y)}$
$\displaystyle=\sum\limits_{\mu_{j}\leq\mu}\mu_{j}^{-3/2}|\alpha_{j}|^{2}\int_{\alpha}^{S_{0}-\alpha}f(\mu_{j}^{1/4}s)^{2}ds.$

Using the change of variable $t=\beta\mu_{j}^{1/4}s,$ it follows that

$\displaystyle\|u\|_{L^{2}(Y)}$
$\displaystyle=\sum\limits_{\mu_{j}\leq\mu}\mu_{j}^{-7/4}|\alpha_{j}|^{2}\int_{\alpha\gamma\mu_{j}^{1/4}}^{(S_{0}-\alpha)\gamma\mu_{j}^{1/4}}f(\gamma^{-1}t)^{2}dt$
$\displaystyle=\sum\limits_{\mu_{j}\leq\mu}\mu_{j}^{-7/4}|\alpha_{j}|^{2}\int_{\alpha\gamma\mu_{j}^{1/4}}^{(S_{0}-\alpha)\gamma\mu_{j}^{1/4}}h(t)^{2}dt$
$\displaystyle\gtrsim\mu^{-7/4}\sum\limits_{\mu_{j}\leq\mu}|\alpha_{j}|^{2}=\mu^{-7/4}\|y\|_{L^{2}(\Omega)};$

we have used the following lemma, whose proof can be found in [9, Lemma 5.7].

######  Lemma 6.3.

Let $0<a<b$ and $t_{0}>0.$ There exists $C>0$ such that
$\int_{at}^{bt}h(s)^{2}ds\geq C_{0}$ for $t\geq t_{0}.$

Hence we obtain

$\|y\|_{L^{2}(\Omega)}\lesssim\mu^{7/4}\|u\|^{1-\delta}_{H^{3}(Z)}\|y\|_{L^{2}(\mathscr{O})}^{\delta}.$
(6.3)

In order to complete the proof, we now estimate $\|u\|_{H^{3}(Z)}$ with the
use of the following lemma, which by estimate (6.3), allows to conclude the
proof of Theorem 1.3. ∎

######  Lemma 6.4.

There exists $K>0$ such that $\|u\|_{H^{3}(Z)}\leq
Ke^{K\mu^{1/4}}\|y\|_{L^{2}(\Omega)}.$

###### Proof.

We have

$\|u\|_{H^{3}(Z)}\asymp\sum\limits_{\ell=0}^{3}\int_{0}^{S_{0}}\|\partial^{\ell}_{s}u(s,\cdot)\|^{2}_{H^{3-\ell}(\Omega)}ds\lesssim\sum\limits_{\ell=0}^{3}\int_{0}^{S_{0}}\|\partial^{\ell}_{s}u(s,\cdot)\|^{2}_{H^{4}(\Omega)}ds.$

Recalling that if $B_{1}u_{|{\partial\Omega}}=B_{2}u_{|{\partial\Omega}}=0,$
by Corollary 6.2 we find

$\displaystyle\|\partial^{\ell}_{s}u(s,\cdot)\|^{2}_{H^{4}(\Omega)}$
$\displaystyle\lesssim\|\Delta^{2}\sum\limits_{\mu_{j}\leq\mu}\alpha_{j}\mu_{j}^{(\ell-3)/4}f^{(\ell)}(\mu_{j}^{1/4}s)\phi_{j}\|^{2}_{L^{2}(\Omega)}$
$\displaystyle=\|\sum\limits_{\mu_{j}\leq\mu}\alpha_{j}\mu_{j}^{(\ell+1)/4}f^{(\ell)}(\mu_{j}^{1/4}s)\phi_{j}\|^{2}_{L^{2}(\Omega)}$
$\displaystyle=\sum\limits_{\mu_{j}\leq\mu}|\alpha_{j}|^{2}\mu_{j}^{(\ell+1)/2}f^{(\ell)}(\mu_{j}^{1/4}s)^{2}\lesssim\mu^{2}e^{S_{0}\mu^{1/4}}\sum\limits_{\mu_{j}\leq\mu}|\alpha_{j}|^{2}.$

Integrating this estimate over $(0,S_{0})$ and summing over $\ell$ yields

$\|u\|_{H^{3}(Z)}\leq Ke^{K\mu^{1/4}}\|y\|_{L^{2}(\Omega)}.$

∎

###  6.3 A null-controllability result

Let $T>0,$ we consider the controlled evolution equation on
$(0,T)\times\Omega$ under general boundary conditions:

$\begin{cases}\partial_{t}y+\Delta^{2}y=\mathbbm{1}_{\Sigma}v~{}~{}~{}~{}\text{in}~{}(t,x)\in(0,T)\times\Omega,\\\
B_{1}y_{|(0,T)\times\partial\Omega}=B_{2}y_{|(0,T)\times\partial\Omega}=0,\\\
y_{|t=0}=y^{0}\in L^{2}(\Omega),\end{cases}$ (6.4)

where $\Sigma$ is an open subset of $\Omega$ and $\mathbbm{1}_{\Sigma}\in
L^{\infty}$ is such that $\mathbbm{1}_{\Sigma}>0$ on $\Sigma.$ We recall that
he function $v\in L^{2}((0,T)\times\Sigma)$ is the control function. One may
ask the following question: can one choose $v$ to drive the solution from its
initial condition $y^{0}$ to zero in final time $T?$ Thanks to the spectral
inequality of Theorem 1.3 one can answer to this null-controllability question
in the affirmative.

######  Theorem 6.5.

(Null-controllability) There exists $C>0$ such that for any $y^{0}\in
L^{2}(\Omega),$ there exists $v\in L^{2}((0,T)\times\Omega)$ such that the
solution to (6.4) vanishes at time $T$ and moreover the control $v$ satisfies
the bound $\|v\|_{L^{2}((0,T)\times\Omega)}\leq C\|y^{0}\|_{L^{2}(\Omega)}.$

The proof can be adapted in a straightforward manner from the proof scheme in
section 6.1 of [8] developed for the heat equation.

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# Control technique for synchronization of selected nodes in directed networks

Bruno Ursino, Lucia Valentina Gambuzza, Vito Latora, Mattia Frasca∗ B. Ursino,
L.V. Gambuzza, and M. Frasca are with the Department of Electrical Electronic
and Computer Science Engineering, University of Catania, Catania, Italy. V.
Latora is with the School of Mathematical Sciences, Queen Mary University of
London, London E1 4NS, UK and with the Department of Physics and Astronomy,
University of Catania and INFN, Catania, Italy. This work was supported by the
Italian Ministry for Research and Education (MIUR) through Research Program
PRIN 2017 under Grant 2017CWMF93. ∗ Email<EMAIL_ADDRESS>
###### Abstract

In this Letter we propose a method to control a set of arbitrary nodes in a
directed network such that they follow a synchronous trajectory which is, in
general, not shared by the other units of the network. The problem is inspired
to those natural or artificial networks whose proper operating conditions are
associated to the presence of clusters of synchronous nodes. Our proposed
method is based on the introduction of distributed controllers that modify the
topology of the connections in order to generate outer symmetries in the nodes
to be controlled. An optimization problem for the selection of the
controllers, which includes as a special case the minimization of the number
of the links added or removed, is also formulated and an algorithm for its
solution is introduced.

###### Index Terms:

Network analysis and control; Control of networks.

##  I Introduction

In the last few decades most of the works on synchronization control in
complex networks have focused on the problem of steering the network towards a
collective state shared by all the units. Such a synchronized state has been
obtained by means of techniques ranging from pinning control [1, 2] to
adaptive strategies [3], discontinuous coupling [4], stochastic broadcasting
[5] and impulsive control [6]. Other studies have focused on the control of a
more structured state where the units split into clusters of synchronized
nodes, and each one of these groups follows a different trajectory [7, 8, 9,
10, 11, 12].

In the above mentioned works, the control action is such that all network
nodes are forced to follow a given dynamical behavior. However, the number of
nodes and links can be very large in real-world systems, so that the question
of whether it is possible to control the state of only a subset of the network
units, disregarding the behavior of the other units, becomes of great
importance. Solving the problem can lead to potentially interesting
applications. Consider a team of mobile agents and the case in which a
particular task can be accomplished by a subset of the agents only. In such a
scenario, one could exploit the relationship between oscillator
synchronization and collective motion [13] and apply control techniques for
synchronizing a subset of nodes to recruit only a group of all the mobile
agents and coordinate them. In a different context, it is well known that
synchronization of the whole brain network is associated to pathological
states, whereas neural areas are actively synchronized to engage in functional
roles [14]. Our approach could provide control techniques supporting
neuromorphic engineering applications relying on the principles of neuronal
computation [15].

Recently, it has been argued that a subset of network nodes can be controlled
by adopting a distributed control paradigm whose formulation relies on the
notion of symmetries in a graph [16, 17]. The approach there presented is
restricted to undirected graphs, whereas here we propose a control technique
for the more general case of directed networks. We find that, in order to form
a synchronous cluster, the nodes to control must have the same set of
successors, and the common value of their out-degree has to be larger than a
threshold, which decreases when the coupling strength in the network is
increased. Both conditions can be matched by a proper design of controllers
adding to or removing links from the original network structure. The selection
of the controllers is addressed by formulating an optimization problem,
minimizing an objective function which accounts for the costs associated to
adding and/or removing links. We show that an exact solution to the problem
can be found, and we propose an algorithm to calculate it.

The rest of the paper is organized as follows: Sec. II contains the
preliminaries; the problem is formulated in Sec. III; a theorem illustrating
how to design the controllers is illustrated in Sec. IV; the optimization
problem and its solution are dealt with in Sec. V; an example of our approach
is provided in Sec. VI and the conclusions are drawn in Sec. VII.

##  II Preliminaries

In this section we introduce notations and definitions used in the rest of the
paper [18]. A graph $\mathcal{G}=\left(\mathcal{V},\mathcal{E}\right)$
consists of set of _vertices_ or _nodes_
$\mathcal{V}=\left\\{v_{1},...,v_{n}\right\\}$ and a set of _edges_ or _links_
$\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$. Network nodes are
equivalently indicated as $v_{i}$ or, shortly, as $i$. If
$\forall\left(v_{i},v_{j}\right)\in\mathcal{E}\Rightarrow\left(v_{j},v_{i}\right)\in\mathcal{E}$,
the graph is _undirected_ , otherwise it is _directed_. Only simple (i.e.,
containing no loops and no multiple edges) directed graphs are considered in
what follows. The set
$\mathcal{S}_{i}=\left\\{v_{j}\in\mathcal{V}|\left(v_{i},v_{j}\right)\in\mathcal{E}\right\\}$
is the set of _successors_ of $v_{i}$ (in undirected graphs $\mathcal{S}_{i}$
coincides with the set of neighbors).

The graph $\mathcal{G}$ can be described through the adjacency matrix
$\mathrm{A}$, a $N\times N$ matrix, with $N=|\mathcal{V}|$, and whose elements
are $a_{ij}=1$ if $v_{j}\in\mathcal{S}_{i}$ and $a_{ij}=0$, otherwise. We
define the _out-degree_ of a node $i$ as the number of its successors,
$k_{i}=|\mathcal{S}_{i}|=\sum_{j=1}^{N}a_{ij}$. The Laplacian matrix,
$\mathcal{L}$, is defined as $\mathcal{L}=D-\mathrm{A}$, where
$D=diag\left\\{k_{1},...,k_{N}\right\\}$. Its elements are:
$\mathcal{L}_{ij}=k_{i}$, if $i=j$, $\mathcal{L}_{ij}=0$, if $i\neq j$ and
$v_{j}\notin\mathcal{S}_{i}$, and $\mathcal{L}_{ij}=-1$, if $i\neq j$ and
$v_{j}\in\mathcal{S}_{i}$. From the definition it immediately follows that
$\mathcal{L}1_{N}=0_{N,1}$ and, so, $0$ is an eigenvalue of the Laplacian
matrix with corresponding eigenvector $1_{N}$.

While arguments based on network symmetries are used for controlling groups of
nodes in undirected networks [17], directed topologies require the notion of
_outer symmetrical nodes_ , here introduced. We define two nodes $v_{i}$ and
$v_{j}$ _outer symmetrical_ if $\mathcal{S}_{i}=\mathcal{S}_{j}$ and
$\left(v_{i},v_{j}\right)\notin\mathcal{E}$. This notion is more restrictive
than that of input equivalence given in [19] for networks including different
node dynamics and coupling functions. In [19] the _input set_ of a node
$v_{i}$ is defined as
$I(v_{i})=\left\\{e\in\mathcal{E}:e=\left(v_{i},v_{j}\right)\mbox{ for some
}v_{j}\in V\right\\}$. Two nodes $v_{i}$ and $v_{j}$ are called _input
equivalent_ if and only if there exists a bijection $\beta:I(v_{i})\rightarrow
I(v_{j})$ such that the _type of connection_ is preserved, that is the
coupling function is the same and the extremes of the edges have the same
dynamics. For networks of identical dynamical units and coupling functions, as
those considered in our work, input equivalent nodes are nodes with the same
out-degree. To be outer symmetrical, a further condition is required: outer
symmetrical nodes are input equivalent nodes where the bijection is the
identity. This property is fundamental for the control problem dealt with in
our paper.

##  III Problem formulation

Let us consider a directed network of $N$ identical, $n$-dimensional units
whose dynamics is given by

$\dot{x_{i}}=f(x_{i})-\sigma\sum_{j=1}^{N}\mathcal{L}_{ij}\mathrm{H}x_{j}+u_{i},\forall
i=1,...,N$ (1)

with $x_{i}=\left(x_{i1},x_{i2},...,x_{in}\right)^{T}$. Here,
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is the uncoupled dynamics of each
unit, $H\in\mathbb{R}^{n\times n}$ is a constant matrix with elements taking
values in $\\{0,1\\}$ that represents inner coupling, i.e., it specifies the
components of the state vectors through which node $j$ is coupled to node $i$,
and $\sigma>0$ is the coupling strength. $u_{i}$ represent the control actions
on the network. Equations (1) with $u_{i}=0$ are extensively used to model
diffusively coupled oscillators in biology, chemistry, physics and engineering
[20].

Equations (1) can be rewritten in compact form as

$\dot{\mathbf{x}}=F(\mathbf{x})-\sigma\mathcal{L}\otimes\mathrm{H}\mathbf{x}+\mathbf{u}$
(2)

where $\mathbf{x}=\left[x_{1}^{T},x_{2}^{T},...,x_{N}^{T}\right]^{T}$,
$F(\mathbf{x})=\left[f^{T}(x_{1}),f^{T}(x_{2}),...,f^{T}(x_{N})\right]^{T}$
and $\mathbf{u}=\left[u_{1}^{T},u_{2}^{T},...,u_{N}^{T}\right]^{T}$. In the
following we use distributed controllers of the form

$u_{i}=-\sigma\sum_{j=1}^{N}\mathcal{L}^{\prime}_{ij}\mathrm{H}x_{j}$ (3)

where $\mathcal{L}^{\prime}$ is a matrix whose elements are $-1$, $0$ or $1$;
if $\mathcal{L}_{ij}=0$, setting $\mathcal{L}^{\prime}_{ij}=-1$ introduces a
link between two nodes, $i$ and $j$, not connected in the pristine network; on
the contrary, setting $\mathcal{L}^{\prime}_{ij}=1$ in correspondence of
$\mathcal{L}_{ij}=-1$ removes the existing edge $(v_{i},v_{j})$ in the
pristine network; finally, setting $\mathcal{L}^{\prime}_{ij}=0$ indicates no
addition or removal of links between $i$ and $j$. The diagonal elements of
$\mathcal{L}^{\prime}$ are $\mathcal{L}^{\prime}_{ii}=-\sum_{j=1,j\neq
i}^{N}\mathcal{L}^{\prime}_{ij}$. Notice that, even if $\mathcal{L}^{\prime}$
is not a Laplacian, the resulting matrix
$\mathcal{L}^{\prime\prime}=\mathcal{L}+\mathcal{L}^{\prime}$ (a matrix
representing the network formed by the original topology and the links added
or removed by the controllers) is instead a Laplacian.

The system with the controllers reads

$\dot{\mathbf{x}}=F(\mathbf{x})-\sigma\mathcal{L}^{\prime\prime}\otimes\mathrm{H}\mathbf{x}$
(4)

The problem tackled in this paper is twofold: i) given an arbitrary subset
$V_{n_{2}}$ of $n_{2}<N$ nodes, to determine a set of controllers $u_{i}$ with
$i=1,\ldots,N$ such that the nodes in $V_{n_{2}}$ synchronizes to each other;
ii) to formulate an optimization problem for the selection of the controllers
$u_{i}$.

Without lack of generality, we relabel the network nodes so that the nodes to
control are indexed as $i=n_{1}+1,\ldots,N$, such that
$V_{n_{2}}=\left\\{v_{n_{1}+1},...,v_{N}\right\\}$. Objective of the
controllers is, therefore, to achieve a synchronous evolution of the type

$\begin{cases}x_{1}(t)=s_{1}(t)\\\ \vdots\\\ x_{n_{1}}(t)=s_{n_{1}}(t)\\\
x_{n_{1}+1}(t)=x_{n_{1}+2}(t)=...=x_{N}(t)=s(t),\quad
t\rightarrow+\infty\end{cases}$ (5)

In compact form the synchronous state is denoted as
$\mathbf{x}^{s}(t)=\left[s_{1}^{T}(t),...,s_{n_{1}}^{T}(t),s^{T}(t),...,s^{T}(t)\right]^{T}$.
In the most general case, the trajectories of the first $n_{1}$ nodes are
different from each other and from $s(t)$, that is, $s_{i}(t)\neq s_{j}(t)\neq
s(t)$ for $i,j=1,\ldots,n_{1}$, but eventually some of them may coincide or
converge to $s(t)$. In the next section, we demonstrate how to select the
controllers such that the state $\mathbf{x}^{s}(t)$ exists and is locally
exponentially stable, while, in the second part of the paper, we consider the
optimization problem.

##  IV Design of the controllers

To achieve a stable synchronous state $\mathbf{x}^{s}(t)$ the controllers
$u_{i}$ as in Eq. (3) have to satisfy the conditions expressed by the
following theorem.

######  Theorem IV.1.

Consider the dynamical network (1) and the controllers (3) such that the
Laplacian $\mathcal{L}^{\prime\prime}$ satisfies the following conditions:

  1. 1.

$\mathcal{L}^{\prime\prime}_{i_{1},j}=\mathcal{L}^{\prime\prime}_{i_{2},j}$
for $i_{1}=n_{1}+1,\ldots,N$, $i_{2}=n_{1}+1,\ldots,N$, $j=1,\ldots,N$ and
$j\neq i_{1}$, $j\neq i_{2}$;

  2. 2.

$\mathcal{L}^{\prime\prime}_{i_{1},i_{2}}=0$ for $i_{1}=n_{1}+1,\ldots,N$,
$i_{2}=n_{1}+1,\ldots,N$ with $i_{1}\neq i_{2}$;

then, a synchronous behavior
$\mathbf{x}^{s}(t)=\left[s_{1}^{T}(t),...,s_{n_{1}}^{T}(t),s^{T}(t),...,s^{T}(t)\right]^{T}$
exists.

In addition, define $k_{i}=\sum_{j}\mathcal{L}^{\prime\prime}_{i,j}$ with
$i=n_{1}+1,\ldots,N$, and, since from hypothesis 1)
$k_{n_{1}+1}=\ldots=k_{N}$, define $k\triangleq k_{n_{1}+1}=\ldots=k_{N}$. If

  1. 3.

there exists a diagonal matrix $\mathrm{L}>0$ and two constants
$\overline{q}>0$ and $\tau>0$ such that the following linear matrix inequality
(LMI) is satisfied $\forall q\geq\overline{q}$ and $t>0$:

$\left[Df(s(t))-qH\right]^{T}L+L\left[Df(s(t))-qH\right]\leq-\tau I_{n},$ (6)

where $Df(s(t))$ is the Jacobian of $f$ evaluated on $s(t)$;

  2. 4.

$k$ is such that $k>\frac{\overline{q}}{\sigma}$;

then, the synchronous state is locally exponentially stable.

###### Proof.

Existence of the synchronous solution. Hypotheses 1) and 2) induce some
structural properties in the new network defined by the original topology and
the controller links. In particular, hypothesis 1) is equivalent to require
that each node in $V_{n_{2}}$ has the same set of successors, that is,
$\mathcal{S}^{\prime\prime}_{n_{1}+1}=\ldots=\mathcal{S}^{\prime\prime}_{N}$,
while hypothesis 2) requires that there are no links between any pair of nodes
in $V_{n_{2}}$, that is, $\forall v_{i},v_{j}\in
V_{n_{2}}\Rightarrow\left(v_{i},v_{j}\right)\notin\mathcal{E}$. Consequently,
selecting the controllers such that hypotheses 1) and 2) hold makes the nodes
in $V_{n_{2}}$ outer symmetrical.

In turns this means that, with reference to the system in Eq. (4), if we
permute the nodes in $V_{n_{2}}$, the dynamical network does not change, and
the $n_{2}$ nodes have the same equation of motion. If the nodes in
$V_{n_{2}}$ start from the same initial conditions, then they remain
synchronized for $t>t_{0}$, and thus a synchronous solution
$\mathbf{x}^{s}(t)$ as in Eq. (5) exists.

Local exponential stability of the synchronous solution. To prove the
stability of $\mathbf{x}^{s}(t)$, we first prove that the synchronous solution
$\mathbf{x}^{s}(t)$ is locally exponentially stable if

$\dot{\zeta}=\left(Df-\sigma kH\right)\zeta$ (7)

is locally exponentially stable.

We first consider Eq. (4) and linearize it around $\mathbf{x}^{s}(t)$. We
define $\eta=\mathbf{x}-\mathbf{x}^{s}$ and calculate its dynamics as

$\dot{\eta}=DF\eta-\sigma\left(\mathcal{L}^{\prime\prime}\otimes\mathrm{H}\right)\eta$
(8)

Let us indicate as $Df_{i}$ the Jacobian of $F$ evaluated on
$\mathbf{x}^{s}_{i}(t)$. Taking into account Eq. (5), it follows that
$Df_{n_{1}+1}=...=Df_{N}\triangleq Df_{s}$ and, hence,
$DF=diag\left\\{Df_{1},...,Df_{n_{1}},Df_{s},...,Df_{s}\right\\}$.

From the structure of $\mathbf{x}^{s}$, it also follows that the synchronous
behavior is preserved for all variations belonging to the linear subspace
$\mathbb{P}$ generated by the column vectors of the following matrix

$\mathrm{M}_{s}=\begin{bmatrix}1&0&...&0&0\\\ 0&1&...&0&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&...&1&0\\\
0&0&...&0&\frac{1}{\sqrt{n_{2}}}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&...&0&\frac{1}{\sqrt{n_{2}}}\\\ \end{bmatrix}\otimes I_{n}$

Such variations in fact occur along the synchronization manifold where all the
last $n_{2}$ units have the same evolution. The column vectors of
$\mathrm{M}_{s}$ represent an orthonormal basis for the considered linear
subspace with $dim(\mathbb{P})=n(n_{1}+1)$. The remaining vectors in
$\mathbb{R}^{nN}\setminus\mathbb{P}$ represent transversal motions with
respect to the synchronization manifold.

An orthonormal basis for $\mathbb{R}^{nN}$ is built by considering a linear
vector space $\mathbb{O}$ of $dim(\mathbb{O})=n(n_{2}-1)$ that is orthogonal
to $\mathbb{P}$. All vectors of $\mathbb{R}^{nN}$ can be thus expressed as
linear combinations of vectors in $\mathbb{P}$ and vectors in $\mathbb{O}$,
that is, $\eta=\left(M\otimes I_{n}\right)\xi$ with

${M}=\begin{bmatrix}1&0&...&0&0&0&...&0\\\ 0&1&...&0&0&0&...&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&...&1&0&0&...&0\\\
0&0&...&0&\frac{1}{\sqrt{n_{2}}}&o_{n_{1}+1,1}&...&o_{n_{1}+1,n_{2}-1}\\\
0&0&...&0&\frac{1}{\sqrt{n_{2}}}&o_{n_{1}+2,1}&...&o_{n_{1}+2,n_{2}-1}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&...&0&\frac{1}{\sqrt{n_{2}}}&o_{N,1}&...&o_{N,n_{2}-1}\\\ \end{bmatrix}$
(9)

For easy of compactness, matrix ${M}$ is rewritten as
$M=\begin{bmatrix}I_{n_{1}}&0\\\ 0&R_{n_{2}}\\\ \end{bmatrix}$.

The evolution of the first $n(n_{1}+1)$ elements of $\xi$ is the evolution of
motions along the synchronization manifold, while the remaining elements of
$\xi$ are transversal to the synchronization manifold. As a consequence of
this, to prove the exponential stability of the synchronization manifold, we
have to prove that the evolution of the last $n(n_{2}-1)$ elements of vector
$\xi$ decays exponentially to $0$ as $t\rightarrow+\infty$.

We now apply the transformation $\xi=\left(M\otimes I_{n}\right)^{-1}\eta$ to
Eqs. (8):

$\small\begin{array}[]{l}\dot{\xi}=\left(M^{-1}\otimes
I_{n}\right)DF\left(M\otimes
I_{n}\right)\xi-\sigma\left(M^{-1}\mathcal{L}^{\prime\prime}M\right)\otimes
H\xi\end{array}$ (10)

Straightforward calculations yield that $\left(M^{-1}\otimes
I_{n}\right)DF\left(M\otimes I_{n}\right)=DF$. Let us now focus on
$M^{-1}\mathcal{L}^{\prime\prime}M$. To calculate this term, we partition
$\mathcal{L}^{\prime\prime}$ in (4) as follows:

$\mathcal{L}^{\prime\prime}=\begin{bmatrix}A_{n_{1}\times
n_{1}}&B_{n_{1}\times n_{2}}\\\ C_{n_{2}\times n_{1}}&D_{n_{2}\times n_{2}}\\\
\end{bmatrix}$ (11)

From hypothesis 2), it follows that $D=kI_{n_{2}}$. Consider now the block
$C_{n_{2}\times n_{1}}$. From hypothesis 1) we have that
$\mathcal{L}^{\prime\prime}_{i_{1},j}=\mathcal{L}^{\prime\prime}_{i_{2},j},\forall
j\leq n_{1},\forall i_{1},i_{2}>n_{1}$. Denoting with $C_{i}$ the $i$-th row
of $C$ we obtain that $C_{i_{1}}=C_{i_{2}},\quad\forall
i_{1},i_{2}=1,...,n_{2}$.

Given that

$\begin{array}[]{c}M^{-1}\mathcal{L}^{\prime\prime}M=\begin{bmatrix}A&BR_{n_{2}}\\\
R_{n_{2}}^{T}C&R_{n_{2}}^{T}DR_{n_{2}}\\\ \end{bmatrix}\end{array}$ (12)

since $D=kI_{n_{2}}$ and all the rows in $C$ are equal, we can rewrite
$\mathcal{L}^{\prime\prime}$ as:

$\mathcal{L}^{\prime\prime}=\begin{bmatrix}A&B\\\
\begin{matrix}\begin{matrix}c_{1}\\\ c_{1}\\\ \vdots\\\
c_{1}\end{matrix}&...&\begin{matrix}c_{n_{1}}\\\ c_{n_{1}}\\\ \vdots\\\
c_{n_{1}}\end{matrix}\\\ \end{matrix}&\begin{matrix}k&0&...&0\\\ 0&k&...&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&...&k\end{matrix}\end{bmatrix}$ (13)

where $c_{i}\in\\{0,1\\}$ if node $i$ is connected or not with the nodes of
$V_{n_{2}}$.

Notice that the first row of $R_{n_{2}}^{T}$ is a vector parallel to
$\left[1,1,...,1\right]$ while the remaining ones are all orthogonal to it,
so:

$R_{n_{2}}^{T}C=\begin{bmatrix}a_{1}\sqrt{n_{2}}&...&a_{n_{1}}\sqrt{n_{2}}\\\
0&...&0\\\ \vdots&\ddots&\vdots\\\ 0&...&0\end{bmatrix}$ (14)

Moreover
$R_{n_{2}}^{T}DR_{n_{2}}=R_{n_{2}}^{T}kI_{n_{2}}R_{n_{2}}=kI_{n_{2}}=D$.

It follows that Eq. (10) becomes:

$\small\begin{array}[]{l}\dot{\xi}=\left[\begin{array}[]{cccc|ccc}Df_{1}&\cdots&0&0&0&\cdots&0\\\
\vdots&&\vdots&\vdots&\vdots&&\vdots\\\ 0&\cdots&Df_{n_{1}}&0&0&\cdots&0\\\
0&\cdots&0&Df_{s}&0&\cdots&0\\\ \hline\cr 0&\cdots&0&0&Df_{s}&\cdots&0\\\
\vdots&&\vdots&\vdots&\vdots&&\vdots\\\
0&\cdots&0&0&0&\cdots&Df_{s}\end{array}\right]\xi-\sigma\cdot\\\
\left[\begin{array}[]{ccc|cc}l_{11}\ldots&l_{1n_{1}}&l_{1n_{1}+1}&l_{1n_{1}+2}\ldots&l_{1N}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
l_{n_{1}1}\ldots&l_{n_{1}n_{1}}&l_{n_{1}n_{1}+1}&l_{n_{1}n_{1}+2}\ldots&l_{n_{1}N}\\\
c_{1}\sqrt{n_{2}}\ldots&c_{n_{1}}\sqrt{n_{2}}&k&0\ldots&0\\\ \hline\cr
0\ldots&0&0&k\ldots&0\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
0\ldots&0&0&0\ldots&k\end{array}\right]\otimes H\xi\end{array}$ (15)

where the lines in the matrices suggest a partition highlighting the last
$n\left(n_{2}-1\right)$ elements of $\xi$.

The system describing the evolution of variations transversal to the
synchronization manifold is uncoupled from the rest of the equations and
composed of identical blocks, taking the following form:

$\dot{\zeta}=\left(Df_{s}-\sigma kH\right)\zeta,\quad\mbox{ with
}\zeta\in\mathbb{R}^{n}$ (16)

It only remains to prove the exponential stability of (16). By hypothesis 4)
we have that $k>\frac{\overline{q}}{\sigma}$, thus $k\sigma>\overline{q}$.
From the LMI (6) we can use the Lyapunov function $V=\zeta^{T}L\zeta$ to prove
exponential stability:

  1. 1.

$V(0)=0$;

  2. 2.

$V(\zeta)=\zeta^{T}L\zeta>0,\;\forall\zeta\neq 0$ because $L>0$;

  3. 3.

$\dot{V}<0,\;\forall\zeta\neq 0$ in fact:

$\begin{array}[]{l}\frac{d}{dt}\left[\zeta^{T}L\zeta\right]=\dot{\zeta^{T}}L\zeta+\zeta^{T}L\dot{\zeta}\\\
=\zeta^{T}\left[\left(Df_{s}-\sigma kH\right)^{T}L+L\left(Df_{s}-\sigma
kH\right)\right]\zeta\\\ \leq-\tau\zeta^{T}\zeta<0.\end{array}$ (17)

∎

We note that, in the application of Theorem IV.1, we can first consider the
set formed by the union of the successors of the nodes to control. If the
cardinality of this set is greater than $\frac{\overline{q}}{\sigma}$, then we
can add links such that the successors of each node of $V_{n_{2}}$ are all the
elements of this set. Otherwise, one needs to expand this set by including
other nodes of the network. Interestingly, the choice of such nodes is totally
arbitrary and any node, not yet included in the set of successors, fits for
the purpose.

The upper bound of $k$ is the cardinality of $\mathcal{V}\setminus V_{n_{2}}$
and, since Theorem IV.1 requires that $k>\frac{\overline{q}}{\sigma}$, a
necessary condition for the application of the proposed technique is that
$\sigma>\frac{\overline{q}}{n_{1}}$: if this condition is not met, then, there
are not enough nodes in $\mathcal{V}\setminus V_{n_{2}}$ to which the nodes of
$V_{n_{2}}$ can be connected by the controllers.

##  V Optimization

In this section we address the problem of optimizing the controllers with
respect to the cost of the links added or removed. Let $w_{ij}^{-}$
($w_{ij}^{+}$) be the cost associated to the removal of an existing link
(addition of a new link) between $i$ and $j$. These parameters account for a
general scenario where different links have different costs to change.

Formally, the following minimization problem is considered:

$\min\limits_{\mathbf{u}\in\mathcal{U}}\sum\limits_{\mathcal{L}^{\prime}_{i,j}=1}\mathcal{L}^{\prime}_{ij}w_{ij}^{-}+\sum\limits_{\mathcal{L}^{\prime}_{i,j}=-1}|\mathcal{L}^{\prime}_{ij}|w_{ij}^{+}$
(18)

where $\mathcal{U}$ is the set of controllers that satisfies Theorem IV.1 and,
thus, ensures the existence and stability of $\mathbf{x}^{s}(t)$. In the
special case, when the costs are equal and unitary, i.e.,
$w_{ij}^{-}=w_{ij}^{+}=1$, the optimization problem reduces to

$\min\limits_{\mathbf{u}\in\mathcal{U}}\sum\limits_{i,j}|\mathcal{L}^{\prime}_{ij}|$
(19)

i.e., minimization of the number of links added or removed by the controllers.

Let $\bar{k}\triangleq\lceil\frac{\bar{q}}{\sigma}\rceil$. Theorem IV.1
requires that the nodes in $V_{n_{2}}$ have a number of successors greater
than or equal to $\bar{k}$, i.e., since $|\mathcal{S}^{\prime\prime}|=k$,
$k\geq\bar{k}$. The optimization problem is thus equivalent to determine the
nodes in $\mathcal{S}^{\prime\prime}$ which minimize the objective function
(18). Consider the set $\bar{\mathcal{S}}=\bigcup_{v_{i}\in
V_{n_{2}}}\mathcal{S}_{i}\setminus V_{n_{2}}$, containing the successors of at
least one node of the pristine network that are not in $V_{n_{2}}$. Depending
on the cardinality of this set we can have two different scenarios: 1) if
$|\bar{\mathcal{S}}|<\overline{k}$, then, ${\mathcal{S}^{\prime\prime}}$ needs
to contain all the nodes in $\bar{\mathcal{S}}$ and some other nodes of the
set $V_{n_{1}}\setminus\bar{\mathcal{S}}$; 2) if
$\left|\bar{\mathcal{S}}\right|\geq\bar{k}$, then, one has to select
${\mathcal{S}^{\prime\prime}}\subset\bar{\mathcal{S}}$. In both cases, the
choice of the nodes in ${\mathcal{S}^{\prime\prime}}$ is accomplished taking
into account the costs associated to the network links.

First, note that, given $V_{n_{2}}$, to fulfill condition 2) of Theorem IV.1
the links between nodes in this set need to be removed. This yields a fixed
cost $\bar{c}=\sum\limits_{i,j\in V_{n_{2}}}a_{ij}w_{ij}^{-}$ such that
$\min\limits_{\mathbf{u}\in\mathcal{U}}\sum\limits_{\mathcal{L}^{\prime}_{i,j}=1}\mathcal{L}^{\prime}_{ij}w_{ij}^{-}+\sum\limits_{\mathcal{L}^{\prime}_{i,j}=-1}|\mathcal{L}^{\prime}_{ij}|w_{ij}^{+}\geq\bar{c}$.

Let $c_{i}^{+}$ be the cost to have node $i$ in ${\mathcal{S}^{\prime\prime}}$
and $c_{i}^{-}$ the cost of not including it in
${\mathcal{S}^{\prime\prime}}$. It follows that $c_{i}^{-}=\sum\limits_{j\in
V_{n_{2}}}a_{ij}w_{ij}^{-}$ and $c_{i}^{+}=\sum\limits_{j\in
V_{n_{2}}}(1-a_{ij})w_{ij}^{+}$. Once calculated $c_{i}^{+}$ and $c_{i}^{-}$,
we reformulate the optimization problem in terms of minimization of the
overall cost of the control:
$Cost=\sum\limits_{v_{i}\in\bar{\mathcal{S}}}\left[c_{i}^{+}x_{i}+c_{i}^{-}(1-x_{i})\right]+\bar{c}$,
where $x_{i}$ ($i=1,\ldots,N$) are decisional variables, such that $x_{i}=1$
if $v_{i}\in{\mathcal{S}^{\prime\prime}}$ and $x_{i}=0$ otherwise. The
optimization problem now reads:

$\begin{cases}\min\sum\limits_{v_{i}\in\bar{\mathcal{S}}}\left[c_{i}^{+}x_{i}+c_{i}^{-}(1-x_{i})\right]+\bar{c}\\\
\sum\limits_{v_{i}\in\bar{\mathcal{S}}}x_{i}\geq\bar{k}\end{cases}$ (20)

where the constraint $\sum\limits_{v_{i}\in\bar{\mathcal{S}}}x_{i}\geq\bar{k}$
guarantees that condition $k\geq\bar{k}$ holds. Since the overall cost can be
rewritten as
$Cost=\sum\limits_{v_{i}\in\bar{\mathcal{S}}}\left(c_{i}^{+}-c_{i}^{-}\right)x_{i}+\sum\limits_{v_{i}\in\bar{\mathcal{S}}}c_{i}^{-}+\bar{c}$
and the terms $\sum\limits_{v_{i}\in\bar{\mathcal{S}}}c_{i}^{-}$ and $\bar{c}$
do not depend on the variables $x_{i}$, the optimization problem becomes

$\begin{cases}\min\sum\limits_{v_{i}\in\bar{\mathcal{S}}}c_{i}x_{i}\\\
\sum\limits_{v_{i}\in\bar{\mathcal{S}}}x_{i}\geq\bar{k}\end{cases}$ (21)

where $c_{i}\triangleq c_{i}^{+}-c_{i}^{-}$.

This formulation prompts the following solution for the optimization problem.
Defining
$k^{\prime}\triangleq\left|\left\\{v_{i}\in\bar{\mathcal{S}}\;|\;c_{i}\leq
0\right\\}\right|$ and sorting the nodes in $\bar{\mathcal{S}}$ in ascending
order with respect to their cost $c_{i}$, we take
$k_{max}=\max\left\\{\bar{k},\;k^{\prime}\right\\}$ and assign $x_{i}=1$ to
the first $k_{max}$ nodes and $x_{i}=0$ to the remaining ones. The overall
cost to achieve synchronization of the nodes in the set $V_{n_{2}}$ is given
by

$Cost=\sum\limits_{v_{i}\in\bar{\mathcal{S}}}c_{i}^{-}+\sum\limits_{v_{i}\in{\mathcal{S}^{\prime\prime}}}c_{i}+\bar{c}$
(22)

Algorithm 1 is based on the above observations and returns the nodes belonging
to ${\mathcal{S}^{\prime\prime}}$. The inputs are $V_{n_{2}}$, $\bar{k}$ and
$A$ (the adjacency matrix of the network) and the outputs are the set
${\mathcal{S}^{\prime\prime}}$ and the overall cost.

Algorithm 1 Algorithm to select the nodes in ${\mathcal{S^{\prime\prime}}}$

0: $V_{n_{2}}$, $\bar{k}$ and $A$

0: ${\mathcal{S}^{\prime\prime}}$, overall cost $Cost$Initialization:

1: Create $\bar{\mathcal{S}}=\bigcup_{v_{i}\in
V_{n_{2}}}\mathcal{S}_{i}\setminus V_{n_{2}}$ and determine its cardinality
$|\bar{\mathcal{S}}|$Procedure:

2: Calculate $c_{i}^{-}$, $c_{i}^{+}$, $\bar{c}$ and
$c_{i}=c_{i}^{+}-c_{i}^{-}$

3: Sort the nodes in $\bar{\mathcal{S}}$ in ascending order and set
$k^{\prime}=\left|\left\\{v_{i}\in{\mathcal{S}}\;|\;c_{i}\leq
0\right\\}\right|$

4: Calculate $k_{max}=\max\left\\{\bar{k},\;k^{\prime}\right\\}$

5: if $|\bar{\mathcal{S}}|\geq\bar{k}$ then

6: return
$Cost=\bar{c}+\sum\limits_{v_{i}\in\bar{\mathcal{S}}}c_{i}^{-}+\sum\limits_{v_{i}\in\bar{\mathcal{S}^{\prime\prime}}}c_{i}$
and build ${\mathcal{S}^{\prime\prime}}$ taking the first $k_{max}$ nodes in
$\bar{\mathcal{S}}$

7: else

8: Build ${\mathcal{S}^{\prime\prime}}$ taking all the elements of
$\bar{\mathcal{S}}$ and add nodes from $V_{n_{1}}\setminus\bar{\mathcal{S}}$,
in ascending order of $c_{i}$, until $|{\mathcal{S^{\prime\prime}}}|=\bar{k}$

9: end if

10: return $Cost$ and ${\mathcal{S}^{\prime\prime}}$

##  VI Examples

We now discuss an example of how the proposed control works in the directed
network with $N=20$ nodes shown in Fig. 1. We refer to several cases,
corresponding to two distinct sets of nodes to control, $V_{n_{2}}$, two
values of $\bar{k}$, and different costs associated to the links. For each of
these cases, the controllers that satisfy Theorem IV.1 and are the result of
the optimization procedure of Sec. V are discussed; we will show that they
depend on the control goal, on the link costs and, through $\bar{k}$, on the
coupling coefficient.

More specifically, we first consider unitary costs for the links and
synchronization of two different triplets of nodes, i.e., either
$V_{n_{2}}=\\{1,4,16\\}$ or $V_{n_{2}}=\\{1,3,19\\}$, with two values of
$\bar{k}$, i.e., $\bar{k}=1$ and $\bar{k}=3$111The value of $\bar{k}$ depends
on the node dynamics considered and strength of the coupling, e.g., with
reference to Chua’s circuit as node dynamics and coupling of the type
$\mathrm{H}=\textrm{diag}\\{1,1,0\\}$, we have $\bar{q}=4.5$ [17], and
$\bar{k}=1$ for $\sigma=5$ while $\bar{k}=3$ for $\sigma=2$.. This leads to
cases 1-4 in Table I. Case 5, instead, refers to a scenario where the costs
are not unitary.

Figure 1: Case study: a directed network of $N=20$ nodes. The spatial position
of the network nodes is used to define costs for link addition proportional to
the Euclidean distance of the nodes to be connected.

_Case 1: $V_{n_{2}}=\\{1,4,16\\}$, $\bar{k}=1$._ Here, we have that
$\bar{\mathcal{S}}=\\{3,6,8,10,15,20\\}$ and $|\bar{\mathcal{S}}|>\bar{k}$.
Following Algorithm 1, we find $k^{\prime}=2$, so that $k^{\prime}>\bar{k}$
and ${\mathcal{S}^{\prime\prime}}=\\{3,8\\}$. Synchronization of the nodes in
$V_{n_{2}}$ is achieved if two links are added to the original network, and
four links are removed.

_Case 2: $V_{n_{2}}=\\{1,4,16\\}$, $\bar{k}=3$._ Here again
$|\bar{\mathcal{S}}|>\bar{k}$, but $k^{\prime}<\bar{k}$. We get
${\mathcal{S}^{\prime\prime}}=\\{3,6,8\\}$, four links to add and three to
remove.

_Case 3: $V_{n_{2}}=\\{1,3,19\\}$, $\bar{k}=1$._ We have
$\bar{\mathcal{S}}=\\{4,10\\}$ and $|\bar{\mathcal{S}}|>\bar{k}$. In this
case, we obtain ${\mathcal{S}^{\prime\prime}}=\\{4\\}$, a single link to add
and three links to remove.

_Case 4: $V_{n_{2}}=\\{1,3,19\\}$, $\bar{k}=3$._ We have
$\left|\bar{\mathcal{S}}\right|<\bar{k}$ thus, following step 8 of the
algorithm, we need to add a node from $V_{n_{1}}\setminus\bar{\mathcal{S}}$,
i.e. a node which is not a successor of any of the nodes to be synchronized.
As the choice is completely arbitrary, we select node $6$. So,
${\mathcal{S}^{\prime\prime}}=\\{4,6,10\\}$. Control is attained by adding
seven links and removing three links.

_Case 5: $V_{n_{2}}=\\{1,4,16\\}$, $\bar{k}=3$, non-unitary costs._ For the
purpose of illustration, here we assume that the cost to add a link is
proportional to the distance between the two nodes, while removing links
always has a unitary cost. We consider the synchronization problem as in case
2. Here, the different costs yield a different result for
${\mathcal{S}^{\prime\prime}}$, i.e.,
${\mathcal{S}^{\prime\prime}}=\\{3,8,10\\}$. In this scenario, optimization
requires to include in ${\mathcal{S}^{\prime\prime}}$ node 10 rather than node
6.

TABLE I: Added and removed links obtained for different $V_{n_{2}}$ and $\bar{k}$ for the network in Fig. 1. Costs associated to links are considered unitary in all cases, except for case 5.  Case | $V_{n_{2}}$ | $\bar{k}$ |  Added links |  Removed links
---|---|---|---|---
1 | {1,4,16} | 1 |  (1,8) (16,3) |  (16,15) (16,20) (1,10) (4,6)
2 | {1,4,16} | 3 |  (1,8) (16,3) (1,6) (16,6) |  (16,15) (16,20) (1,10)
3 | {1,3,19} | 1 |  (1,4) |  (1,10) (1,3) (19,1)
4 | {1,3,19} | 3 |  (1,6) (1,4) (3,10) (3,4) (3,6) (19,20) (19,6) |  (19,1) (1,3) (3,19)
5* | {1,4,16} | 3 |  (1,8) (4,10) (16,3) (16,10) |  (4,6) (16,15) (16,20)

Finally, for case 2 we report the waveforms obtained by simulating the network
with control (Fig. 2). Chua’s circuits starting from random initial conditions
are considered (equations and parameters have been fixed as in [17]). Fig. 2
shows that the nodes in $V_{n_{2}}=\\{1,4,16\\}$ follow the same trajectory,
while the remaining units are not synchronized with them. Similar results are
obtained for the other scenarios.

Figure 2: Evolution of the first state variable for nodes in
$V_{n_{2}}=\\{1,4,16\\}$ (upper plot) and for all the network nodes (bottom
plot).

##  VII Conclusions

In this work we have focused on the problem of controlling synchronization of
a group of nodes in directed networks. The nodes are all assumed to have the
same dynamics and, similarly, coupling is assumed to be fixed to the same
value along all the links of the network. The technique we propose is based on
the use of distributed controllers which add further links to the network or
remove some of the existing ones, creating a new network structure which has
to satisfy two topological conditions. The first condition refers to the fact
that, in the new network, merging the existing links and those of the
controllers, the nodes to control must be outer symmetrical, while the second
condition requires that the out-degree of these nodes has to be higher than a
threshold. Quite interestingly, the threshold depends on the dynamics of the
units and on the coupling strength, in such a way that a higher coupling
strength favors control as it requires a smaller out-degree. It is also worth
noticing that, when the out-degree needs to be increased to exceed the
threshold, this can be obtained by connecting to any of the remaining nodes of
the network.

The selection of the nodes forming the set of successors of the units to
control is carried out by considering an optimization problem and finding the
exact solution that minimizes the cost of the changes (i.e. link additions or
removals). In the case of unitary costs, the problem reduces to minimization
of the number of added or removed links, thereby defining a strategy for the
control of synchronization of a group of nodes in a directed network with
minimal topological changes.

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