# Mean-field and kinetic limit of affine control systems: optimal control
through leaders

Marta Zoppello

###### Abstract

This paper studies a multi-agent system starting from a single agent dynamics
which is a nonlinear affine control system. It analyze what happens when the
number of agents goes to infinity using two different approaches, a granular
mean-field one, trying to control only a finite number of agents while all the
others goes to infinity, and a kinetic one, when in the limit the percentage
of controlled agents is preserved. In both cases the optimal control problem
starting from the solution for the finite dimensional one is stated.

Marta Zoppello111Politecnico di Torino, Corso Duca degli Abruzzi, 24 10129
Torino, Italy<EMAIL_ADDRESS>
##  1 Introduction

In recent years there has been a growing interest in literature in multi-
agents systems [13, 14, 22] for example describing active cells [19, 21],
cooperative robots [10, 20] or animal motion [8, 11, 12]. Most of these models
develop from a single agent dynamics adding an interaction term. The main goal
of these studies is to clarify the relationship between the interplay of
simple binary interaction forces, and the potential emergence of a global
behavior in the form of specific patterns, such as the formation of flocks or
swarms in animal motion.
However for most of the models, the description of the asymptotic behavior of
a very large system of agents can become an hard task. A classical way to
approach the global description of the system is then to focus on its mean
behavior, as in the classical mean-field theory [9]. In certain circumstances,
the formation of a specific pattern is conditional to the position of the
initial datum, i.e if it belongs to a certain basin of attraction we can
characterize certain pattern formation, otherwise we are not able to guarantee
it. Thus it is interesting to wonder whether an external agent can intervene
on the system towards pattern formation, also in those situations where this
phenomenon is not the result of autonomous self-organization. This
intervention may be modeled as an additional control which can be either an
external forcing term or the direct control of some degrees of freedom of some
agents. The standard way, consists in regarding “external forces” as controls,
modeled usually as an additional vector field in the Newton equations, this
problem has been studied for example in [2, 5, 17]. A less explored strategy
is instead to think that the controller acts on the system by directly
assigning the values of the coordinates of some of the agents, regarded as
control parameters. This can happen for example in the collective motion of
the so called robotic locomotion systems, introduced in [15], where even the
single agent dynamics is controlled assigning the evolution of some
coordinates.
In this paper we follow this latter approach focusing our attention on systems
characterized by a binary interaction which depends on the velocity of the
agents. This approach allows us to directly control the velocity of the
coordinates describing some of the agents, ending up with a nonlinear finite
dimensional system affine in these controls. This is the case for example of
spherical particles immersed in a Stokes fluid [18, 23], which can be
considered a first step for modeling a large number of micro-robots swimming
in a highly viscous fluid.
The aim of this paper is to study what happens when the number of agents goes
to infinity. Although we still use a binary interaction approach, we extend
the techniques developed in [2, 17] to systems with a different type of
dynamics, namely nonlinear affine control system. This is a less investigated
type of dynamics, but very interesting for certain applications, for example
micro-robots swimming, swarms, crowd dynamics, vehicular traffic, opinion
formation, wealth distribution and many others, when the interaction between
agents is generated by their velocities.
In this work we analyze two different situations, the fist one in which the
system is composed by a small number of active controlled agents, usually
called leaders [17], that try to influence the dynamics of a huge number of
passive agents, known as followers. This happens for example in opinion
dynamics when a small group of people try to pilot the opinion of a large
crowd, or in biology when only few active cells are able to influence a huge
population. Opinion dynamics in the presence of different populations has been
previously introduced in [6, 7, 24]. We mention here that control through
leaders in self-organized flocking systems has been studied in [1, 25] as well
as in crowd dynamics in [3]. The technical derivation of the mean-field
optimal control is realized by the simultaneous development of the mean-field
limit of the equations governing the followers dynamics, while the leaders
dynamics remains finite dimensional, together with the $\Gamma$-limit of the
finite dimensional optimal control problems. Moreover we also derive the
corresponding first order optimality conditions, resulting in a coupled system
of forward/backward time-dependent ODE’s for the finite dimensional optimal
control problem or PDEs for the infinite dimensional one.
The second situation instead is when the overall number of agents goes to
infinity, preserving the percentage of controlled ones. This is the case when
the number of controls should be a fraction of the total number of agents in
order to be effective. We develop a Boltzmann-type control approach following
the ideas recently presented in [4]. It yields an approximation of the mean-
field dynamics by means of an iterative sampling of 2-agent microscopic
dynamics (binary dynamics). This same principle allows us to generate control
signals for the mean-field model by means of solving optimal control problems
associated to the binary dynamics.

##  2 The finite dimensional system of equations and mathematical
preliminaries

In several mechanical control systems, called “hard-device” systems, the
controls are precisely given by further degrees of freedom and the evolution
of the remaining coordinates can be determined by solving the control
equations. More precisely especially in systems driven by viscous forces, like
for example spherical particles immersed in Stokes fluid, we can suppose to
control directly the velocity of some of them, denoted by “leaders” or “active
agents” , and determine the evolution of the “followers” or“passive agents”
through their interaction with the active ones. This interaction, due to the
viscosity of the system, is linear in the velocity of the active agents,
therefore it is natural that in the equations the control appears both in the
leaders equations and also in the followers one, and turn out to be a finite
dimensional dynamics affine in the $m$ controls [23] . In this paper we focus
on this kind of multi-agents systems in which the individual dynamics is an
“hard-device” control system, the overall finite-dimensional dynamics is in
$d\times(N+m)$ variables, where $N$ is the number of passive agents, $m$ are
the controlled ones, and $d$ is the dimension of the space in which the motion
of such individuals evolves. More precisely the equation of motion of the
$N+m$ agents are

$\small\begin{cases}\dot{Y}_{i}=\mathbf{u}_{i}\qquad i=1,\cdots m\\\
\dot{X}_{i}=\frac{1}{N}\sum_{j=m+1}^{N}H(X_{i}-X_{j})+\sum_{\ell=1}^{d}\frac{1}{m}\sum_{k=1}^{m}G^{\ell}(X_{i}-Y_{k})u_{k\,\ell}\quad
i=1\cdots N\end{cases}$ (1)

where $Y_{i}\in\mathbb{R}^{d}$ is the state of the $i$-th controlled agent,
$X_{i}\in\mathbb{R}^{d}$ represents the state of the $i-$th passive agent and
the control $\mathbf{u}_{i}=(u_{i\,1},\,\cdots,\,u_{i\,d})\in\mathbb{R}^{d}$
is the velocity of the $i$-th active agent for $i=1,\cdots m$ and are
measurable functions of time. Moreover $H$ and $G^{\ell}$ are vectored valued
functions: $H$ represents the interaction between non-controlled agents and
$G^{\ell}$ $\ell=1\cdots d$ account for the interaction that the non-
controlled agent $i$ feels when the $k$-th controlled agent moves with a
velocity $u_{k\ell}$ along the $\ell$-th direction of $\mathbb{R}^{d}$.
Moreover we make the following hypothesis on $H$ and $G^{\ell}$

  * (H)

$H:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and
$G^{\ell}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ for $\ell=1,\cdots,d$ are, bounded,
locally Lipschitz and with sub-linear growth, i.e.

$|H(\xi)|\leq C_{H}(1+|\xi|)\qquad|G^{\ell}(\xi)|\leq
C^{\ell}_{G}(1+|\xi|)\,,$

where $|\cdot|$ is the euclidean norm in $\mathbb{R}^{d}$ and
$\xi=(\xi_{1},\cdots,\xi_{d})$.

Furthermore we are interested in solving the following optimal control problem

$\min_{(\mathbf{u}_{1},\cdots,\mathbf{u}_{m})\in
L^{\infty}([0,T],U)}\int_{0}^{T}\Bigg{(}L(Y(t),X(t))+\sum_{k=1}^{m}|\mathbf{u}_{k}(t)|^{2}\Bigg{)}\,dt\,,$

with $\mathcal{U}\subset B(0,U)\,U>0$ a a fixed nonempty compact subset of
$\mathbb{R}^{md}$ and $(Y,X)$ subjected to the evolution equation (1). For
example

$L(Y(t),X(t)):=\frac{1}{2}\sum_{i=1}^{N}|X_{i}(t)-X^{*}|^{2},$

where $X^{*}$ is a desired configuration of agents, can be used if we want to
minimize the distance from a fixed state configuration.
Our aim is to see what happens to these equations when the number of agents
tents to infinity. We will treat different cases, more precisely

  * a)

Only the number of non-controlled agents tends to infinity and $m$ remains
finite. In this case we use a mixed granular mean-field approach.

  * b)

$N\to\infty$ preserving the percentage of controlled agents. In this case we
will use the kinetic limit which tries to preserve the percentage of
controlled agents

###  2.1 Mathematical preliminaries

In this section we present some basic definitions and some mathematical
results that we will need to prove the main theorems.
In the following we consider the space $\mathcal{P}_{1}(\mathbb{R}^{n})$,
consisting of all probability measures on $\mathbb{R}^{n}$ of finite first
moment. If we denote $\Pi(\mu,\nu)$ the set of probability measures on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$ with first and second marginals equal to
$\mu$ and $\nu$ respectively, then we have that the Wasserstein distance
between the measures $\mu$ and $\nu$ is given by

$\mathcal{W}_{1}(\mu,\nu):=\inf_{\pi\in\Pi(\mu,\nu)}\Bigl{\\{}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}|x-y|d\pi(x,y)\Bigr{\\}}.$
(2)

Let us define the following

$\mu_{N}=\frac{1}{N}\sum_{j=1}^{N}\delta_{X_{j}}\qquad\mu_{m\,\ell}=\frac{1}{m}\sum_{k=1}^{m}u_{k\,\ell}\delta_{Y_{k}},\quad\ell=1,\cdots\ell\,.$
(3)

with $\mathbf{u}\in\mathcal{U}$, a nonempty compact subset of
$\mathbb{R}^{md}$.
Our system of equations (1) can be rewritten as

$\begin{cases}\dot{Y}_{k}=\mathbf{u}_{k}\quad k=1\ldots m\\\
\dot{X}_{i}=H*\mu_{N}(X_{i})+\sum_{\ell=1}^{d}G^{\ell}*\mu_{m\,\ell}(X_{i})\quad
i=1\ldots N\,.\end{cases}$ (4)

Therefore we have the following proposition

######  Proposition 1.

Given $H$ and $G^{\ell}$ $\ell=1,\cdots,d$ satisfying hypothesis (H), and
$\mu_{N}$, $\mu_{m}$ defined in (3), we have

$\displaystyle|H*\mu_{N}(X_{i})|\leq
C_{H}\bigl{(}1+|X_{i}|+\frac{1}{N}\sum_{j=1}^{N}|X_{j}|\bigr{)}\,,$
$\displaystyle\quad|G^{\ell}*\mu_{m}(X_{i})|\leq
C^{\ell}_{G}\bigl{(}1+|X_{i}|+\frac{1}{m}\sum_{k=1}^{m}|u_{k\,\ell}||Y_{k}|\bigr{)}\leq
C^{\ell}_{G}\bigl{(}1+|X_{i}|+\frac{1}{m}\sum_{k=1}^{m}|Y_{k}|\bigr{)}\,.$

###### Proof.

It follows immediately from the sub-linear growth (H) of $H$ and $G^{\ell}$
for $\ell=1,\cdots d$. ∎

Moreover if $\zeta=(X,Y)\in\mathbb{R}^{dN}\times\mathbb{R}^{d\,m}$, we denote
by

$||\zeta||:=||Y||_{m}+||X||_{N}:=\frac{1}{m}\sum_{i=k}^{m}|Y_{k}|+\frac{1}{N}\sum_{i=1}^{N}|X_{i}|\,.$
(5)

We can therefore state the following existence and uniqueness of the solution

######  Proposition 2.

Given a control function $\mathbf{u}=(\mathbf{u}_{1},\ldots\mathbf{u}_{m})\in
L^{\infty}([0,T],\mathbb{R}^{md})$ and an initial datum
$\mathrm{bar}{\zeta}=(\,\mathrm{bar}{X}_{1},\ldots,\mathrm{bar}{X}_{N},\mathrm{bar}{Y}_{1},\ldots,\mathrm{bar}{Y}_{m})$
there exists a unique Caratheodory solution
$\zeta(t)=(X_{1}(t),\ldots,X_{N}(t),\,Y_{1}(t),\ldots Y_{m}(t))$ of (4) such
that

$\int_{0}^{T}||\zeta(t)||\leq(||\mathrm{bar}{\zeta}||+\tilde{C}T)e^{\tilde{C}T}\,.$
(6)

Moreover, the trajectory is Lipschitz continuous in time

$||\zeta(t_{1})-\zeta(t_{2})||\leq\mathcal{L}|t_{1}-t_{2}|\,.$ (7)

###### Proof.

Thanks to the explicit form (4), to Proposition 1 and by the boundedness of
$\mathbf{u}_{k}$ the right-hand side of (4) fulfills the linear growth
condition. Thus the solution is uniquely determined moreover we have

$||\zeta(t_{1})-\zeta(t_{2})||\leq\int_{t_{1}}^{t_{2}}\tilde{C}(1+||\zeta||)\,ds\leq\tilde{C}(1+||\zeta_{0}||+\tilde{C}T)e^{\tilde{C}T}|t_{1}-t_{2}|\,.$

Thus the conclusion follows. ∎

Furthermore we have also the continuous dependence on the data holds also in
the Caratheodory setting

######  Lemma 2.1.

Let $g_{1}$ and $g_{2}$ be Caratheodory functions both satisfying

$|g_{i}(t,y)|\leq C(1+|y|)\qquad i=1,2.$

Let $r>0$ and define

$\rho_{r,m,T}:=\bigl{(}r+CT\bigr{)}e^{CT}\,,$

assume also that $g_{1}$ is Lipschitz function with respect to the $y$
variable, for all $|y_{i}|\leq\rho_{r,m,T}\,,i=1,2$ . Set

$q(t):=||g_{1}(t,\cdot)-g_{2}(t,\cdot)||_{L^{\infty}(B(0,\rho_{r,m,T}))}\,.$

Then if $\dot{y}_{1}=g_{1}(t,y_{1})$ and $\dot{y}_{2}=g_{2}(t,y_{2})$,
$|y_{1}(0)|\leq r$ and $|y_{2}(0)|\leq r$, one has

$y_{1}(t)-y_{2}(t)\leq
e^{Lt}|y_{1}(0)-y_{2}(0)|+\int_{0}^{t}e^{\int_{s}^{t}L\,d\sigma}q(s)\,ds\,.$

###### Proof.

It follows from [17] and Gronwall’s lemma ∎

Here we recall the statement and proof of Lemmas and Theorems concerning
Lipschitz continuity estimates for transport flows induced by Caratheodory
dynamics, that are an adjustment to the ones in [16, 17] to the dynamics (4)
which is affine in the controls. They are preparatory for the main theorems of
the next sections regarding the existence of a solution in the limit for the
number of agents that tend to infinity.

######  Lemma 2.2.

Let $H:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and
$G^{\ell}:\mathbb{R}^{d}\to\mathbb{R}^{d}$, $\ell=1,\cdots,d$ be locally
Lipschitz functions such that

$|H(y)|\leq C_{H}(1+|y|),\qquad|G^{\ell}(y)|\leq
C_{G^{\ell}}(1+|y|)\quad\text{for $\ell=1,\cdots,d$ and for all
$y\in\mathbb{R}^{d}$}\,,$

moreover let $\mu:[0,T]\to\mathcal{P}_{1}(\mathbb{R}^{d})$ be a continuous map
with respect to $\mathcal{W}_{1}$, and
$\mu_{m\,\ell}:[0,T]\to\mathcal{P}_{1}(\mathbb{R}^{d})$, time-dependent
measure supported on the respective absolutely continuous trajectory $t\mapsto
Y_{k}(t)$, i.e
$\mu_{m\,\ell}=\frac{1}{m}\sum_{k=1}^{m}u_{k\,\ell}\,\delta_{Y_{k}}$ with
$u_{k}\in L^{\infty}[0,T]$. Then there exist constants $C^{{}^{\prime}}_{H}$
and $C^{{}^{\prime}}_{G^{\ell}}$ such that

$\displaystyle|H*\mu(t)(y)|\leq C^{{}^{\prime}}_{H}(1+|y|)\,,$
$\displaystyle|G^{\ell}*\mu_{m,\,\ell}(t)(y)|\leq
C^{{}^{\prime}}_{G^{\ell}}(1+|y|)\,\,\text{for $\ell=1,\cdots,d$ and all
$t\in[0,T]$.}$

Furthermore, if

$\mathrm{supp}\,\,\mu(t)\subset B(0,R),$

for all $t\in[0,T]$, then for every compact subset $K\subset\mathbb{R}^{d}$
there exist a constants $L_{{R,K}_{H}}$ $L_{{R,K}_{G^{\ell}}}$ such that

$\displaystyle|H*\mu(t)(y_{1})-H*\mu(t)(y_{2})|\leq
L_{{R,K}_{H}}|y_{1}-y_{2}|\,,$
$\displaystyle|G^{\ell}*\mu_{m,\,\ell}(t)(y_{1})-G^{\ell}*\mu_{m,\,\ell}(t)(y_{2})|\leq
L_{R,K_{G^{\ell}M}}|y_{1}-y_{2}|\,,$

where $M$ is the maximum of the norms of the $\mathbf{u}_{k}$. For every
$t\in[0,T]$ and every $y_{1},\,y_{2}\in K$.

###### Proof.

$\displaystyle|H*\mu(t)(y)|=|$
$\displaystyle\int_{\mathbb{R}^{d}}H(y-\xi)\,d\mu(\xi)|\leq
C_{H}(1+|y|)+\int_{\mathbb{R}^{d}}|\xi|\,d\mu(\xi)$ $\displaystyle\leq
C^{{}^{\prime}}_{H}(1+|y|)\,,$

where in the last inequality we have used the uniform boundedness of the last
term thanks to our continuity assumption. Analogously

$\displaystyle|G^{\ell}*\mu_{m\,\ell}(t)(y)|=|$
$\displaystyle\int_{\mathbb{R}^{d}}G^{\ell}(y-\xi)\,d\mu_{m\,\ell}(\xi)|\leq
C_{G^{\ell}}(1+|y|+\frac{1}{m}\sum_{k=1}^{m}|u_{k\,\ell}||\xi_{k}|)$
$\displaystyle\leq C^{{}^{\prime}}_{G^{\ell}M}(1+|y|)\,,$

where for the last inequality we have used the fact that $u\in L^{\infty}$ and
the uniform boundedness of the sum.
Finally if $\mathrm{supp}\,\,\mu(t)\subset B(0,R),$ we have

$\displaystyle|H*\mu(t)(y_{1})-H*\mu(t)(y_{2})|=|\int_{\mathbb{R}^{d}}\left(H(y_{1}-\xi)-H(y_{2}-\xi)\right)\,d\mu(\xi)|\leq
L_{{R,K}_{H}}|y_{1}-y_{2}|\,,$
$\displaystyle|G^{\ell}*\mu_{m\,\ell}(t)(y_{1})-G^{\ell}*\mu_{m\,\ell}(t)(y_{2})|=|\int_{\mathbb{R}^{d}}\left(G^{\ell}(y_{1}-\xi)-G^{\ell}(y_{2}-\xi)\right)\,d\mu_{m\,\ell}(\xi)|$
$\displaystyle\phantom{|G^{\ell}*\mu_{m}(t)}\leq\frac{1}{m}\sum_{k=1}^{m}|u_{k\,\ell}||G^{\ell}(y_{1}-\xi_{k})-G^{\ell}(y_{2}-\xi_{k})|\leq\frac{1}{m}\sum_{k=1}^{m}|u_{k\,\ell}|L_{R,K_{G^{\ell}}}|y_{1}-y_{2}|$
$\displaystyle\phantom{|G^{\ell}*\mu_{m}(t)(y_{1})|}\leq
L_{{R,K_{G^{\ell}}}}|y_{1}-y_{2}|\,.$

∎

The following regards the regularity with respect to the initial condition

######  Lemma 2.3.

Let $H:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and
$G^{\ell}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ $\ell=1,\cdots,d$ be locally
Lipschitz functions such that

$|H(y|\leq C_{H}(1+|y|),\qquad|G^{\ell}(y)|\leq
C_{G^{\ell}}(1+|y|)\quad\text{for all $y\in\mathbb{R}^{d}$}\,,$

and let $\mu^{1}:[0,T]\to\mathcal{P}_{1}(\mathbb{R}^{d})$ and
$\mu^{2}:[0,T]\to\mathcal{P}_{1}(\mathbb{R}^{d})$ be continuous maps with
respect to $\mathcal{W}_{1}$, both satisfying

$\mathrm{supp}\,\,\mu^{1}(t)\subset
B(0,R),\qquad\mathrm{supp}\,\,\mu^{2}(t)\subset B(0,R),$

and $\mu^{1}_{m}\,,\,\mu^{2}_{m}$ two time-dependent measures supported on the
respective absolutely continuous trajectories $t\mapsto X^{i}_{k}(t)$, i.e
$\mu_{m\,\ell}=\frac{1}{m}\sum_{k=1}^{m}u^{i}_{k\,\ell}\,\delta_{X^{i}_{k}}\,,i=1,2$
with $u^{i}_{k\,\ell}\in L^{\infty}[0,T]$. Consider the flow map
$\mathcal{T}^{\mu^{i},\mu_{m}^{i}},i=1,2$ associated to the system

$\dot{X}=H*\mu^{i}(t)(X(t))+\sum_{\ell=1}^{d}G^{\ell}*\mu^{i}_{m\,\ell}(t)(X(t))\,.$
(8)

Fix $r>0$: then there exist a constant $\rho$, a constant $L>0$ both depending
only on $r,\,C,\,R,\,T$ and
$M=\max_{\ell,k,i}||u^{i^{\ell}}_{k}||_{L^{\infty}}$ such that

$\displaystyle|\mathcal{T}^{\mu^{1},\mu_{m\,\ell}^{1}}_{t}$
$\displaystyle(P_{1})-\mathcal{T}^{\mu^{2},\mu_{m\,\ell}^{2}}_{t}(P_{2})|\leq
e^{Lt}|P_{1}-P_{2}|$
$\displaystyle+\int_{0}^{t}e^{L(t-s)}||H*\mu^{1}(s)-H*\mu^{2}(s)+\sum_{\ell=1}^{d}(G^{\ell}*\mu^{1}_{m\,\ell}(s)-G^{\ell}*\mu^{2}_{m\,\ell}(s))||_{L^{\infty}}\,,$

whenever $|P_{1}|\leq r$ and $|P_{2}|\leq r$.

###### Proof.

Let $g_{i}:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the right hand side
of (8). As in Lemma 2.2 we can find a constant $\mathrm{bar}{C}$ such that

$|H*\mu^{i}(t)(P)|+\sum_{\ell=1}^{d}|G^{\ell}*\mu^{i}_{m\,\ell}(t)(P)|\leq\mathrm{bar}{C}(1+|P|)\,.$

Therefore, for every $P_{1},\,P_{2}\in\mathbb{R}^{d}$ such that $|P_{i}|\leq
r$ applying Lemma 2.1 we have

$|\mathcal{T}^{\mu^{i},\mu_{m\,\ell}^{i}}_{t}(P_{i})|\leq\bigl{(}r+\mathrm{bar}{C}^{{}^{\prime}}T\bigr{)}e^{\mathrm{bar}{C}^{{}^{\prime}}T}\,,$

and setting
$\rho:=\bigl{(}r+\mathrm{bar}{C}^{{}^{\prime}}T\bigr{)}e^{\mathrm{bar}{C}^{{}^{\prime}}T}$
we have

$\displaystyle||g_{1}(t,\cdot)-g_{2}(t,\cdot)||_{L^{\infty}(B(0,\rho))}=$
$\displaystyle||H*\mu^{1}(s)-H*\mu^{2}(s)+$
$\displaystyle\sum_{\ell=1}^{d}(G^{\ell}*\mu^{1}_{m\,\ell}(s)-G^{\ell}*\mu^{2}_{m\,\ell}(s))||_{L^{\infty}(B(0,\rho))}\,.$

∎

The following lemma gives a result on the transportation of a measure by two
different functions

######  Lemma 2.4.

Let $E_{1}$ and $E_{2}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be two borel
measurable functions. Then, for every $\mu\in\mathcal{P}_{1}(\mathbb{R}^{d})$
one has

$\mathcal{W}_{1}((E_{1})_{\sharp}\mu,(E_{2})_{\sharp}\mu)\leq||E_{1}-E_{2}||_{L^{\infty}(\mathrm{supp}\mu)}.$

If in addition $E_{1}$ is locally Lipschitz continuous, and
$\mu,\,\nu\in\mathcal{P}_{1}(\mathbb{R}^{n})$ are both compactly supported on
a ball then

$\mathcal{W}_{1}((E_{1})_{\sharp}\mu,(E_{1})_{\sharp}\nu)\leq
L_{r}\mathcal{W}_{1}(\mu,\nu)\,.$

###### Proof.

It can be found in [17]. ∎

##  3 Mean-field limit

In this section we will derive a system of equations governing the dynamics of
an infinite number of agents when only a finite number $m$ of them is
controlled. In particular, differently from what already done in literature,
we focus of the case when the finite dimensional dynamics turns out to be a
nonlinear affine control system.
As $N\to\infty$ we claim that the ODEs system (4) turns to

$\begin{cases}Y_{k}=\mathbf{u}_{k}\qquad\text{for}\,i=1\ldots m\\\
\partial_{t}\mu+\nabla\cdot\Bigl{[}\bigl{(}H*\mu+\sum_{\ell=1}^{d}G^{\ell}*\mu_{m\,\ell}\bigr{)}\mu\Bigr{]}=0\,.\end{cases}$
(9)

######  Definition 3.1.

Let $u\in L^{\infty}([0,T],\mathcal{U})$ be given. We say that a map
$(Y,\mu):[0,T]\to\chi:=\mathbb{R}^{dm}\times\mathcal{P}_{1}(\mathbb{R}^{d})$
is a solution of the controlled system (9) if

  * •

The measure $\mu$ is equi-compactly supported in time, i.e., there exists
$R>0$ such that $\mathrm{supp}(\mu(t))\subset B(0,R)$ for all $t\in[0,T]$

  * •

the solution is continuous in time with respect to the following metric

$||(Y,\mu)-(Y^{{}^{\prime}},\mu^{{}^{\prime}})||_{\chi}:=\frac{1}{m}\sum_{k=1}^{m}|Y_{k}-Y_{k}^{{}^{\prime}}|+\mathcal{W}_{1}(\mu,\mu^{{}^{\prime}})\,,$

where $\mathcal{W}_{1}$ is the $1$-Wasserstein distance

  * •

The $Y$ coordinates define a Caratheordory solution of the following
controlled problem with control $u$

$\dot{Y}_{k}=\mathbf{u}_{k}\,.$

  * •

The $\mu$ component satisfies

$\frac{d}{dt}\int_{\mathbb{R}^{d}}\varphi(x)\,d\mu(t)(x)=\int_{\mathbb{R}^{d}}\nabla\varphi\cdot\bigl{(}H*\mu(t)(x)+\sum_{\ell=1}^{d}G^{\ell}*\mu_{m\,\ell}(t)(x)\bigr{)}\,d\mu(t)(x)\,,$

for every $\varphi\in C^{\infty}_{c}(\mathbb{R}^{d})$.

Fist of all it will be convenient to address the stability of the system (9).
Let us assume the following:

  * (A)

Let $H:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and
$G^{\ell}:\mathbb{R}^{d}\to\mathbb{R}^{d},\ \ell=1,\cdots d$ be locally
Lipschitz functions with sub-linear growth, let
$\mu:[0,T]\to\mathcal{P}_{1}(\mathbb{R}^{d})$ and
$\nu:[0,T]\to\mathcal{P}_{1}(\mathbb{R}^{d})$ be continuous maps with respect
to $\mathcal{W}_{1}$ both satisfying

$\mathrm{supp}\mu(t)\subset B(0,R)\qquad\mathrm{supp}\nu(t)\subset B(0,R),$

Let $\mathbf{u}\in L^{\infty}$ be fixed and
$\mu_{m\ell}=\frac{1}{m}\sum_{k=1}^{m}u_{k\ell}\delta_{Y_{k}}$ and
$\mu^{{}^{\prime}}_{m\ell}=\frac{1}{m}\sum_{k=1}^{m}u_{k\ell}^{\prime}\delta_{Y^{{}^{\prime}}_{k}}$
time dependent measures supported on the respective absolutely continuous
trajectories $t\mapsto Y_{k}(t)$ and $t\mapsto Y_{k}^{\prime}(t)$. Assume that
for every $\rho>0$ there exist constants $L_{{\rho,R}_{H}}$ and
$L_{{\rho,R,M}_{G}}$ such that

$\displaystyle||H*\mu(t)-H*\nu(t)||_{L^{\infty}(B(0,\rho))}\leq
L_{{\rho,R}_{H}}\mathcal{W}_{1}(\mu(t),\nu(t))\,,$
$\displaystyle||G^{\ell}*\mu_{m\,\ell}(t)-G^{\ell}*\mu^{{}^{\prime}}_{m\,\ell}(t)||_{L^{\infty}(B(0,\rho))}\leq
L_{{\rho,R,M}_{G^{\ell}}}\frac{1}{m}\sum_{k=1}^{m}|Y_{k}-Y^{{}^{\prime}}_{k}|\,,$

where $M$ is the maximum of the absolute value of $u_{k}$ and
$u_{k}^{\prime}$.

######  Remark 1.

Note that the first inequality in Hypothesis (A) is satisfied for all
probability measures $\mu$ and $\nu$ absolutely continuous with respect to
$\mathcal{W}_{1}$ and with compact support (see [17]). Moreover the second
inequality it is satisfied for example when the controls are positive and
their sum is one, namely if $\mu_{m\,\ell}$ and $\mu_{m\,\ell}^{\prime}$ are
probability measures.

######  Theorem 3.2.

Suppose that Hypothesis (A) holds and let $u\in L^{\infty}([0,T],\mathcal{U})$
be a given fixed control for (9), and two solutions $(Y^{1},\mu^{1})$ and
$(Y^{2},\mu^{2})$ of (9) relative to the control $u$ and given respective
initial data $(Y^{0,i},\mu^{0,i})\in\chi$ with $\mu^{0,i}$ compactly
supported, $i=1,2$. Then there exists a constant $C_{T}>0$ such that

$||(Y^{1},\mu^{1})-(Y^{2},\mu^{2})||_{\chi}\leq
C_{T}||(Y^{0,1},\mu^{0,1})-(Y^{0,2},\mu^{0,2})||_{\chi}\,.$

###### Proof.

First of all, since the control $u$ is the same for both solutions we have
that

$|Y^{1}_{k}(t)-Y^{2}_{k}(t)|\leq|Y^{0,1}_{k}-Y^{0,2}_{k}|$

Thus let us focus on the $\mu$-part of the solution. Given
$\mu^{1}_{m},\,\mu^{2}_{m}$, in view of the representation of the solution by
means of mass transportation there exist constants $\mathcal{L}>0,\,L_{R}>0$
and $\rho>0$ such that

$\displaystyle\mathcal{W}_{1}(\mu^{1}(t),\mu^{2}(t))$
$\displaystyle\leq\mathcal{W}_{1}((\mathcal{T}^{\mu^{1},\mu_{m}^{1}}_{t})_{\sharp}\mu^{0,1},(\mathcal{T}^{\mu^{2},\mu_{m}^{2}}_{t})_{\sharp}\mu^{0,2})$
$\displaystyle\leq\mathcal{W}_{1}((\mathcal{T}^{\mu^{1},\mu_{m}^{1}}_{t})_{\sharp}\mu^{0,1},(\mathcal{T}^{\mu^{1},\mu_{m}^{1}}_{t})_{\sharp}\mu^{0,2})+\mathcal{W}_{1}((\mathcal{T}^{\mu^{1},\mu_{m}^{1}}_{t})_{\sharp}\mu^{0,2},(\mathcal{T}^{\mu^{2},\mu_{m}^{2}}_{t})_{\sharp}\mu^{0,2})$
$\displaystyle\leq\mathcal{L}\mathcal{W}_{1}(\mu^{0,1},\mu^{0,2})+\mathcal{W}_{1}((\mathcal{T}^{\mu^{1},\mu_{m}^{1}}_{t})_{\sharp}\mu^{0,2},(\mathcal{T}^{\mu^{2},\mu_{m}^{2}}_{t})_{\sharp}\mu^{0,2})$
$\displaystyle\leq\mathcal{L}\mathcal{W}_{1}(\mu^{0,1},\mu^{0,2})+||(\mathcal{T}^{\mu^{1},\mu_{m}^{1}}_{t})-\mathcal{T}^{\mu^{2},\mu_{m}^{2}}_{t})||_{L^{\infty}(B(0,R))}$
$\displaystyle\leq\mathcal{L}\mathcal{W}_{1}(\mu^{0,1},\mu^{0,2})+\int_{0}^{t}e^{\mathcal{L}(t-s)}||H*\mu^{1}(s)-H*\mu^{2}(s)$
$\displaystyle\phantom{\leq\mathcal{L}\mathcal{W}_{1}(\mu^{0,1},\mu^{0,2})+}\qquad+\sum_{\ell=1}^{3}(G^{\ell}*\mu^{1}_{m\,\ell}(s)-G^{\ell}*\mu^{2}_{m\,\ell}(s))||_{L^{\infty}(B(0,\rho))}\,ds$
$\displaystyle\leq\mathcal{L}\mathcal{W}_{1}(\mu^{0,1},\mu^{0,2})+L_{R,M}\int_{0}^{t}e^{\mathcal{L}(t-s)}\Bigl{(}\frac{1}{m}\sum_{k=1}^{m}|Y^{1}_{k}(s)-Y^{2}_{k}(s)|$
$\displaystyle\phantom{\leq\mathcal{L}\mathcal{W}_{1}(\mu^{0,1},\mu^{0,2})+}\qquad+\mathcal{W}_{1}(\mu^{1}(s)-\mu^{2}(s))\Bigr{)}\,ds\,,$

where we first applied the triangular inequality, in the second inequality we
used the Lipschitz continuity of the map
$\mathcal{T}_{t}^{\mu^{1},\mu_{m}^{1}}$ given by Lemma 2.3 for
$\mu^{1}=\mu^{2}$ and $\mu_{m}^{1}=\mu_{m}^{2}$ and Lemma 2.4 also for the
third inequality. Fourth inequality is ahgain a consequence of Lemma 2.3, and
the last one due to hypotesys (A). Combining all together we have

$\displaystyle||(Y^{1},\mu^{1})-(Y^{2}-\mu^{2})||_{\chi}\leq$ $\displaystyle
C_{0}\Big{(}||(Y^{0,1},\mu^{0,1})-(Y^{0,2},\mu^{0,2})||_{\chi}+$
$\displaystyle\int_{0}^{t}||(Y^{1},\mu^{1})-(Y^{2},\mu^{2})||_{\chi}\,ds\Big{)}\,,$

for a suitable constant $C_{0}$ depending on $\mathcal{L},\,L_{R,M},\,T$. ∎

We will now derive the existence of solution of (9) by a limit process for
$N\to\infty$ where we allow for a variable control $\mathbf{u}^{N}$ depending
on $N$

######  Theorem 3.3.

Assume Hypothesis (A) holds and et $(Y^{0},\mu^{0})\in\chi$ be given, with
$\mu^{0}$ of bounded support in $B(0,R)$. Define a sequence
$(\mu^{0}_{N})_{N\in\mathbb{N}}$ of atomic probability measures equi-compactly
supported in $B(0,R)$, such that each $\mu^{0}_{N}$ is given by
$\mu^{0}_{N}=\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}}$ and
$\lim_{N\to\infty}\mathcal{W}_{1}(\mu^{0}_{N},\mu^{0})=0$. Fix now a
convergent sequence $(\mathbf{u}_{N})_{N\in\mathbb{N}}\subset
L^{\infty}([0,T],\mathcal{U})$ of control functions, i.e.
$\mathbf{u}_{N}\to\mathbf{u}^{*}$ in $L^{\infty}([0,T],\mathcal{U})$, with
$\mathcal{U}$ a nonempty compact subset of $\mathbb{R}^{md}$. For each initial
datum $\zeta^{0}_{N}=(Y^{0},X^{0}_{N})$, denote with
$\zeta_{N}(t)=(Y_{N},\mu_{N}):=(Y_{N},X_{N})$ the unique solution of the
finite dimensional problem (4) with control $\mathbf{u}_{N}$. Then the
sequence $(Y_{N},\mu_{N})$ converges $C^{0}([0,T],\chi)$ to some
$(Y^{*},\mu^{*})$ which is a solution of (9) with initial data
$(Y^{0},\mu^{0})$ and control $\mathbf{u}^{*}$ in the sense of Definition 3.1.

###### Proof.

Since the initial data $\zeta^{0}_{N}$ are equi-compactly supported, taking
into account Hypothesis (A), and the properties (6) and (7), the trajectories
$\zeta_{N}=(Y_{N},\mu_{N})$ are equibounded and equi-Lipschitz continuous in
$C^{0}([0,T],\chi)$. Thus, by an application of the ascoli-Arzelá theorem for
functions on $[0,T]$ and values in the complete space $\chi$, there exists a
subsequence $\zeta_{N}=(Y_{N},\mu_{N})$ converging uniformly to a limit
$\zeta^{*}=(Y^{*},\mu^{*})$, which is also equi-compactly supported in a ball
$B(0,R_{T})$.
We now show that $\zeta^{*}$ is a solution of (9) in the sense of Definition
3.1. First of all we observe that $\zeta_{N}\to\zeta^{*}$ implies that $Y_{N}$
converges uniformly to $Y^{*}$ and $\mu_{N}$ converges to $\mu^{*}$ in the
Wasserstein metric, i.e.

$Y_{N}\rightarrow
Y^{*}\qquad\text{and}\qquad\mathcal{W}_{1}(\mu_{N}(t),\mu^{*}(t))=0\,.$

Thus the first equation is easily satisfied by the uniform convergence of the
controls. We are now left with verifying that $\mu^{*}$ is a solution of the
second equation of (9). To this end, let us denote

$\mu_{m\ell,N}=\frac{1}{m}\sum_{k=1}^{m}u_{k\ell,N}\delta_{Y_{k,N}}\qquad\text{and}\qquad\mu_{m\ell,*}=\frac{1}{m}\sum_{k=1}^{m}u_{k\,\ell}^{*}\delta_{Y_{k}^{*}}$

For all $\hat{t}\in[0,T]$and for all $\phi\in C^{1}_{C}(\mathbb{R}^{d})$ we
say that

$\langle\phi,\mu_{N}(\hat{t})-\mu_{N}(0)\rangle=\int_{0}^{\hat{t}}\Big{[}\int_{\mathbb{R}^{d}}\nabla\phi(x)\cdot
w_{H,G,\mu_{N},Y_{N}}(t,x)\,d\mu_{N}(t,x)\Big{]}\,dt\,,$

which is verified considering the differentiation

$\frac{d}{dt}\langle\phi,\mu_{N}(t)\rangle=\frac{1}{N}\frac{d}{dt}\sum_{i=1}^{N}\phi(X_{i}(t))=\frac{1}{N}\sum_{i=1}^{N}\nabla\phi(X_{i}(t))\cdot\dot{X}_{i}(t)\,,$

and applying the substitution for the followers variables $X$ as in (4).
Moreover

$\lim_{N\to\infty}\langle\phi,\mu_{N}(\hat{t})-\mu_{N}(0)\rangle=\langle\phi,\mu_{*}(\hat{t})-\mu^{0}\rangle\,,$
(10)

for all $\phi\in C^{1}_{C}(\mathbb{R}^{d})$, and by Assumption (A) we have
that for every $\rho>0$

$\lim_{N\to\infty}||(H*\mu_{N}(t)+\sum_{\ell=1}^{d}G^{\ell}*\mu_{m\ell,N}(t))-(H*\mu_{*}(t)+\sum_{\ell=1}^{d}G^{\ell}*\mu_{m\ell,*}(t))||_{L^{\infty}(B(0,\rho))}=0.$

As $\phi\in C^{1}_{C}(\mathbb{R}^{d})$ has compact support, it follows that

$\lim_{N\to\infty}||\nabla\phi\cdot\Big{(}(H*\mu_{N}(t)+\sum_{\ell=1}^{d}G^{\ell}*\mu_{m\ell,N}(t))-(H*\mu_{*}(t)+\sum_{\ell=1}^{d}G^{\ell}*\mu_{m\ell,*}(t))\Big{)}||_{\infty}=0.$

Denoting with $\Lambda^{1}_{[0,\hat{t}]}$ the lebesgue measure on
$[0,\hat{t}]$, since the product measures
$\Lambda^{1}_{[0,\hat{t}]}\times\frac{1}{\hat{t}}\mu_{N}(t)$ converge in
$\mathcal{P}_{1}([0,\hat{t}]\times\mathbb{R}^{d})$ to
$\Lambda^{1}_{[0,\hat{t}]}\times\frac{1}{\hat{t}}\mu_{*}(t)$ we finally get

$\displaystyle\lim_{N\to\infty}$
$\displaystyle\int_{0}^{\hat{t}}\int_{\mathbb{R}^{d}}\Big{(}\nabla\phi(x)\cdot\bigl{(}H*\mu_{N}(t)+\sum_{\ell=1}^{d}G^{\ell}*\mu_{m\ell,N}(t)\bigr{)}\Big{)}\,d\mu_{N}(t)\,dt$
(11)
$\displaystyle=\int_{0}^{\hat{t}}\int_{\mathbb{R}^{d}}\Big{(}\nabla\phi(x)\cdot\bigl{(}H*\mu_{*}(t)+\sum_{\ell=1}^{d}G^{\ell}*\mu_{m\ell,*}(t)\bigr{)}\Big{)}\,d\mu_{*}(t)\,dt\,.$

The thesis follows combining (10) and (11). ∎

###  3.1 Mean field optimal control

In this section we focus on a mean field optimal control for the dynamic
system with a finite number of controlled agents, in particular when the
finite dynamics is a nonlinear affine control system. Here we will give the
solution to the infinite optimal control problem as the limit, for the number
of followers that goes to infinity, of the solution of the finite dimensional
one.

####  3.1.1 The finite dimensional optimal control problem

In the space $\mathbb{X}:=\mathbb{R}^{dm}\times\mathbb{R}^{dN}$ equipped with
the norm in (5), given an initial datum
$(Y_{1}(0),\cdots,Y_{m}(0),X_{1}(0),\cdots,X_{N}(0))$, we consider the
following optimal control problem

$\min_{(\mathbf{u}_{1},\cdots,\mathbf{u}_{m})\in
L^{\infty}([0,T],\mathcal{U})}\int_{0}^{T}\Bigg{(}L(Y(t),X(t))+\sum_{k=1}^{m}|\mathbf{u}_{k}(t)|^{2}\Bigg{)}\,dt\,,$
(12)

with $L:\mathbb{X}\to\mathbb{R}$ a continuous function with respect to the
distance induced on $\mathbb{X}$ by the norm (5), for example

$L(Y(t),X(t)):=\frac{1}{2}\sum_{i=1}^{N}|X_{i}(t)-X^{*}|^{2},$ (13)

where $X^{*}$ is a desired configuration of agents, can be used if we want to
minimize the distance from a fixed state configuration. With this particular
choice of L, it does not depend on $Y(t)$ since we are mainly interested in
getting closer to a fixed configuration $X^{*}$ only in the $X$ variables,
since the $Y$ ones are a finite number and directly controlled.
Recall that $(Y,X)\in\mathbb{R}^{dm}\times\mathbb{R}^{dN}$ are governed by the
following affine control equation

$\begin{cases}\dot{Y}_{k}=\mathbf{u}_{i}\quad k=1\ldots m\\\
\dot{X}_{i}=H*\mu_{N}(X_{i})+\sum_{\ell=1}^{d}G^{\ell}*\mu_{m\,\ell}(X_{i})\quad
i=1\ldots N\,,\end{cases}$ (14)

thus we want to find the optimal control that steers the system as close as
possible to the desired configuration of the followers $X^{*}$.
First of all we need to prove the existence of a solution.

######  Theorem 3.4.

The optimal control problem (12)-(14) with initial datum
$(Y_{1}(0),\cdots,Y_{m}(0),X_{1}(0),\cdots,X_{N}(0))$ has a solution.

###### Proof.

Using the fact that a Caratheodory solution of (14) exists, the fact that the
controls are bounded and the continuity of $L$ in $\mathbb{X}$ the statement
follows from classical tools in optimal control (Filippov theorem) and the
idea of the proof is similar to the one [17] ∎

The solution of the finite dimensional optimal control problem (12),(14) can
be characterized in the following way.
Let us denote by $Z=H^{1}([0,T],\mathbb{R}^{dm})\times
H^{1}([0,T],\mathbb{R}^{dN}$ the state space of system (14). Further we define
$W=L^{2}([0,T],\mathbb{R}^{dm})\times L^{2}([0,T],\mathbb{R}^{dN})$.

######  Theorem 3.5.

Let
$(\mathrm{bar}Y_{1},\cdots,\mathrm{bar}Y_{m},\mathrm{bar}X_{1},\cdots,\mathrm{bar}X_{N},\mathrm{bar}\mathbf{u})$
be an optimal solution for the optimal control problem (12),(14). Then there
exist $(\xi_{Y},\xi_{X})\in W$ such that the the first order optimality
conditions read

$\displaystyle\mathrm{bar}u_{k\,\ell}=\mathrm{Proj}_{\
\mathcal{U}}\Big{(}-\mathrm{bar}\xi_{X_{i}}\cdot(G^{\ell}*\tilde{\mu}_{m})-\mathrm{bar}\xi_{Y_{k}}^{\ell}\Big{)}\quad\text{with}\quad\tilde{\mu}_{m}=\sum_{k=1}^{m}\delta_{\mathrm{bar}Y_{k}}$
(15)
$\displaystyle\begin{cases}\dot{\mathrm{bar}Y}_{k}=\mathrm{bar}\mathbf{u}_{k}\quad
k=1\ldots m\\\
\dot{\mathrm{bar}X}_{i}=H*\mu_{N}(\mathrm{bar}X_{i})+\sum_{\ell=1}^{d}(G^{\ell}*\mu_{m\,\ell})(\mathrm{bar}X_{i})\quad
i=1\ldots N\\\ \mathrm{bar}Y(0)=\mathrm{bar}Y_{0}\\\
\mathrm{bar}X(0)=\mathrm{bar}X_{0}\end{cases}$ (16)
$\displaystyle\begin{cases}\dot{\mathrm{bar}\xi}_{Y_{k}}=\mathrm{bar}\xi_{X_{i}}\cdot\partial_{Y_{k}}\Big{[}\sum_{\ell=1}^{d}(G^{\ell}*\mu_{m\,\ell})(\mathrm{bar}X_{i})\Big{]}\\\
\dot{\mathrm{bar}\xi}_{X_{i}}=|\mathrm{bar}X_{i}-X^{*}|+\mathrm{bar}\xi_{X_{i}}\cdot\partial_{X_{i}}\Big{[}(H*\mu_{N})(\mathrm{bar}X_{i})+\sum_{\ell=1}^{d}(G^{\ell}*\mu_{m\,\ell})(\mathrm{bar}X_{i})\Big{]}\,,\end{cases}$
(17)

supplemented by the terminal conditions $\mathrm{bar}{\xi}_{X_{i}}(T)=0$,
$\mathrm{bar}{\xi}_{Y_{k}}(T)=0$.

###### Proof.

Let $(\mathrm{bar}\xi_{X},\mathrm{bar}\xi_{Y},\eta_{X},\eta_{Y})\in
W\times(\mathbb{R}^{dm}\times\mathbb{R}^{dN})$ denote the Lagrange
multipliers, which are in fact the adjoint states. Then the extended Lagrange
functional corresponding to the optimal control problem (12),(14) is

$\displaystyle\int_{0}^{T}$
$\displaystyle\tilde{L}(\mathrm{bar}Y_{k},\mathrm{bar}X_{i},\mathbf{u},\mathrm{bar}\xi_{X_{i}},\mathrm{bar}\xi_{Y_{k}},\eta_{X},\eta_{Y})\,dt=\int_{0}^{T}\Bigg{(}L(\mathrm{bar}Y,\mathrm{bar}X)+\sum_{k=1}^{m}|\mathrm{bar}\mathbf{u}_{k}(t)|^{2}+$
(18)
$\displaystyle\sum_{i=1}^{N}\Big{\langle}\mathrm{bar}\xi_{X_{i}},\dot{\mathrm{bar}X}_{i}-(H*\mu_{N}(\mathrm{bar}X_{i})+\sum_{\ell=1}^{d}G^{\ell}*\mu_{m\,\ell}(\mathrm{bar}X_{i}))\Big{\rangle}$
$\displaystyle+\sum_{k=1}^{m}\Big{\langle}\mathrm{bar}\xi_{Y_{k}},\dot{\mathrm{bar}Y}_{k}-\mathrm{bar}\mathbf{u}_{k}\Big{\rangle}\Bigg{)}\,dt+\eta_{X}\cdot(\mathrm{bar}X(0)-\mathrm{bar}X_{0})+\eta_{Y}\cdot(\mathrm{bar}Y(0)-\mathrm{bar}Y_{0}).$

As usual the first order optimality conditions are derived solving

$d\tilde{L}(\mathrm{bar}Y_{k},\mathrm{bar}X_{i},\mathrm{bar}\mathbf{u},\mathrm{bar}\xi_{X_{i}},\mathrm{bar}\xi_{Y_{k}},\eta_{X},\eta_{Y})=0.$

The derivative with respect to the adjoint state results in the state
equations, while the derivative with respect to the state gives the adjoint
system, while the optimality condition is obtained by the derivative with
respect to the control.
Indeed formal calculation give that for any
$\mathbf{h}=(h_{X},h_{Y},h_{\mathbf{u}})$

$\displaystyle
d_{\mathrm{bar}u_{k}^{\ell}}\tilde{L}[h_{u_{k}^{\ell}}]=\int_{0}^{T}\Big{(}2\mathrm{bar}u_{k}^{\ell}+\mathrm{bar}\xi_{X_{i}}\cdot(G^{\ell}*\tilde{\mu}_{m})(\mathrm{bar}X_{i})+\mathrm{bar}\xi_{Y_{k}}^{\ell}\Big{)}h_{u_{k}^{\ell}}\,dt,$
(19) $\displaystyle
d_{\mathrm{bar}X_{i}}\tilde{L}[h_{X_{i}}]=\int_{0}^{T}\frac{dh_{X_{i}}}{dt}\cdot\mathrm{bar}\xi_{X_{i}}+|\mathrm{bar}X_{i}-X^{*}|+$
(20)
$\displaystyle\phantom{d_{\mathrm{bar}X_{i}}\tilde{L}[h_{X_{i}}]=}\mathrm{bar}\xi_{X_{i}}\cdot\partial_{\mathrm{bar}X_{i}}\Big{[}(H*\mu_{N})(\mathrm{bar}X_{i})+\sum_{\ell=1}^{d}(G^{\ell}*\mu_{m\,\ell})(\mathrm{bar}X_{i})\Big{]}h_{X_{i}}\,dt+h_{X_{i}}(0)\cdot\eta_{X_{i}},$
$\displaystyle
d_{\mathrm{bar}Y_{k}}\tilde{L}[h_{Y_{k}}]=\int_{0}^{T}\frac{dh_{Y_{k}}}{dt}\cdot\mathrm{bar}\xi_{Y_{k}}+\mathrm{bar}\xi_{X_{i}}\cdot\partial_{\mathrm{bar}Y_{k}}\Big{[}\sum_{\ell=1}^{d}(G^{\ell}*\mu_{m\,\ell})(\mathrm{bar}X_{i})\Big{]}h_{Y_{k}}\,dt+h_{Y_{k}}(0)\cdot\eta_{Y_{k}}.$
(21)

Integrating by parts, taking into account the terminal conditions
$\mathrm{bar}{\xi}_{X_{i}}(T)=0$, $\mathrm{bar}{\xi}_{Y_{k}}(T)=0$ and that
the adjoint variables are in $H^{1}$ we get the strong formulation of the
adjoint system. ∎

Thus we have proved what are the necessary conditions that the optimal
solution should satisfy.

####  3.1.2 The infinite dimensional optimal control problem

We now consider the infinite dimensional version of the cost functional (12)

$\min_{(\mathbf{u}_{1},\cdots,\mathbf{u}_{m})\in
L^{\infty}([0,T],\mathcal{U})}\int_{0}^{T}\Bigg{(}\mathcal{L}(Y(t),\mu(t))+\sum_{k=1}^{m}|\mathbf{u}_{k}(t)|^{2}\Bigg{)}\,dt\,,$
(22)

with

(L) $\mathcal{L}$ a nonnegative continuous function such that if
$(\mu_{j})_{j\in\mathbb{N}}$ is a sequence converging narrowly to $\mu$ then
$\mathcal{L}(Y,\mu_{j})\to\mathcal{L}(Y,\mu)$ uniformly.

For example

$\mathcal{L}(Y(t),\mu):=\frac{1}{2}\int_{\mathbb{R}^{3}}|x-x^{*}|^{2}\mu(x,t)\,dx\,,$
(23)

if in the finite dimensional case it is (13), where $x^{*}$ represent the
desired state.
The optimization problem should be solved under the dynamic constraint

$\begin{cases}Y_{k}=\mathbf{u}_{k}\qquad\text{for}\,i=1\ldots m\\\
\partial_{t}\mu+\nabla\cdot\Bigl{[}\bigl{(}H*\mu+\sum_{\ell=1}^{d}G^{\ell}*\mu_{m\,\ell}\bigr{)}\mu\Bigr{]}=0\,,\end{cases}$
(24)

where $\mu_{m\,\ell}=\frac{1}{m}\sum_{k=1}^{m}\delta_{Y_{k}}u_{k\,\ell}$. This
is nothing but the infinite dimensional version of the finite dimensional
optimal control problem in which we are trying to minimize the distance from a
fixed configuration.

We can now state the convergence result both for the functionals and for the
solutions.

######  Theorem 3.6.

Let $H$ and $G^{\ell}$, $\ell=1,\cdots d$ satisfying assumption (H) and
$\mathcal{L}$ satisfying assumption (L). Given an initial datum
$(Y^{0},\mu^{0})\in\chi$ and an approximating sequence $\mu^{0}_{N}$, with
$\mu^{0},\,\mu^{0}_{N}$ equicompactly supported for all $N\in\mathbb{N}$, then
the sequence of functionals

$J_{N}(\mathbf{u})=\int_{0}^{T}\Bigg{(}\mathcal{L}(Y_{N}(t),\mu_{N}(t))+\sum_{k=1}^{m}|\mathbf{u}_{k}(t)|^{2}\Bigg{)}\,dt\,,$

$\Gamma-$converges to the functional

$J(\mathbf{u})=\int_{0}^{T}\Bigg{(}\mathcal{L}(Y(t),\mu(t))+\sum_{k=1}^{m}|\mathbf{u}_{k}(t)|^{2}\Bigg{)}\,dt.$

###### Proof.

Let us start by the $\Gamma-\liminf$ condition. Let us fix a sequence of
controls $\mathbf{u}_{N}\to\mathbf{u}^{*}$ in $L^{\infty}$. As done in the
proof of Theorem 3.3 we can associate with each of these controls a sequence
of solutions $\zeta_{N}(t)=(Y_{N}(t),\mu_{N}(t)):=(Y_{N}(t),X_{N}(t))$ of
(14), uniformly converging to a solution $\zeta^{*}(t)=(Y^{*}(t),\mu^{*}(t))$
of (24) in the sense of definition 3.1 with control $\mathbf{u}^{*}$ and
initial datum $(Y^{0},\mu^{0})$. Since solutions $\zeta_{N}$ and $\zeta^{*}$
have supports uniformly bounded with respect to $N$ and by the uniform
convergence of trajectories $Y_{N}(t)\to Y^{*}(t)$ in $L^{\infty}$ and
$\mathcal{W}_{1}(\mu_{n}(t),\mu(t))\to 0$, from condition (L) it follows that

$\lim_{N\to+\infty}\int_{0}^{T}\mathcal{L}(Y_{N}(t),\mu_{N}(t))=\int_{0}^{T}\mathcal{L}(Y^{*}(t),\mu^{*}(t)).$
(25)

Moreover by the $L^{\infty}$ convergence of $\mathbf{u}_{N}$ to
$\mathbf{u}^{*}$ ve have

$\liminf_{N\to+\infty}\int_{0}^{T}\sum_{k=1}^{m}|\mathbf{u}_{k,N}(t)|^{2}\,dt\geq\int_{0}^{T}\sum_{k=1}^{m}|\mathbf{u}_{k}^{*}(t)|^{2}\,dt\,.$

Thus we get

$\liminf_{N\to+\infty}J_{N}(\mathbf{u}_{N})\geq J(\mathbf{u}^{*})\,.$

We now want to address the $\Gamma-\limsup$ condition. Let us fix
$\mathbf{u}^{*}$ and consider the trivial sequence
$\mathbf{u}_{N}\equiv\mathbf{u}^{*}$ for all $N\in\mathbb{N}$. As we have done
before we can associate with each of these controls a sequence of solutions
$\zeta_{N}(t)=(Y_{N}(t),\mu_{N}(t)):=(Y_{N}(t),X_{N}(t))$ of (14) uniformly
converging to $\zeta^{*}(t)=(Y^{*}(t),\mu^{*}(t))$ of (24) with control
$\mathbf{u}^{*}$ and initial datum $(Y^{0},\mu^{0})$. Therefore we can
similarly conclude the limit (25). In addition, since the sequence
$\mathbf{u}_{N}$ is a constant sequence we have

$\liminf_{N\to+\infty}\int_{0}^{T}\sum_{k=1}^{m}|\mathbf{u}_{k,N}(t)|^{2}\,dt\geq\int_{0}^{T}\sum_{k=1}^{m}|\mathbf{u}_{k}^{*}(t)|^{2}\,dt\,.$

Combyining the two limits we easily get

$\limsup_{N\to+\infty}J_{N}(\mathbf{u}_{N})=\lim_{N\to+\infty}J_{N}(\mathbf{u}^{*})=J(\mathbf{u}^{*}).$

∎

######  Corollary 1.

Let $H$ and $G^{\ell}$ $\ell=1,\cdots d$ satisfying assumption (H) and
$\mathcal{L}$ satisfy hypothesis (L). Given an initial datum
$(Y^{0},\mu^{0})\in\chi$ with $\mu^{0}$ compactly supported, the optimal
control problem

$\min_{(\mathbf{u}_{1},\cdots,\mathbf{u}_{m})\in
L^{\infty}([0,T],\mathcal{U})}\int_{0}^{T}\Bigg{(}\mathcal{L}(Y(t),\mu(t))+\sum_{k=1}^{m}|\mathbf{u}_{k}(t)|^{2}\Bigg{)}\,dt\,,$

has solution, where $(Y,\mu)$ defines the solution of (24) with initial datum
$(Y^{0},\mu^{0})$ and control $\mathbf{u}$. Moreover this solution can be
built as limits $\mathbf{u}^{*}$ of sequences of optimal controls
$\mathbf{u}_{N}^{*}$ of the finite dimensional problems

$\min_{(\mathbf{u}_{1},\cdots,\mathbf{u}_{m})\in
L^{\infty}([0,T],\mathcal{U})}\int_{0}^{T}\Bigg{(}\mathcal{L}(Y_{N}(t),\mu_{N}(t))+\sum_{k=1}^{m}|\mathbf{u}_{k}(t)|^{2}\Bigg{)}\,dt\,,$

where $\mu_{N}=\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i}_{N}}$ and
$\mu_{m,N}^{\ell}=\frac{1}{m}\sum_{k=1}^{m}u_{k}^{\ell}\delta_{Y_{k}}$ are
measures supported on the trajectories of the system

$\begin{cases}\dot{Y}_{k}=\mathbf{u}_{k}\quad k=1\ldots m\\\
\dot{X}_{i}=H*\mu_{N}(X_{i})+\sum_{\ell=1}^{d}(G^{\ell}*\mu_{m\,\ell})(X_{i})\quad
i=1\ldots N\,,\end{cases}$ (26)

with initial datum $(Y^{0},X^{0}_{N})$, control $\mathbf{u}$ and $\mu^{0}_{N}$
is such that $\mathcal{W}_{1}(\mu^{0}_{N},\mu^{0})\to 0$ for $N\to+\infty$.

###### Proof.

First of all note that the optimal controls $\mathbf{u}^{*}_{N}$ of the finite
dimensional optimal control problem belong to $L^{\infty}([0,T],\mathcal{U})$
hence $\mathbf{u}_{N}^{*}$ admits a subsequence converging to some
$\mathbf{u}^{*}\in L^{\infty}([0,T],\mathcal{U})$. As we have done previously
in the proof of Theorem 3.3, we can associate with each of these controls a
sequence of solutions $\zeta_{N}(t)=(Y_{N}(t),\mu_{N}(t))$ of (26) uniformly
converging to a solution $\zeta^{*}(t)=(Y(t),\mu(t))$ in the sense of
Definition 3.1 with control $\mathbf{u}^{*}$. In order to conclude that
$\mathbf{u}^{*}$ is an optimal control we need to show that it is actually a
minimizer of $J$. Let $\mathbf{u}$ be an arbitrary control and
$\mathbf{u}_{N}$ be a recovery sequence given by the $\Gamma-\limsup$
condition, so that

$J(\mathbf{u})\geq\limsup_{N\to+\infty}J_{N}(\mathbf{u}_{N})\,.$

By using the optimality of $\mathbf{u}_{N}^{*}$ we have

$\limsup_{N\to+\infty}J_{N}(\mathbf{u}_{N})\geq\limsup_{N\to+\infty}J_{N}(\mathbf{u}_{N}^{*})\geq\liminf_{N\to+\infty}J_{N}(\mathbf{u}_{N}^{*})\,,$

and applying the $\Gamma-\liminf$ condition we get

$\liminf_{N\to+\infty}J_{N}(\mathbf{u}_{N}^{*})\geq
J(\mathbf{u}^{*})\,\,\Rightarrow\,\,J(\mathbf{u})\geq J(\mathbf{u}^{*}).$

∎

After proving the convergence of the optimal solution for the finite
dimensional optimal control problem to the infinite dimensional one, we are
interested in finding necessary conditions that the optimal solution should
satisfy. Also in this case we start from the necessary conditions for the
finite dimensional one recovering then the infinite dimensional ones.
Let us denote by $\mathcal{Z}=H^{1}([0,T],\mathbb{R}^{dm})\times
H^{1}([0,T],L^{2}(\mathbb{R}^{dN}))$ the state space of system (24). Further
we define

$\mathscr{W}=L^{2}([0,T],\mathbb{R}^{dm})\times\bigl{(}H^{1}([0,T],L^{2}(\mathbb{R}^{dN}))\cap
L^{2}([0,T],H^{1}(\mathbb{R}^{dN}))\bigr{)}\,.$

######  Theorem 3.7.

Let $(\mathrm{bar}Y,\mathrm{bar}\mu,\mathrm{bar}\mathbf{u})$ be an optimal
solution for the infinite dimensional optimal control problem (22)(24). Then
there exist $(\mathrm{bar}\phi_{Y},\mathrm{bar}\phi_{\mu})\in\mathscr{W}$ such
that the optimality conditions read

$\displaystyle\mathrm{bar}\mathbf{u}_{k}^{\ell}=\mathrm{Proj}_{\
\mathcal{U}}\Big{(}-\mathrm{bar}\phi_{\mu}\cdot(G^{\ell}*\tilde{\mu}_{m})-\mathrm{bar}\phi_{Y_{k}}^{\ell}\Big{)}\quad\text{with}\quad\tilde{\mu}_{m}=\sum_{k=1}^{m}\delta_{\mathrm{bar}Y_{k}}\,,$
(27)
$\displaystyle\begin{cases}\dot{\mathrm{bar}Y}_{k}=\mathrm{bar}\mathbf{u}_{k}\qquad\text{for}\,\,i=1\ldots
m\\\ \
\partial_{t}\mathrm{bar}\mu+\nabla\cdot\Bigl{[}\bigl{(}H*\mathrm{bar}\mu+\sum_{\ell=1}^{d}G^{\ell}*\tilde{\mu}_{m\,\ell}\bigr{)}\mathrm{bar}\mu\Bigr{]}=0\\\
\mathrm{bar}Y(0)=\mathrm{bar}Y_{0}\\\
\mathrm{bar}\mu(0)=\mathrm{bar}\mu_{0}\end{cases}$ (28)
$\displaystyle\begin{cases}\partial_{t}\mathrm{bar}\phi_{\mu}=|x-x^{*}|-\nabla\mathrm{bar}\phi_{\mu}\cdot\Big{[}(H*\mathrm{bar}\mu+\sum_{\ell=1}^{d}G^{\ell}*\tilde{\mu}_{m\,\ell})+D_{\mu}((H*\mathrm{bar}\mu)\mathrm{bar}\mu)\Big{]}\\\
\dot{\mathrm{bar}\phi}_{Y_{k}}=\mathrm{bar}\phi_{\mu}\cdot\partial_{\mathrm{bar}Y_{k}}\Big{[}\sum_{\ell=1}^{d}(G^{\ell}*\tilde{\mu}_{m\,\ell})\Big{]}\,.\end{cases}$
(29)

Supplemented by the terminal conditions $\mathrm{bar}\phi_{\mu}(T,\cdot)=0$
and $\mathrm{bar}\phi_{Y_{k}}(T)=0$.

###### Proof.

The proof is similar to the one for the finite dimensional case, using the
extended Lagrange functional

$\displaystyle\int_{0}^{T}\tilde{\mathcal{L}}(\mathrm{bar}Y,\mathrm{bar}\mu,\mathrm{bar}\mathbf{u},\mathrm{bar}\phi_{\mu},\mathrm{bar}\phi_{Y},\eta_{\mu},\eta_{Y})\,dt=\int_{0}^{T}\Bigg{(}\mathcal{L}(\mathrm{bar}Y(t),\mathrm{bar}\mu(t))+\sum_{k=1}^{m}|\mathrm{bar}\mathbf{u}_{k}(t)|^{2}+(\dot{\mathrm{bar}}Y_{k}-\mathrm{bar}\mathbf{u}_{k})\cdot\mathrm{bar}\phi_{Y_{k}}$
$\displaystyle+\int_{\mathbb{R}^{d}}\bigl{(}\partial_{t}\mathrm{bar}\phi_{\mu}+\nabla\mathrm{bar}\phi_{\mu}\cdot(H*\mathrm{bar}\mu+\sum_{\ell=1}^{d}G^{\ell}*\tilde{\mu}_{m\,\ell})\bigr{)}\mathrm{bar}\mu\,dx$
$\displaystyle+\int_{\mathbb{R}^{d}}\mathrm{bar}\phi_{\mu}(T,x)\mathrm{bar}\mu(T,x)-\mathrm{bar}\phi_{\mu}(0,x)\mathrm{bar}\mu(0,x)\,dx-\int_{\mathbb{R}^{d}}(\mathrm{bar}\mu(0)-\mathrm{bar}\mu_{0})\eta_{\mu}\,dx+(\mathrm{bar}Y(0)-\mathrm{bar}Y_{0})\cdot\eta_{Y}\,,$

where
$(\mathrm{bar}\phi_{\mu},\mathrm{bar}\phi_{Y},\eta_{\mu},\eta_{Y})\in\mathscr{W}\times\Bigl{(}L^{2}(\mathbb{R}^{dN})\times\mathbb{R}^{dm}\Bigr{)}$
are Lagrange multipliers or coadjoint variables. Analogously to the finite
dimensional case we derive the adjoint system and the optimality conditions by
calculating the derivatives of $\tilde{\mathcal{L}}$ with respect to the state
variable $(\mathrm{bar}Y,\mathrm{bar}\mu)$ in direction $(h_{Y},h_{\mu})$ and
the control $\mathrm{bar}\mathbf{u}$ in direction $h_{\mathbf{u}}$. ∎

##  4 Kinetic limit

In this section, differently from what we have done in the previous section,
we will investigate what happens when the number of agents goes to infinity
preserving the percentage of controlled particles. Therefore the number of
controlled agents will not remain finite but goes to infinity as the non
controlled ones. To this end we will use a Boltzmann approach. Let us consider
$\alpha$ as a characteristic time of interaction, supposing that only two
agents $x$ and $y$ are interacting with each other, we have that the state
$\hat{x}$ of the particle $x$ after the interaction with the particle $y$ is
an affine function of the controls

$\displaystyle\hat{x}=x+\alpha
H(x-y)(1-\Theta)(1-\Theta^{*})+\alpha\sum_{\ell=1}^{d}\Big{[}G^{\ell}(x-y)(u_{\alpha}^{*})_{\ell}(1-\Theta)\Theta^{*}+\mathbf{u}_{\alpha}\Theta\Big{]}\,,$
(30)

where $\mathbf{u}_{\alpha}$ and $\mathbf{u}^{*}_{\alpha}$ are measuarable
functions of time, $\Theta$ is a random variable which characterizes the
probability that the agent $x$ is controlled through the control
$\mathbf{u}_{\alpha}$ and $\Theta^{*}$ a random variable which characterizes
the probability that the agent $y$ is controlled via the control
$\mathbf{u}^{*}_{\alpha}$. More precisely their law is a Bernoulli one, so
that we have

$\mathcal{P}\\{\Theta=1\\}=p\in[0,1]\qquad\mathcal{P}\\{\Theta^{*}=1\\}=p\in[0,1].$

Thus we now show which is the continuity equation that the density of agents
should satisfy, starting from a dynamics affine in the controls.
Consider a kinetic model for the evolution of the density $\mu=\mu(x,t)$ of
agents, with $x\in\mathbb{R}^{d}$ at time $t$, ruled by the following
Boltzmann-type equation

$\partial_{t}\mu(x,t)=\mathbb{Q}_{\alpha}(\mu,\mu)(x,t)\,,$ (31)

where $\mathbb{Q}_{\alpha}$ is an interaction operator which accounts the gain
and loss of agents in position $x$.

$\mathbb{Q}_{\alpha}=\eta\Big{\langle}\int_{\mathbb{R}^{d}}\frac{1}{\mathcal{J}_{\alpha}}\mu(\hat{x},t)\mu(\hat{y},t)-\mu(x,t)\mu(y,t)\,dy\Big{\rangle}\,,$
(32)

where $\mathcal{J}_{\alpha}$ represents the Jacobian of the transformation
$(x,y)\to(\hat{x},\hat{y})$.
In order to state the main result, namely performing the quasi invariant limit
for interaction strength low and frequency high, for a binary dynamics affine
in the controls, we give the following definition

######  Definition 4.1.

The density $\mu$ is called weak solution of the Boltzmann equation (31), with
initial datum $\mu_{0}\in\mathcal{P}_{1}(\mathbb{R}^{d})$ if $\mu\in
L^{2}([0,T],\mathcal{P}_{1}(\mathbb{R}^{d}))$ such that $\mu(x,0)=\mu_{0}$ and
$\mu$ satisfies the weak form of (31), i.e

$\frac{d}{dt}\bigl{\langle}\mu,\varphi\bigr{\rangle}=\bigl{\langle}\mathbb{Q}_{\alpha}(\mu,\mu),\varphi\bigr{\rangle}\,,$

for all $t\in[0,T]$ and $\varphi\in C^{\infty}_{C}(\mathbb{R}^{d},\mathbb{R})$
and where

$\bigl{\langle}\mathbb{Q}_{\alpha}(\mu,\mu),\varphi\bigr{\rangle}=\Big{\langle}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}(\varphi(\hat{x})-\varphi(x))\mu(x)\mu(y)\,dx\,dy\Big{\rangle}.$

We are now ready to state the main result, namely to prove which is the
infinite dimensional equation that the density of agents satisfy when the
interaction strength is low and the frequency is high

######  Theorem 4.2.

Let us fix controls $\mathbf{u}_{\alpha}$ and $\mathbf{u}^{*}_{\alpha}$,
measurable functions of time which stay in a compact set of $\mathbb{R}^{md}$
and suppose that

$\displaystyle\lim_{\alpha\to
0}\mathbf{u}_{\alpha}(t)=\mathrm{bar}\mathbf{u}(t)\,,$
$\displaystyle\lim_{\alpha\to
0}\mathbf{u}^{*}_{\alpha}(t)=\mathrm{bar}\mathbf{u}^{*}(t).$

Consider a weak solution $\mu$ of (31) with initial datum $\mu_{0}(x)$. Thus
introducing the following scaling

$\alpha=\epsilon\qquad\eta=\frac{1}{\epsilon}$

for the binary interaction (30), and defining by $\mu^{\epsilon}(x,t)$ a
solution for the scaled equation (31), for $\epsilon\to 0$
$\mu^{\epsilon}(x,t)$ converges pointwise, up to a subsequence, to $\mu(x,t)$
where $\mu$ satisfies

$\partial_{t}\mu+\mbox{\rm
div\,}\Big{[}\Big{(}\int_{\mathbb{R}^{3}}\Big{\langle}K(x,y,\mathrm{bar}\mathbf{u},\mathrm{bar}\mathbf{u}^{*})\Big{\rangle}\mu(y,t)\,dy\Big{)}\mu\Big{]}=0\,,$
(33)

where

$\Big{\langle}K(x,y,\mathrm{bar}\mathbf{u},\mathrm{bar}\mathbf{u}^{*})\Big{\rangle}=(1-p)^{2}H(x-y)+p(1-p)\sum_{\ell=1}^{d}G^{\ell}(x-y)\mathrm{bar}u_{\ell}^{*}+\mathrm{bar}\mathbf{u}p\,.$
(34)

###### Proof.

The proof follows the ideas of the one in [5] but now the binary dynamics is a
bit different, i.e. it is an affine function of the controls.
Since $\mu=\mu(x,t)$ is a weak solution of (31), pick a test function
$\varphi\in C^{\infty}_{c}(\mathbb{R}^{d},\mathbb{R})$ we want to compute

$\frac{d}{dt}\int_{\mathbb{R}^{d}}\varphi(x(t))\mu(x(t),t)\,dx=\frac{1}{\alpha}\Big{\langle}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}(\varphi(\hat{x})-\varphi(x))\mu(x,t)\mu(y,t)\,dx\,dy\Big{\rangle}\,.$
(35)

For $\alpha=\epsilon$ small, using Taylor expansion we have

$\varphi(\hat{x})-\varphi(x)=\nabla\varphi(x)(\hat{x}-x)+\frac{1}{2}\mathscr{H}(x)(\hat{x}-x)\cdot(\hat{x}-x)+o(|\hat{x}-x|^{2})\,.$
(36)

Taking $\mathrm{bar}\mathbf{u}$ and $\mathrm{bar}\mathbf{u}^{*}$ in a compact
set, using the boundedness of the vector field in (30) and substituting (36)
in (35), we can neglect the term
$\frac{1}{2}\mathscr{H}(x)(\hat{x}-x)\cdot(\hat{x}-x)$ as $\epsilon\to 0$
since it is bounded and, in (35), is of order $\epsilon$, which goes to zero
as $\epsilon\to 0$. Therefore we obtain

$\small\frac{d}{dt}\int_{\mathbb{R}^{d}}\varphi(x(t))\mu(x(t),t)\,dx=\int_{\mathbb{R}^{d}}\nabla\varphi(x)\cdot\Big{[}\int_{\mathbb{R}^{d}}\Big{\langle}K(x,y,\mathrm{bar}\mathbf{u},\mathrm{bar}\mathbf{u}^{*})\Big{\rangle}\mu(y,t)\,dy\Big{]}\mu(x,t)\,dx\,,$
(37)

where is the right hand side of (30), i.e.

$\displaystyle
K(x,y,\mathbf{u},\mathbf{u}^{*})=H(x-y)(1-\Theta)(1-\Theta^{*})+\sum_{\ell=1}^{d}G^{\ell}(x-y)\mathrm{bar}u_{\ell}^{*}(1-\Theta)\Theta^{*}+\mathrm{bar}\mathbf{u}\Theta\,.$

This equation turns out to be the weak formulation of

$\partial_{t}\mu+\mbox{\rm
div\,}\Big{[}\Big{(}\int_{\mathbb{R}^{d}}\Big{\langle}K(x,y,\mathrm{bar}\mathbf{u},\mathrm{bar}\mathbf{u}^{*})\Big{\rangle}\mu(y,t)\,dy\Big{)}\mu\Big{]}=0.$
(38)

∎

######  Remark 2.

Note that splitting (33), it reads also

$\partial_{t}\mu+(1-p)^{2}\mbox{\rm
div\,}\bigl{[}(H*\mu)\mu\bigr{]}+p(1-p)\sum_{\ell=1}^{d}u_{\ell}^{*}\mbox{\rm
div\,}\bigl{[}(G^{\ell}*\mu)\mu\bigr{]}+p\mbox{\rm div\,}(\mathbf{u}\mu)=0.$
(39)

###  4.1 The optimal control problem: binary control and Boltzmann approach

In this section, starting from the finite dimensional solution of an optimal
control problem, derive the infinite dimensional one, exploiting a Boltzmann
approach. We will give a sub optimal solution to the infinite dimensional
optimal control problem starting from the optimal solution of the finite
dimensional one

####  4.1.1 The binary optimal control

Let us consider the evolution equation (1), with the control variables
$\mathbf{u}$ with values in a compact $U\subset\mathbb{R}^{dm}$. The
controllers are obtained as the solution of the following optimal control
problem

$\min_{\mathbf{u}\in
L^{\infty}([0,T],\mathcal{U})}\mathcal{J}(\mathbf{u},X_{0})=\int_{0}^{T}e^{-\lambda
t}\mathscr{L}(X(t),\mathbf{u}(t))\,dt\,,$ (40)

with

$\mathscr{L}(X(t),\mathbf{u}(t))=||\mathrm{bar}{X}-X(t)||_{2}^{2}+\gamma||\mathbf{u}||_{2}^{2}\,,$

with $\mathrm{bar}X$ a desired fixed configuration.
We first focus our analysis on the optimal control problem when $N=2$, so
there are only two agents $i$ and $j$. We shall assume that system dynamics
have been discretized in time in intervals $\Delta t=[t_{k},t_{k+1}]$ with a
first-order approximation

$\begin{cases}X_{i}^{k+1}=X_{i}^{k}+\frac{\Delta
t}{2}\Big{(}H(X^{k}_{i}-X^{k}_{j})(1-\Theta_{i})(1-\Theta_{j})+\\\
\phantom{X_{i}^{k+1}=}\Theta_{j}(1-\Theta_{i})\displaystyle{\sum_{\ell=1}^{d}}G^{\ell}(X^{k}_{i}-X^{k}_{j})u_{j\,\ell}^{k}+\mathbf{u}_{i}^{k}\Theta_{i}\Big{)}\\\
X_{j}^{k+1}=X_{j}^{k}+\frac{\Delta
t}{2}\Big{(}H(X^{k}_{j}-X^{k}_{i})(1-\Theta_{j})(1-\Theta_{i})+\\\
\phantom{X_{j}^{k+1}=}\Theta_{i}(1-\Theta_{j})\displaystyle{\sum_{\ell=1}^{d}}G^{\ell}(X^{k}_{j}-X^{k}_{i})u_{i\,\ell}^{k}+\mathbf{u}_{j}^{k}\Theta_{j}\Big{)}\,,\end{cases}$
(41)

where $H$ and $G^{\ell}$ $\ell=1,\cdots d$ satisfy assumption $(H)$,
$\Theta_{i}$ is a Bernoulli random variable which describes the probability
that the agent $i$ is controlled and we have assumed that the control is
piecewise constant in each time interval $[t_{k},t_{k+1}]$, i.e
$\mathbf{u}_{i}(t)=\sum_{k=1}^{N_{T}}\mathbf{u}_{i}^{k}\chi_{[t_{k},t_{k+1}]}$.
In this setting the discretized functional becomes

$\min_{\mathbf{u}_{ij}1\in L^{\infty}([0,T],\mathcal{U})}\mathcal{J}_{\Delta
t}(\mathbf{u}_{ij},X^{0}_{ij})=\min_{\mathbf{u}_{ij}1\in
L^{\infty}([0,T],\mathcal{U})}\sum_{k=0}^{N_{T}}\beta^{k}\mathscr{L}(X^{k}_{ij},\mathbf{u}_{ij}^{k})\,.$
(42)

The strategy is now to find the values of
$\mathbf{u}_{ij}^{k}=(\mathbf{u}_{i}^{k},\mathbf{u}_{j}^{k})$ in each time
interval solving an instantaneous optimal control problem. For example in the
time interval $[0,t_{1}]$ the instantaneous control problem is further
simplified to minimizing the functional

$\mathcal{J}_{\Delta
t}(\mathbf{u}_{ij},X^{0}_{ij})=\frac{\beta}{2}||\mathrm{bar}{X}-X_{ij}^{1}||_{2}^{2}+\gamma||\mathbf{u}^{0}||_{2}^{2}\,,$

with $X_{ij}^{1}$ given by

$\begin{cases}X_{i}^{1}=X_{i}^{0}+\frac{\Delta
t}{2}\Big{(}H(X^{0}_{i}-X^{0}_{j})(1-\Theta_{i})(1-\Theta_{j})+\\\
\phantom{X_{i}^{1}=}\Theta_{j}(1-\Theta_{i})\displaystyle{\sum_{\ell=1}^{d}}G^{\ell}(X^{0}_{i}-X^{0}_{j})u_{j\,\ell}^{0}+\mathbf{u}_{i}^{0}\Theta_{i}\Big{)}\\\
X_{j}^{1}=X_{j}^{0}+\frac{\Delta
t}{2}\Big{(}H(X^{0}_{j}-X^{0}_{i})(1-\Theta_{j})(1-\Theta_{i})+\\\
\phantom{X_{j}^{1}=}\Theta_{i}(1-\Theta_{j})\displaystyle{\sum_{\ell=1}^{d}}G^{\ell}(X^{0}_{j}-X^{0}_{i})u_{i\,\ell}^{0}+\mathbf{u}_{j}^{0}\Theta_{j}\Big{)}\end{cases}$
(43)

Since $X_{ij}^{1}$ depend linearly on
$(\mathbf{u}_{i}^{0},\mathbf{u}_{j}^{0})$ the functional is clearly convex and
the optimal control is the solution of the following linear system

$\mathbf{D}_{\Delta t}\begin{pmatrix}\mathbf{u}_{i}^{0}\\\
\mathbf{u}_{j}^{0}\end{pmatrix}=\mathbf{C}_{\Delta t}\,,$

where $\mathbf{D}_{\Delta t}$ is a $2d\times 2d$ square matrix and
$\mathbf{C}_{\Delta t}$ a vector in $\mathbb{R}^{2d}$.
The solution $\mathbf{U}^{*^{0}}_{\Delta
t}=(\mathbf{u}_{i}^{0},\mathbf{u}_{j}^{0})$ is

$\displaystyle\begin{pmatrix}\mathbf{u}_{i}^{0}\\\
\mathbf{u}_{j}^{0}\end{pmatrix}=\Pi_{\mathcal{U}}\Big{(}(\mathbf{D}_{\Delta
t})^{-1}\mathbf{C}_{\Delta t}\Big{)}\,,$ (44)

where $\Pi_{\mathcal{U}}$ is the projection of the compact $\mathcal{U}$.
Iterating his procedure in each time interval we generate an optimal control
in feedback form, i.e. at a given discrete instant $k$, the instantaneous
optimal action is a non linear mapping only depending on the current state
$X^{k}$ and model parameters.

####  4.1.2 The Boltzmann approach for the infinite dimensional optimal
control

For a large ensemble of agents, the microscopic optimal control problem is
well-approximated by the following mean-field optimal control problem,

$\min_{\mathbf{u}\in
L^{\infty}([0,T],\mathcal{U})}J(\mu,\mathbf{u}):=\min_{\mathbf{u}\in
L^{\infty}([0,T],\mathcal{U})}\int_{0}^{T}\int_{\mathbb{R}^{d}}e^{-\lambda
t}\mathcal{L}(X,t,\mu(X,t),\mathbf{u}(X,t))\,d\mu(X,t)\,dt\,,$ (45)

constrained to the mean-field multi-agent dynamical system for the agents’
density distribution $\mu$

$\partial_{t}\mu+\mbox{\rm
div\,}\Big{[}\Big{(}\int_{\mathbb{R}^{d}}\Big{\langle}K(X,Y,\mathbf{u}(X),\mathbf{u}(Y))\Big{\rangle}\mu(Y,t)\,dY\Big{)}\mu\Big{]}=0.$
(46)

with

$\Big{\langle}K(X,Y,\mathbf{u},\mathbf{u}^{*})\Big{\rangle}=(1-p)^{2}H(X,Y)+p(1-p)\sum_{\ell=1}^{d}G^{\ell}(X,Y)u^{*}_{\ell}+\mathbf{u}p\,,$

where $p$ is the probability that the particle $X$ or $Y$ is controlled. The
mean-field functional $J(\mu,\mathbf{u})$ is defined accordingly to the finite
dimensional cost $\mathscr{L}$ in this way:

$J(\mu,\mathbf{u}):=\int_{0}^{T}\Bigg{(}\int_{\mathbb{R}^{d}}||\mathrm{bar}{X}-X||_{2}^{2}\mu(X,t)\,dX+\int_{\mathbb{R}^{d}}\gamma||\mathbf{u}||_{2}^{2}\mu(X,t)\,dX\Bigg{)}\,dt\,,$
(47)

where $\mathrm{bar}X$ is the desired fixed configuration.
We propose a sub-optimal solution to this problem using a Boltzmann-type
equation to model the evolution of a system of agents ruled by a binary
interactions.
For $\mu=\mu(X,t)$ denoting the kinetic probability density of agents in
position $X\in\mathbb{R}^{d}$ at time $t>0$, the time evolution of the density
$\mu$ is given as a balance between bilinear gains and losses of the agents’
position, due to the following constrained binary interaction,

$\begin{cases}X^{*}=X+\alpha\Big{(}H(X-Y)(1-\Theta_{X})(1-\Theta_{Y})+\\\
\phantom{X^{*}=}\Theta_{Y}(1-\Theta_{X})\displaystyle{\sum_{\ell=1}^{d}}G^{\ell}(X-Y)U^{Y*}_{\alpha_{\ell}}(Y,X)+\mathbf{U}^{X*}_{\alpha}(X,Y)\Theta_{X}\Big{)}\\\
Y^{*}=Y+\alpha\Big{(}H(Y-X)(1-\Theta_{Y})(1-\Theta_{X})+\\\
\phantom{Y^{*}=}\Theta_{X}(1-\Theta_{Y})\displaystyle{\sum_{\ell=1}^{d}}G^{\ell}(Y-X)U^{X*}_{\alpha_{\ell}}(X,Y)+\mathbf{U}^{Y*}_{\alpha}(Y,X)\Theta_{Y}\Big{)}\,,\end{cases}$
(48)

where $(X^{*},Y^{*})$ are the post-interaction states, the parameter $\alpha$
measures the strength of the interaction and the feedback
$\mathbf{U}^{i*}_{\alpha}(X,Y)$ for $i=X,Y$, indicates the control of the
dynamics. For $\alpha=\frac{\Delta t}{2}$ and
$\mathbf{U}^{i*}_{\alpha}(X^{1}_{i},X^{1}_{j})$ given by (44) the resulting
dynamics is equivalent to the expression (43).
We now proceed similarly to Theorem 4.2, making the quasi-invariant limit
supposing that the density $\mu$ satisfies a Boltzmann-type equation and the
agents dynamic is the one in (48) considering a regime where interaction
strength is low and frequency high. More precisely we consider that the
density $\mu$ satisfies the Boltzmann type equation (31) with the interaction
operator $\mathbb{Q}_{\alpha}$ given by (32) where $\mathbb{J}_{\alpha}$
represents the Jacobian of the transformation $(X,Y)\to(X^{*},Y^{*})$ given in
(48).

######  Theorem 4.3.

For $\alpha\geq 0$ and $t\geq 0$, assume that $H(\cdot)$ and $G(\cdot)\in
L^{2}_{loc}$ and for $\alpha\to 0$ we assume that
$\mathbf{U}^{i*}_{\alpha}(X,Y)\to K^{i}(X,Y)$ for $i=1,2$. Then we consider a
weak solution $\mu$ of the Boltzmann-type equation (31) with initial datum
$\mu_{0}(X)$. Setting $\alpha=\epsilon$ and $\eta=\frac{1}{\epsilon}$ for the
binary interaction (48), and defining $\mu^{\epsilon}(X,t)$ a solution for the
scaled equation (31), when $\epsilon\to 0$, $\mu^{\epsilon}(X,t)$ converges
point-wise, up to a subsequence, to $\mu(X,t)$ satisfying the following
nonlinear equation

$\partial_{t}\mu+\mbox{\rm
div\,}\Big{[}\Big{(}\int_{\mathbb{R}^{d}}\Big{\langle}K(X,Y,K^{1}(X,Y),K^{2}(X,Y))\Big{\rangle}\mu(Y,t)\,dY\Big{)}\mu\Big{]}=0\,,$
(49)

with

$\displaystyle K(X,Y,K^{1}(X,Y),$ $\displaystyle
K^{2}(X,Y))=H(X-Y)(1-\Theta_{X})(1-\Theta_{Y})$
$\displaystyle+\sum_{\ell=1}^{d}G^{\ell}(X-Y)K^{2}_{\ell}(X,Y)(1-\Theta_{X})\Theta_{Y}+K^{1}(X,Y)\Theta_{X}\,.$

###### Proof.

The proof is similar to the one of Theorem 4.2 in Section 4 and follows the
one in [5] ∎

This is the suboptimal solution to the infinite dimensional optimal control
problem.

##  5 Comparison between the two approaches

In this section we make some comparisons between the infinite dimensional
dynamics obtained by the kinetic limit and the mean filed granular one. First
of all note that in both cases the finite dimensional dynamics in the case
$N=2$ is the same, therefore passing to the limit for the number of agents
tending to infinity, the two infinite dimensional equations are similar. More
precisely looking at the PDE (46)

$\partial_{t}\mu+\mbox{\rm
div\,}\Big{[}\Big{(}\int_{\mathbb{R}^{d}}\Big{\langle}K(X,Y,\mathbf{u}(X),\mathbf{u}(Y))\Big{\rangle}\mu(Y,t)\,dY\Big{)}\mu\Big{]}=0.$

with

$\Big{\langle}K(X,Y,\mathbf{u},\mathbf{u}^{*})\Big{\rangle}=(1-p)^{2}H(X,Y)+p(1-p)\sum_{\ell=1}^{d}G^{\ell}(X,Y)u^{*}_{\ell}+\mathbf{u}p\,,$

and at the second PDE equation in (9), namely

$\partial_{t}\mu+\nabla\cdot\Bigl{[}\bigl{(}H*\mu+\sum_{\ell=1}^{d}G^{\ell}*\mu_{m\,\ell}\bigr{)}\mu\Bigr{]}=0\,.$

the fields of which we are making the divergence are very similar, they both
contain the convolution of the density $\mu$ with the kernel $H$, which
describes the non interacting dynamics. Moreover in them both there is the
action of the kernel $G^{\ell}$ which describes the interaction between the
controlled and non controlled agents.
However, we notice that despite this similarity it is not possible to recover
one PDE starting from the other. Indeed if in the kinetic framework we
consider to control only a finite number of agents independently from the
total number, it is not possible to choose a suitable bernoulli parameter $p$
to describe this situation. Controlling a finite number of agents would mean
to make $p$ tending to zero. Unfortunately in this limit the resulting PDE
does not coincide with (9), since the interaction term $G^{\ell}$ would
disappear making the system not controlled.
Nevertheless the two approaches are both valid depending of the problem we are
interested in. For example if in the finite dimensional system we are
interested in controllability issues, i.e the possibility of finding controls
that move the system between two fixed configurations, is reasonable to need a
number of controlled agents which is proportional to their total number. If we
want to preserve this property also in the limit, the kinetic approach surely
would be the more suitable. In this situation, if the finite dimensional
system is controllable when the number of controls is a fraction of the total
number of agents we are sure, in the finite dimensional optimal control
problem (40) with the current cost
$\mathscr{L}(X(t),\mathbf{u}(t))=||\mathrm{bar}{X}-X(t)||_{2}^{2}+\gamma||\mathbf{u}||_{2}^{2}$,
to find a control which allow us to reach the desired configuration
$\mathrm{bar}X$. Using the kinetic limit we preserve the fraction of
controlled agents keeping the controllability property, crucial to ensure that
we are able to reach also in the infinite dimensional case the desired
configuration $\mathrm{bar}X$. This kind of problems are typical for swarms of
robots or animal collective dynamics.
On the contrary in situations in which in the finite dimensional system the
same amount of controls are sufficient to control all the others or in
situations in which is preferable to identify only a finite number of leaders
to convince all the others to have a certain behavior, like in opinion
dynamics or crowd motion, the mean field granular approach is more suitable.
In this case in the infinite dimensional limit the number of controlled agents
remains finite, of course we cannot ensure to be able to reach exactly the
desired configuration but only to move towards it, however we use a little
amount of control energy.

##  6 Conclusions

In this paper we studied an individual-based model for a multi agent system.
The originality of the work is the fact that the individual dynamics is a
nonlinear affine control system, where the controls are the velocities of the
coordinates of some of the agents and the interaction force between controlled
and non-controlled agents occurs through the velocity of the leader agents,
like in systems of particles immersed in Stokes fluid. We analyze what happen
when the number of agents goes to infinity and we studied two different
situations. The first one in which the number of controlled agents is kept
fixed and finite, whereas the non-controlled or passive ones goes to infinity.
The second instead where the overall number of agents goes to infinity
preserving the percentage of controlled ones. In both cases we perform a limit
procedure, using a granular mean-field limit for the first situation and a
kinetic approach for the second one, and obtain a PDE for the evolution of the
agents’ density. Moreover in both cases we state the optimal control problem
starting from the solution of the finite dimensional one and derive the
corresponding first order optimality conditions.

## References

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