# A single player and a mass of agents:
a pursuit evasion-like game

Fabio Bagagiolo This author was partially supported by MUR-PRIN2020 Project
(No. 2020JLWP23) “Integrated mathematical approaches to socio-epidemiological
dynamics”.  Rossana Capuani This author was supported by the co-financing of
European Union-FSE-REACT-UE, PON “Ricerca e Innovazione 2014-2020, Azione IV.6
Contratti di ricerca su tematiche Green”.  Luciano Marzufero Member of INdAM,
GNAMPA research group.

###### Abstract

We study a finite-horizon differential game of pursuit-evasion like, between a
single player and a mass of agents. The player and the mass directly control
their own evolution, which for the mass is given by a first order PDE of
transport equation type. Using also an adapted concept of non-anticipating
strategies, we derive an infinite dimensional Isaacs equation, and by dynamic
programming techniques we prove that the value function is the unique
viscosity solution on a suitable invariant subset of a Hilbert space.

Keywords. pursuit-evasion games; differential games; infinite-dimensional
Isaacs equation; continuity equation; mass transportation; mean-field,
viscosity solution.

1991 Mathematics Subject Classification. Primary: 49N70; Secondary: 49N75;
49L12; 49L25; 49N80; 35Q49.

##  1 Introduction

The goal of this work is the study of a zero-sum differential game between a
single player and a population of agents. In particular, we are interested to
the corresponding Hamilton-Jacobi-Isaacs equation.

Both the single player and the mass of the agents are moving in
$\mathbb{R}^{d}$, by the following controlled dynamics (respectively: ordinary
differential equation for the single player and continuity partial
differential equation for the mass):

$y^{\prime}(t)=f(y(t),\alpha(t)),\qquad
m_{t}(x,t)+\mbox{div}(\beta(x,t)m(x,t))=0,$ (1)

where the dynamics $f:\mathbb{R}^{d}\times A\longrightarrow\mathbb{R}^{d}$,
and the controls $\alpha:[0,T]\longrightarrow A$,
$\beta:\mathbb{R}^{d}\times[0,T]\longrightarrow\mathbb{R}^{d}$ are suitable
functions. Here, $m:\mathbb{R}^{d}\times[0,T]\longrightarrow[0,1]$ stands for
the time-dependent density distribution of the mass of the agents. The payoff
function is described as

$J(x,m_{0},t,\alpha,\beta)=\int_{t}^{T}\ell\big{(}y(s),m(\cdot,s),s,\alpha(s),\beta(\cdot,s)\big{)}ds+\psi\big{(}y(T),m(\cdot,T)\big{)},$
(2)

where the running cost $\ell$ and the final cost $\psi$ are suitable functions
whose dependence on $m$ and $\beta$ is given by their actual interpretations
as state-dependent functions $m(\cdot,s):\mathbb{R}^{d}\longrightarrow[0,1]$
and $\beta(\cdot,s):\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d}$.

The goal of the single player is to minimize $J$ and the goal of the mass of
the agents is to maximize $J$.

Our main motivation for this kind of study is the possible future application
to a differential game between two distinct masses of agents, setting the
problem in the framework of the mean-field games theory. In such a
differential game, the single agent of one of the two masses should play a
zero-sum differential game against the other mass, say $m_{Y}$, and a mean-
field game looking at the evolution of the mass it belongs to, say $m_{X}$.
Hence, it should infer the evolution of $m_{X}$ and, giving it as a datum,
play a zero-sum differential game against the other mass $m_{Y}$, with payoff

$\int_{t}^{T}\ell\big{(}y(s),m_{X}(\cdot,s),m_{Y}(\cdot,s),\alpha(s),\beta(\cdot,s)\big{)}ds+\psi\big{(}y(T),m_{X}(\cdot,T),m_{Y}(\cdot,T)\big{)}.$

Being $m_{X}$ as given, the payoff of the game played by the single agent
against the mass $m_{Y}$ is of the type (2). However, a possible formulation
and study of such a mean-field (differential) game problem go beyond the
purposes of the present article and seem to present other several crucial
issues, both from a modeling and analytical point of view.

Then it is natural to start such an investigation project by the study of the
game between a single player and a mass. Actually, such a problem is not well
investigated in the literature, especially as concerns the derivation and the
study of the corresponding Isaacs equation which naturally turns out to be an
infinite dimension equation, due to the presence of the state-space dependent
function $m(\cdot,s)$. Hence, its mathematical interest goes beyond the
possible application to the mean-field games framework, and moreover it has
interesting applicative motivations.

In view of the Dynamic Programming Principle, as it is standard for
differential games, we are going to use the concept of non-anticipating
strategies ([18, 6]). In particular, we are concerned with the “$\inf\sup$”
problem only, that is we study the lower value function

$\underline{V}(x,m_{0},t)=\inf_{\gamma}\sup_{\beta}J(x,m_{0},y,\gamma[\beta],\beta).$

Here, $\gamma$ is any non-anticipating strategy for the single player, that is
a function from the set of the controls $\beta$ for the mass to the set of the
controls $\alpha$ for the single player, such that

$\beta_{1}(\cdot,s)=\beta_{2}(\cdot,s)\ \mbox{a.e.
}s\in[t^{\prime},t^{\prime\prime}]\ \mbox{as functions from }\mathbb{R}^{d}\
\mbox{to }\mathbb{R}^{d}$ $\Downarrow$
$\gamma[\beta_{1}](s)=\gamma[\beta_{2}](s)\ \mbox{a.e.
}s\in[t^{\prime},t^{\prime\prime}].$

Denoting by $\tilde{\cal B}$ a suitable set of functions from $\mathbb{R}^{d}$
to itself, and assuming that, for all controls $\beta$, it is
$\beta(\cdot,t)\in\tilde{\cal B}$ for almost every $t$, then, under suitable
hypotheses, the Isaacs equation turns out to be

$-u_{t}+\min_{b\in\tilde{\cal B}}\max_{a\in A}\left\\{-f(x,a)\cdot
D_{x}u-\langle
D_{m}u,\operatorname{div}(bm)\rangle_{L^{2}}-\ell(x,m,t,a,b)\right\\}=0,$ (3)

where the unknown function $u$ is a function of
$(x,m,t)\in\mathbb{R}^{d}\times L^{2}(\mathbb{R}^{d})\times[0,T]$, and
$\langle\cdot,\cdot\rangle_{L^{2}}$ is the scalar product in $L^{2}$.

In this paper, we prove that $\underline{V}$ is the unique viscosity solution
$u$ of (3) satisfying $u(x,m,T)=\psi(x,m)$.

We notice that with similar techniques we can also study the problem for

$\overline{\text{V}}(x,m_{0},t)=\sup_{{\delta}}\inf_{{\alpha}}J(x,m_{0},y,{\alpha},{\delta}[{\alpha}]),$

where ${\alpha}$ is the control for the single player and ${\delta}$ are the
non-anticipating strategies for the mass. However, from a modeling point of
view, it seems less reasonable because it probably gives too much emphasis in
the possible centralized feature of the decisions of the mass.

As far as the domain of the equation (3) is concerned, with respect to the
triple $(x,m,t)$, we are able to restrict it to a suitable set $\tilde{X}$
which is viable under the evolution (1) and, more important, is compactly
embedded in $\mathbb{R}^{d}\times L^{2}(\mathbb{R}^{d})\times[0,T]$. To this
end, we first need a careful analysis of the solution of the continuity
equation in (1). Indeed, it is usual in the literature to interpret it in the
sense of measures: a solution is a time dependent measure $\mu$ on
$\mathbb{R}^{d}$, whose evolution is continuous in a suitable metric space of
measures. Here instead, we want to interpret the solution as a function, i.e.
the density of the measure $\mu$ when it is absolutely continuous with respect
to the Lebesgue measure. It is known, indeed, that whenever the initial datum
$\bar{\mu}$ is a measure with density $\bar{m}$, then the solution $\mu(t)$ is
also a measure with density $m(\cdot,t)$. An important part of the present
paper is then a preliminary direct and deep analysis of the evolution
$t\longmapsto m(\cdot,t)$, together with the right choice of the admissible
set of controls $\beta$ in view of the subsequent study of the infinite
dimensional Isaacs equation (3). Hence Section 2 contains all such necessary
results that, even if somehow and sometimes not surprising, we account
interesting by themselves and are not well found in the literature.

The originality and the interest of the Isaacs equation (3) is given by its
infinite dimensional feature which, to the best of the authors’ knowledge, was
not studied yet in the framework of the viscosity solution theory (we refer
the reader to [6] for a comprehensive account of viscosity solutions theory in
finite dimension and to the seminal papers [14, 15] for the beginning of the
theory in infinite dimension). We also notice that first order Hamilton-Jacobi
equations in infinite dimension, arising from optimal control problems with
controlled dynamics given by a first order PDE as the one in (1), are not so
widely studied in the literature. Finally, we have to strongly point out that
the finite horizon/time dependence feature of the differential game being
accounted, even in the simpler finite dimensional case, is rather a newness as
for dynamic programming and Isaacs equation.

The outline of the paper is the following: in Section 2, we study the
properties of the evolution $t\longmapsto m(\cdot,t)$. In section 3, we
introduce the differential game, write down the Dynamic Programming Principle
and study the regularity properties of the lower value function. In Section 4,
we derive the Isaacs equation, we prove that the lower value function is a
viscosity solution of the corresponding final value problem and we finally
prove the comparison theorem which gives the uniqueness of the solution. In
Section 5 we add some comments on a one-dimensional example.

The present Introduction then briefly concludes with a related literature
review and comparisons with it.

Recently there has been some interest on Hamilton-Jacobi equations arising
from optimal control problems and differential games of pursuit-evasion type,
which concern masses and/or measures evolutions and the corresponding
controlled continuity equation, see for instance [9, 8, 25, 22, 11]. In
particular in [25] a Mayer-type differential game between two masses, somehow
similar to the present one (even if with just one player and one mass), is
studied. In that paper, as in [9, 8, 22, 11], the authors work in the space of
probability measures with Wasserstein distance. This fact poses the use of
some weakened concepts like, among the others, the evolutions of measures
concentrated on the trajectories and the definitions of the solution of the
continuity equation and of the Hamilton-Jacobi equation. Despite the
generality of such concepts, we believe that our model, although less
weakened, together with the evolutions of functions in suitable Hilbert
spaces, may be anyway useful especially for possible applications and the
synthesis of the optimal feedback. We notice that our stronger regularity of
the solutions just comes from the regularity assumptions (in space but not in
time) on the available controls for the mass, which anyway do not affect the
assumptions on the equations themselves. As far as the authors know, this is
the first time that a similar approach is adopted. Compared to [25], our model
also differs from the presence of a running cost depending on all the state,
time and control variables.

Other models of differential games, also in a mean-field type framework,
dealing with continuity equations and measures evolutions can be found in [1,
12, 5, 13, 10]. We also refer the reader to [27, 19, 20] for further pursuit-
evasion like games of multiple pursuers and one evader. Possible applications
are investigated in [23, 24] as regards computer science and cyber-security,
and in [29, 28] as regards engineering and machine learning.

##  2 On the continuity equation for the mass

The evolution of a mass on ${\mathbb{R}}^{d}$ can be represented by the
evolution $t\longmapsto\mu(t)$ of a measure on ${\mathbb{R}}^{d}$, where the
quantity of mass in $A\subset{\mathbb{R}}^{d}$ at time $t$ is the measure of
$A$ at the same time: $\mu(t)(A)$. When the mass population is moving
according to a given time-dependent vector field
$\beta:{\mathbb{R}}^{d}\times[0,T]\longrightarrow{\mathbb{R}}^{d}$, then, at
least formally, the evolution of the measure $\mu$ satisfies the following so-
called continuity equation

$\begin{cases}\frac{\partial\mu}{\partial
s}(x,s)+\operatorname{div}(\beta(x,s)\mu(x,s))=0,&(x,s)\in\mathbb{R}^{d}\times]t,T[\\\
\mu(\cdot,t)=\bar{\mu}\end{cases},$ (4)

where $T>0$ is the final time, $\bar{\mu}$ is the value of the measure at the
initial time $t$ and, here and in the sequel, $\operatorname{div}$ means the
spatial divergence. In (4) we have included the variable $x$ in the notation
of $\mu$ with the meaning that for any time $s$, it is a measure on
${\mathbb{R}}^{d}$ whose elements are denoted by $x$. Equation (4) comes from
conservation of mass arguments and for a general account, in particular for
its possible weak formulations in the sense of measures, we refer the reader
to [17] and [2, 3, 4].

The well-posedness of (4) is strongly related to well-posedness of the Cauchy
problem

$\begin{cases}y^{\prime}(s)=\beta(y(s),s),&s\in]t,T[\\\ y(t)=x\end{cases}.$
(5)

In the sequel, we will denote the flow associated to (5) by $\Phi(x,t,s)$,
which is defined as $y(s)$ where $y$ is the solution of (5) with the given
$(x,t)\in{\mathbb{R}}^{d}\times[0,T]$. As it is known, if $\beta\in
L^{1}([0,T],W^{1,\infty}({\mathbb{R}}^{d},{\mathbb{R}}^{d}))$, then system (5)
has a unique solution for any initial data $(x,t)$. For our future purposes,
here we assume the stronger hypothesis

$\beta\in{\mathcal{B}}:=\\{\beta\in
L^{2}([0,T],W^{2,\infty}({\mathbb{R}}^{d},{\mathbb{R}}^{d})\cap
H^{1}({\mathbb{R}}^{d},{\mathbb{R}}^{d})):\beta(\cdot,t)\in\tilde{\mathcal{B}}\
\text{ for a.e. }t\\},$ (6)

where, for a given fixed $M>0$,

$\tilde{\mathcal{B}}:=\left\\{b\in
W^{2,\infty}({\mathbb{R}}^{d},{\mathbb{R}}^{d})\cap
H^{1}({\mathbb{R}}^{d},{\mathbb{R}}^{d}):\right.\\\
\left.\|b\|_{L^{\infty}({\mathbb{R}}^{d},{\mathbb{R}}^{d})}\leq M,\
\|b\|_{H^{1}({\mathbb{R}}^{d},{\mathbb{R}}^{d})}\leq M,\
\|\operatorname{div}b\|_{W^{1,\infty}({\mathbb{R}}^{d})}\leq M\right\\}.$ (7)

######  Remark 1.

By hypothesis (6), we have $\operatorname{div}\beta\in
L^{2}([0,T],W^{1,\infty}({\mathbb{R}}^{d}))$. Moreover, since
$W^{2,\infty}({\mathbb{R}}^{d},{\mathbb{R}}^{d})\subset
C^{1}({\mathbb{R}}^{d},{\mathbb{R}}^{d})$, for any $(t,s)\in[0,T]\times]t,T]$
the function $\Phi(\cdot,t,s):{\mathbb{R}}^{d}\longrightarrow{\mathbb{R}}^{d}$
is invertible, Lipschitz continuous and $C^{1}$, together with its inverse,
with the Lipschitz constant independent on $t$ and $\beta\in{\mathcal{B}}$.

We recall that, for every $t\in[0,T]$ and $s\in[t,T]$, the inverse of the flow
$\Phi(\cdot,t,s)$ is defined as $\Phi^{-1}(x,t,s)=y$, where $y$ is such that,
starting from $y$ at time $t$ with the forward flow $\Phi$, we arrive in $x$
at time $s$. Still holding our hypotheses, in the sequel we will use the fact
that $\Phi^{-1}(\cdot,t,s)$ is Lipschitz continuous w.r.t. $s$ independently
on $x,t,\beta$.

Furthermore, if all the fields $\beta$ have the support contained in the same
bounded subset of ${\mathbb{R}}^{d}$, then the $L^{2}$-closure
$\overline{\tilde{\mathcal{B}}}$ of (7) is a compact subset of $L^{2}$.

######  Proposition 1.

Under hypothesis (6), for any Borel measure $\bar{\mu}$ the solution of the
continuity equation (4) in the sense of distribution is given by the push-
forward

$\mu(\cdot,s)=\Phi(\cdot,t,s)\sharp\bar{\mu},$ (8)

where, for any Borel measurable set $A\subset{\mathbb{R}}^{d}$,
$\mu(\cdot,s)(A)=\bar{\mu}(\Phi^{-1}(\cdot,t,s)(A))$.

###### Proof.

See [4], Proposition 4. ∎

In the case when $\bar{\mu}$ is absolutely continuous w.r.t. the Lebesgue
measure ${\mathcal{L}}^{d}$, that is $\bar{\mu}=\bar{m}\mathcal{L}^{d}$, where
$\bar{m}:{\mathbb{R}}^{d}\longrightarrow{\mathbb{R}}$ is the density, all the
measures $\mu(\cdot,s)$ are absolutely continuous w.r.t. $\mathcal{L}^{d}$
too, and their density $m(\cdot,s)$ can be explicitly computed as (see [4])

$m(\cdot,s)=\Phi(\cdot,t,s)\sharp\bar{m}(\cdot)=\frac{\bar{m}(\cdot)}{{|\rm{det}}J\Phi(\cdot,t,s)|}\circ\Phi^{-1}(\cdot,t,s),$
(9)

where $J\Phi$ is the Jacobian matrix of $\Phi(\cdot,t,s)$. Still referring to
[4], and also using our hypothesis (6), we have the following inequalities

$0<e^{-MT}\leq{\rm{det}}J\Phi(x,t,s)\leq e^{MT}\quad\text{for all }t\in[0,T],\
s\in[t,T]\ \text{and}\ x\in{\mathbb{R}}^{d}.$ (10)

Such estimates come from the fact that the function $s\longmapsto\det
J\Phi(x,t,s)$ solves

$\begin{cases}\frac{d}{ds}{\rm{det}}J\Phi(x,t,s)=({\rm{div}}\beta)(\Phi(x,t,s),s){\rm{det}}J\Phi(x,t,s),&s\in]t,T]\\\
{\rm{det}}J\Phi(x,t,t)=1\end{cases},$

where in the first line $(\operatorname{div}\beta)(\Phi(x,t,s),s)$ stands for
the spatial divergence of $\beta$ calculated in the point
$(\Phi(x,t,s),s)\in{\mathbb{R}}^{d}\times]t,T]$. In particular it is

${\rm{det}}J\Phi(x,t,s)=e^{\left(\int_{t}^{s}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau\right)}.$

In the following, by $m(x,s;\beta,t,\bar{m})$ we denote the value at
$(x,s)\in{\mathbb{R}}^{d}\times]t,T]$ of the density of the solution (8) of
(4) with $\beta\in{\mathcal{B}}$, where the initial datum at time $t$,
$\bar{\mu}$, has density $\bar{m}$.

From now on we assume

$\bar{m}\in H^{1}(\mathbb{R}^{d})\cap W^{1,\infty}({\mathbb{R}}^{d}).$ (11)

###  2.1 Spatial and time estimates on the density $m$

In this subsection we prove that, under hypotheses (11) and (6),

$\displaystyle x\longmapsto m(x,s;\beta,t,\bar{m})\text{ belongs to
}H^{1}(\mathbb{R}^{d})\cap W^{1,\infty}({\mathbb{R}}^{d}),$ (12)
$\displaystyle m(\cdot,\cdot;\beta,t,\bar{m})\in
C^{0}([t,T],H^{1}({\mathbb{R}}^{d})\cap W^{1,\infty}({\mathbb{R}}^{d})),$ (13)

for every $t\in[0,T]$ and $s\in[t,T]$, where in (13) the modulus of continuity
is independent of $t,\beta$ and only depends on the norms of $\bar{m}$.
Moreover we prove that

$\|m(\cdot,s;\beta,t_{1},\bar{m}^{1})-m(\cdot,s;\beta,t_{2},\bar{m}^{2})\|_{H^{1}(\mathbb{R}^{d})}\leq\tilde{L}\|\bar{m}^{1}(\cdot)-\bar{m}^{2}(\cdot)\|_{H^{1}(\mathbb{R}^{d})},$
(14)

for every $t_{1},t_{2}\in[0,T]$, $s\in[\min\\{t_{1},t_{2}\\},T]$ and for some
$\tilde{L}$ independent of $t_{1},t_{2}$ and $\beta$.

For (12), if we prove that $\det J\Phi(\cdot,t,s)\in
W^{1,\infty}({\mathbb{R}}^{d})$, by the estimates (10) and by (11), we then
have that

$\frac{\bar{m}(\cdot)}{\det J\Phi(\cdot,t,s)}\in H^{1}({\mathbb{R}}^{d})\cap
W^{1,\infty}({\mathbb{R}}^{d}).$

Hence, since $\Phi^{-1}(\cdot,t,s)$ is a $C^{1}$ invertible function, by a
known result (see for example [7]), (9) gives (12).

So we prove that $\det J\Phi(\cdot,t,s)\in W^{1,\infty}({\mathbb{R}}^{d})$:

$\|\det J\Phi(\cdot,t,s)\|_{W^{1,\infty}({\mathbb{R}}^{d})}=\|\det
J\Phi(\cdot,t,s)\|_{L^{\infty}({\mathbb{R}}^{d})}+\|\partial_{x}\det
J\Phi(\cdot,t,s)\|_{L^{\infty}({\mathbb{R}}^{d})}\\\
=\operatorname*{ess\,sup}_{x\in{\mathbb{R}}^{d}}|\det
J\Phi(x,t,s)|+\operatorname*{ess\,sup}_{x\in{\mathbb{R}}^{d}}|\partial_{x}\det
J\Phi(x,t,s)|\\\
=\operatorname*{ess\,sup}_{x\in{\mathbb{R}}^{d}}\left|e^{\left(\int_{t}^{s}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau\right)}\right|+\operatorname*{ess\,sup}_{x\in{\mathbb{R}}^{d}}\left|\partial_{x}\left(e^{\left(\int_{t}^{s}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau\right)}\right)\right|\\\
=\operatorname*{ess\,sup}_{x\in{\mathbb{R}}^{d}}\left|e^{\left(\int_{t}^{s}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau\right)}\right|\\\
+\operatorname*{ess\,sup}_{x\in{\mathbb{R}}^{d}}\left|e^{\left(\int_{t}^{s}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau\right)}\int_{t}^{s}\partial_{x}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)\partial_{x}(\Phi(x,t,\tau))d\tau\right|,$

where, in general, here and sometimes in the sequel, by $\partial_{x}$ we
denote both the Jacobian matrix $d\times d$ and, when the function is scalar,
the gradient $1\times d$. By (10) and Remark 1, we have that
$e^{\left(\int_{t}^{s}(\operatorname{div}\beta)(\Phi(x.t,\tau),\tau)d\tau\right)}$,
$\partial_{x}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)$ and
$\partial_{x}(\Phi(x,t,\tau))$ are bounded, and hence $\|\det
J\Phi(\cdot,t,s)\|_{W^{1,\infty}({\mathbb{R}}^{d})}<+\infty$. Note also that
the derivation w.r.t. $x$ of $(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)$
is possible because $(\operatorname{div}\beta)(\cdot,\tau)\in
W^{1,\infty}({\mathbb{R}}^{d})$ and $\Phi(\cdot,t,\tau)$ is a $C^{1}$
invertible function, and the (weak) derivative can be computed with the usual
differentiation rules (see again [7]).

Furthermore, after similar calculations one can show that for the density $m$
we have the following estimate

$\|m(\cdot,s;\beta,t,\bar{m})\|_{W^{1,\infty}({\mathbb{R}}^{d})}\leq
e^{MT}\sup\\{1+\tilde{M},L_{\Phi^{-1}}\\}\|\bar{m}(\cdot)\|_{W^{1,\infty}({\mathbb{R}}^{d})},$
(15)

where $\tilde{M}>0$ is a constant enclosing the bounds for the spatial
derivative of $\operatorname{div}\beta$, $\Phi$ and $\Phi^{-1}$, independent
of $t,\beta$, and $L_{\Phi^{-1}}$ is the Lipschitz constant of
$\Phi^{-1}(\cdot,t,s)$.

Now we prove (13). At first we show that, for every $t\in[0,T]$ and $s\geq t$,
the function

$\psi:s\longmapsto\det J\Phi(\cdot,t,s)\ \text{ belongs to
}\operatorname{Lip}([t,T],W^{1,\infty}({\mathbb{R}}^{d}))$ (16)

with Lipschitz constant independent of $t,\beta$. In this way (9) gives (13)
(see also Remark 1). Then for every $s_{1},s_{2}\in[t,T]$ we have

$\|\psi(s_{1})-\psi(s_{2})\|_{W^{1,\infty}({\mathbb{R}}^{d})}=\left\|e^{\int_{t}^{s_{1}}(\operatorname{div}\beta)(\Phi(\cdot,t,\tau),\tau)d\tau}-e^{\int_{t}^{s_{2}}(\operatorname{div}\beta)\Phi(\cdot,t,\tau),\tau)d\tau}\right\|_{W^{1,\infty}({\mathbb{R}}^{d})}\\\
=\operatorname*{ess\,sup}_{x\in{\mathbb{R}}^{d}}\left|e^{\int_{t}^{s_{1}}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau}-e^{\int_{t}^{s_{2}}(\operatorname{div}\beta)\Phi(x,t,\tau),\tau)d\tau}\right|\\\
+\operatorname*{ess\,sup}_{x\in{\mathbb{R}}^{d}}\left|\partial_{x}\left(e^{\int_{t}^{s_{1}}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau}\right)-\partial_{x}\left(e^{\int_{t}^{s_{2}}(\operatorname{div}\beta)\Phi(x,t,\tau),\tau)d\tau}\right)\right|\\\
\leq
L_{1}\operatorname*{ess\,sup}_{x\in{\mathbb{R}}^{d}}\left|\int_{t}^{s_{1}}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau-\int_{t}^{s_{2}}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau\right|\\\
+\operatorname*{ess\,sup}_{x\in{\mathbb{R}}^{d}}\left|e^{\int_{t}^{s_{1}}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau}\int_{t}^{s_{1}}\partial_{x}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)\partial_{x}(\Phi(x,t,\tau))d\tau\right.\\\
\left.-e^{\int_{t}^{s_{2}}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau}\int_{t}^{s_{2}}\partial_{x}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)\partial_{x}(\Phi(x,t,\tau))d\tau\right|\\\
\leq
L_{1}\operatorname*{ess\,sup}_{x\in{\mathbb{R}}^{d}}\left|\int_{s_{2}}^{s_{1}}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau\right|\\\
+\operatorname*{ess\,sup}_{x\in{\mathbb{R}}^{d}}\left|\left(\int_{t}^{s_{1}}\partial_{x}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)\partial_{x}(\Phi(x,t,\tau))d\tau\right)\left(e^{\int_{t}^{s_{1}}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau}\right.\right.\\\
\left.\left.-e^{\int_{t}^{s_{2}}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau}\right)\right.\\\
\left.+\left(e^{\int_{t}^{s_{2}}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)d\tau}\right)\left(\int_{s_{2}}^{s_{1}}\partial_{x}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)\partial_{x}(\Phi(x,t,\tau))d\tau\right)\right|\\\
\leq(L_{1}M+M_{1}TL_{1}M+e^{MT}M_{1})|s_{1}-s_{2}|,$

where $L_{1}>0$ is the Lipschitz constant of $e^{(\cdot)}$ and $M_{1}>0$ is
the bound for the term
$\partial_{x}(\operatorname{div}\beta)(\Phi(x,t,\tau),\tau)\partial_{x}(\Phi(x,t,\tau))$.
Hence (16) holds with a Lipschitz constant independent of $t,\beta$. For
proving (14), we observe that, similarly as the computations for (15), we have

$\|m(\cdot,s;\beta,t,\bar{m})\|^{2}_{H^{1}({\mathbb{R}}^{d})}\leq
3e^{2MT}\sup\\{1+\tilde{M}^{2},L^{2}_{\Phi^{-1}}\\}\|\bar{m}(\cdot)\|^{2}_{H^{1}({\mathbb{R}}^{d})}$
(17)

and then, by the linearity of the continuity equation for $m$, (17) gives the
desired inequality.

######  Remark 2.

By (12) and (7), $\operatorname{div}(\beta m)\in
L^{2}({\mathbb{R}}^{d}\times[0,T])$. Given the Lipschitz continuity of
$s\longmapsto\Phi^{-1}(x,t,s)$ (see Remark 1), similar computations as above
give $\frac{\partial m}{\partial t}\in L^{2}({\mathbb{R}}^{d}\times[0,T])$.
Moreover, since $\mu$, given by (8), is the solution of (4) in the sense of
measures, we have that its density $m$ is solution, in the sense of
distributions, of $\frac{\partial m}{\partial t}+\operatorname{div}(\beta
m)=0$. Hence, since $\frac{\partial m}{\partial t},\operatorname{div}(\beta
m)\in L^{2}({\mathbb{R}}^{d}\times[0,T])$, we have that $m$ is the unique
solution in $L^{2}({\mathbb{R}}^{d}\times[0,T])$ (see [16]) and, in
particular, $\frac{\partial m}{\partial t}=-\operatorname{div}(\beta m)$ as
functions in $L^{2}({\mathbb{R}}^{d}\times[0,T])$. From this, it follows that:
if $\varphi\in C^{1}(L^{2}({\mathbb{R}}^{d}))$, then the map

$\displaystyle\xi$ $\displaystyle:[t,T]\longrightarrow{\mathbb{R}}$
$\displaystyle\tau$
$\displaystyle\longmapsto\xi(\tau)=\varphi(m(\cdot,\tau;\beta,t,\bar{m}))$

is differentiable and $\xi^{\prime}(\tau)=-\langle
D_{m}\varphi,\operatorname{div}(\beta m)\rangle_{L^{2}({\mathbb{R}}^{d})}$,
where $D_{m}\varphi$ denotes the Fréchet derivative of $\varphi$ w.r.t. $m$
and $\langle\cdot,\cdot\rangle_{L^{2}({\mathbb{R}}^{d})}$ the scalar product
in $L^{2}({\mathbb{R}}^{d})$.

##  3 The differential game model: first results

In this section, we introduce the model of the differential game between a
single player and a mass of agents.

The controlled dynamics of the single player is given by

$\begin{cases}y^{\prime}(s)=f(y(s),{\alpha}(s)),&s\in]t,T]\\\
y(t)=x\in\mathbb{R}^{d}\end{cases},$ (18)

where $T>0$ is given, $t\in[0,T]$, $f:{\mathbb{R}}^{d}\times
A\longrightarrow{\mathbb{R}}^{d}$ ($A\subset{\mathbb{R}}^{m}$ compact) is
continuous, bounded and Lipschitz continuous w.r.t. $x\in{\mathbb{R}}^{d}$
uniformly w.r.t. $a\in A$, i.e. there exists $L>0$ such that

$\|f(x,a)-f(z,a)\|\leq L\|x-z\|\quad\text{for every}\ x,z\in{\mathbb{R}}^{d}\
\text{and}\ a\in A.$

The pair $(x,t)$ is the initial datum and the control is

${\alpha}\in{\mathcal{A}}(t):=\\{{\alpha}:[t,T]\longrightarrow
A:{\alpha}\text{ is measurable}\\}.$

Given the control and the initial datum, we denote by
$y_{(x,t)}(\cdot;{\alpha})$ the unique solution of (18) in $[t,T]$, and, if no
ambiguity arises, we do not display the dependence on the control ${\alpha}$
using the notation $y_{(x,t)}(\cdot)$.

Coherently with Remark 2, the controlled equation for the evolution of the
mass is

$\begin{cases}m_{s}(\cdot,s)+\operatorname{div}(\beta(\cdot,s)m(\cdot,s))=0,&s\in]t,T]\\\
m(\cdot,t)=\bar{m}\end{cases},$ (19)

where $t\in[0,T]$ and, using the same notation as in the previous section,
$\bar{m}\in H^{1}({\mathbb{R}}^{d})\cap W^{1,\infty}({\mathbb{R}}^{d})$ and
the control $\beta$ belongs to

${\mathcal{B}}(t):=\\{\beta\in
L^{2}([t,T],W^{2,\infty}({\mathbb{R}}^{d},{\mathbb{R}}^{d})\cap
H^{1}({\mathbb{R}}^{d},{\mathbb{R}}^{d})):\beta(\cdot,s)\in\tilde{\mathcal{B}}\
\text{ for a.e. }s\\}.$

We consider a running cost

$\displaystyle\ell:{\mathbb{R}}^{d}\times
H^{1}({\mathbb{R}}^{d})\times[0,T]\times{\mathbb{R}}^{m}\times
L^{2}({\mathbb{R}}^{d})$ $\displaystyle\longrightarrow[0,+\infty[$
$\displaystyle(x,m,s,a,b)$ $\displaystyle\longmapsto\ell(x,m,s,a,b),$

(recall that $A\subset{\mathbb{R}}^{m}$ and $\tilde{\mathcal{B}}\subset
L^{2}({\mathbb{R}}^{d})$) bounded, strongly continuous and uniformly strongly
continuous w.r.t. $(x,m,t)$ uniformly w.r.t. $(a,b)$, that is there exists a
modulus of continuity $\omega_{\ell}$ such that, for any fixed $a,b$ it is

$|\ell(x_{1},m_{1},s_{1},a,b)-\ell(x_{1},m_{2},s_{2},a,b)|\leq\omega_{\ell}\left(\|x_{1}-x_{2}\|+\|m_{1}-m_{2}\|_{H^{1}({\mathbb{R}}^{d})}+|s_{1}-s_{2}|\right)$

for all $(x_{1},m_{1},s_{1}),(x_{2},m_{2},s_{2})\in{\mathbb{R}}^{d}\times
H^{1}({\mathbb{R}}^{d})\times[0,T]$.

Similarly we consider a final cost

$\displaystyle\psi:{\mathbb{R}}^{d}\times H^{1}({\mathbb{R}}^{d})$
$\displaystyle\longrightarrow[0,+\infty[$ $\displaystyle(x,m)$
$\displaystyle\longmapsto\psi(x,m),$

bounded and uniformly strongly continuous w.r.t. $(x,m)$.

The corresponding cost functional $J$, for
$(x,\bar{m},t,{\alpha},\beta)\in{\mathbb{R}}^{d}\times
H^{1}({\mathbb{R}}^{d})\times[0,T]\times{\mathcal{A}}(t)\times{\mathcal{B}}(t)$,
is given by

$J(x,\bar{m},t,{\alpha},\beta)\\\
=\int_{t}^{T}\ell(y_{(x,t)}(s),m(\cdot,s;\beta,t,\bar{m}),s,{\alpha}(s),\beta(\cdot,s))ds+\psi(y_{(x,t)}(T),m(\cdot,T;\beta,t,\bar{m})).$

The single player wants to minimize the cost $J$ and the mass $m$ wants to
maximize it. As argued in the Introduction, here we consider only the
$\inf\sup$ best-worst case for the single player. Hence we introduce the non-
anticipating strategies for the single player

$\Gamma(t)=\\{\gamma:{\mathcal{B}}(t)\longrightarrow{\mathcal{A}}(t):\beta\longmapsto\gamma[\beta]\
\text{non-anticipating}\\},$

where “non-anticipating” means

$\beta_{1}(\cdot,s)=\beta_{2}(\cdot,s)\ \text{in $\tilde{\mathcal{B}}$ a.e.
$s$}\in[t,T]\ \Rightarrow\gamma[\beta_{1}](s)=\gamma[\beta_{2}](s)\
\text{a.e.}\ s\in[t,T].$

We then consider the lower value function

$\underline{V}(x,\bar{m},t)=\inf_{\gamma\in\Gamma(t)}\sup_{\beta\in{\mathcal{B}}(t)}J(x,\bar{m},t,\gamma[\beta],\beta).$
(20)

###  3.1 Dynamic Programming Principle and regularity of the value function

######  Proposition 2.

Under the previous hypotheses and (6), the lower value function
$\underline{V}$ (20) satisfies the Dynamic Programming Principle: for all
$(x,\bar{m},t)\in\mathbb{R}^{d}\times H^{1}(\mathbb{R}^{d})\times[0,T]$ and
for all $\tau\in[t,T]$,

$\underline{V}(x,\bar{m},t)=\inf_{\gamma\in\Gamma(t)}\sup_{\beta\in{\mathcal{B}}(t)}\Bigg{(}\int_{t}^{\tau}\ell(y_{(x,t)}(s),m(\cdot,s;\beta,t,\bar{m}),s,\gamma[\beta](s),\beta(\cdot,s))ds\\\
+\underline{V}(y_{(x,t)}(\tau),m(\cdot,\tau;\beta,t,\bar{m}),\tau)\Bigg{)},$
(21)

where $s\longmapsto y_{(x,t)}(s):=y_{(x,t)}(s;{\alpha})$ is the corresponding
solution to (18) with control ${\alpha}=\gamma[\beta]$.

###### Proof.

For all $(\xi,u,\tau)\in\mathbb{R}^{d}\times H^{1}(\mathbb{R}^{d})\times[0,T]$
and for all $\varepsilon>0$ we take $\gamma_{(\xi,u,\tau)}\in\Gamma(\tau)$
such that

$\underline{V}(\xi,u,\tau)\geq\sup_{\beta\in{\mathcal{B}}(\tau)}J(\xi,u,\tau,\gamma_{(\xi,u,\tau)}[\beta],\beta)-\varepsilon.$
(22)

Denote by $w(x,\bar{m},t)$ the right-hand side of (21). Then we have to prove
that

  * $(i)$

$\underline{V}(x,\bar{m},t)\leq w(x,\bar{m},t)$;

  * $(ii)$

$\underline{V}(x,\bar{m},t)\geq w(x,\bar{m},t)$.

For $(i)$, take $\bar{\gamma}\in\Gamma(t)$ such that

$w(x,\bar{m},t)\geq\sup_{\beta\in{\mathcal{B}}(t)}\Bigg{(}\int_{t}^{\tau}\ell(y_{(x,t)}(s),m(\cdot,s;\beta,t,\bar{m}),s,\bar{\gamma}[\beta](s),\beta(\cdot,s))ds\\\
+\underline{V}(y_{(x,t)}(\tau),m(\cdot,\tau;\beta,t,\bar{m}),\tau)\Bigg{)}-\varepsilon.$

Taken $\beta\in{\mathcal{B}}(t)$ and still denoting by $\beta$ its restriction
to $[\tau,T]$, we have $\beta\in{\mathcal{B}}(\tau)$. We then define
$\tilde{\gamma}\in\Gamma(t)$ as

$\tilde{\gamma}[\beta](s)=\begin{cases}\bar{\gamma}[\beta](s),&s\in[t,\tau]\\\
\gamma_{(y_{(x,t)}(\tau),m(\cdot,\tau;\beta,t,\bar{m}),\tau)}[\beta](s),&s\in[\tau,T]\end{cases},$

where $y_{(x,t)}(\tau)$ is the position reached at time $\tau$ by the solution
of (18) with control ${\alpha}=\bar{\gamma}[\beta]$. The strategy
$\tilde{\gamma}$ is well-defined and non-anticipating, i.e., it belongs to
$\Gamma(t)$. We then have

$w(x,\bar{m},t)\geq\sup_{\beta\in{\mathcal{B}}(t)}\Bigg{(}\int_{t}^{\tau}\ell(y_{(x,t)}(s),m(\cdot,s;\beta,t,\bar{m}),s,\bar{\gamma}[\beta](s),\beta(\cdot,s))ds\\\
+\underline{V}(y_{(x,t)}(\tau),m(\cdot,\tau;\beta,t,\bar{m}),\tau)\Bigg{)}-\varepsilon\\\
\geq\sup_{\beta\in{\mathcal{B}}(t)}\Bigg{(}\int_{t}^{\tau}\ell(y_{(x,t)}(s),m(\cdot,s;\beta,t,\bar{m}),s,\bar{\gamma}[\beta](s),\beta(\cdot,s))ds\\\
+J(y_{(x,t)}(\tau),m(\cdot,\tau;\beta,t,\bar{m}),\tau,\gamma_{(y_{(x,t)}(\tau),m(\cdot,\tau;\beta,t,\bar{m}),\tau)}[\beta],\beta)\Bigg{)}-2\varepsilon\\\
=\sup_{\beta\in{\mathcal{B}}(t)}J(x,\bar{m},t,\tilde{\gamma}[\beta],\beta)-2\varepsilon\geq\underline{V}(x,\bar{m},t)-2\varepsilon,$

and we conclude.

For $(ii)$, take $\varepsilon>0$ and $\beta_{1}\in{\mathcal{B}}(t)$ such that

$w(x,\bar{m},t)\leq\int_{t}^{\tau}\ell(y_{(x,t)}(s),m(\cdot,s;\beta_{1},t,\bar{m}),s,\gamma_{(x,\bar{m},t)}[\beta_{1}](s),\beta_{1}(\cdot,s))ds\\\
+\underline{V}(y_{(x,t)}(s),m(\cdot,\tau;\beta_{1},t,\bar{m}),\tau)+\varepsilon,$

where $y_{(x,t)}(\cdot):=y_{(x,t)}(\cdot;{\alpha})$ is the solution of (18)
with control ${\alpha}=\gamma_{(x,\bar{m},t)}[\beta_{1}]$. We define
$\tilde{\gamma}\in\Gamma(\tau)$ as

$\tilde{\gamma}[\beta](s)=\gamma_{(x,\bar{m},t)}[\tilde{\beta}](s),$

where $\tilde{\beta}\in{\mathcal{B}}(t)$ is defined as

$\tilde{\beta}(\cdot,s)=\begin{cases}\beta_{1}(\cdot,s),&s\in[t,\tau],\\\
\beta(\cdot,s),&\text{otherwise}\end{cases}.$

By the non-anticipating feature of $\gamma_{(x,\bar{m},t)}$ and by
construction, also $\tilde{\gamma}$ is non-anticipating, i.e.,
$\tilde{\gamma}\in\Gamma(\tau)$. Now, take $\beta_{2}\in{\mathcal{B}}(\tau)$
such that

$\underline{V}(y_{(x,t)}(\tau),m(\cdot,\tau;\beta_{1},t,\bar{m}),\tau)\leq
J(y_{(x,t)}(\tau),m(\cdot,\tau;\beta_{1},t,\bar{m}),\tau,\tilde{\gamma}[\beta_{2}],\beta_{2})+\varepsilon,$

and we get (also recall (22))

$\underline{V}(x,\bar{m},t)\geq
J(x,\bar{m},t,\gamma_{(x,\bar{m},t)}[\tilde{\beta}],\tilde{\beta})-\varepsilon\\\
=\int_{t}^{\tau}\ell(y_{(x,t)}(s),m(\cdot,s;\beta_{1},t,\bar{m}),s,\gamma_{(x,\bar{m},t)}[\beta_{1}](s),\beta_{1}(\cdot,s))ds\\\
+J(y_{(x,t)}(\tau),m(\cdot,\tau;\beta_{1},t,\bar{m}),\tau,\tilde{\gamma}[\beta_{2}],\beta_{2})-\varepsilon\\\
\geq\int_{t}^{\tau}\ell(y_{(x,t)}(s),m(\cdot,s;\beta_{1},t,\bar{m}),s,\gamma_{(x,\bar{m},t)}[\beta_{1}](s),\beta_{1}(\cdot,s))ds\\\
+\underline{V}(y_{(x,t)}(\tau),m(\cdot,\tau;\beta_{1},t,\bar{m}),\tau)-2\varepsilon\geq
w(x,\bar{m},t)-3\varepsilon,$

and we conclude. ∎

We now address the regularity properties of $\underline{V}$. Here, and in the
following, we denote by $\omega_{\psi}$ the modulus of continuity of $\psi$
and we set $C:=\max\\{\|f(z,a)\|:(z,a)\in{\mathbb{R}}^{d}\times A\\}$.

######  Proposition 3.

Under the hypotheses of Proposition 2, the lower value function
$\underline{V}$ is bounded and uniformly continuous.

###### Proof.

For the boundedness, for all $(x,\bar{m},t)$ we get

$\underline{V}(x,\bar{m},t)\leq G_{1}T+G_{2},$

where $G_{1}>0$ and $G_{2}>0$ are the bounds for $\ell$ and $\psi$
respectively.

We then consider
$(x_{1},m_{1},t_{1}),(x_{2},m_{2},t_{2})\in\mathbb{R}^{d}\times
H^{1}(\mathbb{R}^{d})\times[0,T]$, and take $\varepsilon>0$. Let
$\gamma_{2}\in\Gamma(t_{2})$ be such that

$\underline{V}(x_{2},m_{2},t_{2})\geq\sup_{\beta\in{\mathcal{B}}(t_{2})}J(x_{2},m_{2},t_{2},\gamma_{2}[\beta],\beta)-\varepsilon.$

We define $\gamma_{2}^{1}\in\Gamma(t_{1})$ as

$\gamma_{2}^{1}[\beta](s)=\begin{cases}\gamma_{2}[\tilde{\beta}](s)&\text{for
any }s\ \text{if }t_{2}\leq t_{1}\\\ \bar{a}&\text{for }t_{1}\leq s\leq t_{2}\
\text{if }t_{1}\leq t_{2}\\\ \gamma_{2}[\beta_{|[t_{2},T]}](s)&\text{for
}s\geq t_{2}\ \text{if }t_{1}\leq t_{2}\end{cases},$

where $\bar{a}\in\mathbb{R}^{d}$ is any a priori fixed vector and, if
$t_{2}\leq t_{1}$,

$\tilde{\beta}(\cdot,s)=\begin{cases}\beta(\cdot,t_{1})&\text{if }t_{2}\leq
s\leq t_{1}\\\ \beta(\cdot,s)&\text{if }s\geq t_{1}\end{cases}.$

We then take $\beta_{1}\in{\mathcal{B}}(t_{1})$ such that

$J(x_{1},m_{1},t_{1},\gamma_{2}^{1}[\beta_{1}],\beta_{1})\geq\sup_{\beta\in{\mathcal{B}}(t_{1})}J(x_{1},m_{1},t_{1},\gamma_{2}^{1}[\beta],\beta)-\varepsilon.$

Finally, we define $\beta_{1}^{2}\in{\mathcal{B}}(t_{2})$ such that

$\beta_{1}^{2}(\cdot,s)=\beta_{1}(\cdot,s)\ \text{for }s\geq
t_{2}+|t_{1}-t_{2}|.$

Hence we have

$\underline{V}(x_{1},m_{1},t_{1})-\underline{V}(x_{2},m_{2},t_{2})\\\
\leq\sup_{\beta\in{\mathcal{B}}(t_{1})}J(x_{1},m_{1},t_{1},\gamma_{2}^{1}[\beta],\beta)-\sup_{\beta\in{\mathcal{B}}(t_{2})}J(x_{2},m_{2},t_{2},\gamma_{2}[\beta],\beta)+\varepsilon\\\
\leq
J(x_{1},m_{1},t_{1},\gamma_{2}^{1}[\beta_{1}],\beta_{1})-J(x_{2},m_{2},t_{2},\gamma_{2}[\beta_{1}^{2}],\beta_{1}^{2})+2\varepsilon.$

Now, if

$|J(x_{1},m_{1},t_{1},\gamma_{2}^{1}[\beta_{1}],\beta_{1})-J(x_{2},m_{2},t_{2},\gamma_{2}[\beta_{1}^{2}],\beta_{1}^{2})|$
(23)

is infinitesimal as
$|t_{1}-t_{2}|+\|x_{1}-x_{2}\|+\|m_{1}(\cdot)-m_{2}(\cdot)\|_{H^{1}(\mathbb{R}^{d})}$
goes to zero, then we conclude.

Assume $t_{1}\leq t_{2}$. Then (23) is equal to

$\Bigg{|}\int_{t_{1}}^{T}\ell(y_{(x_{1},t_{1})}(s),m(\cdot,s;\beta,t_{1},m_{1}),s,\gamma_{2}^{1}[\beta_{1}],\beta_{1}(\cdot,s))ds+\psi(y_{(x_{1},t_{1})}(T),m(\cdot,T;\beta,t_{1},m_{1}))\\\
-\int_{t_{2}}^{T}\ell(y_{(x_{2},t_{2})}(s),m(\cdot,s;\beta,t_{2},m_{2}),s,\gamma_{2}[\beta_{1}^{2}](s),\beta_{1}^{2}(\cdot,s))ds\\\
-\psi(y_{(x_{1},t_{1})}(T),m(\cdot,T;\beta,t_{2},m_{2}))\Bigg{|}\\\
=\Bigg{|}\int_{t_{1}}^{t_{2}}\ell(y_{(x_{1},t_{1})}(s),m(\cdot,s;\beta,t_{1},m_{1}),s,\bar{a},\beta_{1}(\cdot,s))ds\\\
+\int_{t_{2}}^{T}\ell(y_{(x_{1},t_{1})}(s),m(\cdot,s;\beta,t_{1},m_{1}),s,\gamma_{2}[\beta_{|[t_{2},T]}](s),\beta_{1}^{2}(\cdot,s))ds\\\
-\int_{t_{2}}^{T}\ell(y_{(x_{2},t_{2})}(s),m(\cdot,s;\beta,t_{2},m_{2}),s,\gamma_{2}[\beta_{|[t_{2},T]}](s),\beta_{1}^{2}(\cdot,s))ds\\\
+\psi(y_{(x_{1},t_{1})}(T),m(\cdot,T;\beta,t_{1},m_{1}))-\psi(y_{(x_{2},t_{2})}(T),m(\cdot,T;\beta,t_{2},m_{2}))\Bigg{|}\\\
\leq
G_{1}|t_{1}-t_{2}|+|\psi(y_{(x_{1},t_{1})}(T),m(\cdot,T;\beta,t_{1},m_{1}))-\psi(y_{(x_{2},t_{2})}(T),m(\cdot,T;\beta,t_{2},m_{2}))|\\\
+\int_{t_{2}}^{T}\left|\ell(y_{(x_{1},t_{1})}(s),m(\cdot,s;\beta,t_{1},m_{1}),s,\gamma_{2}[\beta_{|[t_{2},T]}](s),\beta_{1}^{2}(\cdot,s))\right.\\\
\left.-\ell(y_{(x_{2},t_{2})}(s),m(\cdot,s;\beta,t_{2},m_{2}),s,\gamma_{2}[\beta_{|[t_{2},T]}](s),\beta_{1}^{2}(\cdot,s))\right|ds\\\
\leq G_{1}|t_{1}-t_{2}|\\\
+\omega_{\psi}\left(e^{LT}(\|x_{1}-x_{2}\|+C|t_{1}-t_{2}|)+\|m(\cdot,T;\beta,t_{1},m_{1})-m(\cdot,T;\beta,t_{2},m_{2})\|_{H^{1}(\mathbb{R}^{d})}\right)\\\
+T\omega_{\ell}\left(e^{LT}(\|x_{1}-x_{2}\|+C|t_{1}-t_{2}|)+\|m(\cdot,s;\beta,t_{1},m_{1})-m(\cdot,s;\beta,t_{2},m_{2})\|_{H^{1}(\mathbb{R}^{d})}\right)\\\
\leq
G_{1}|t_{1}-t_{2}|+\omega_{\psi}\left(e^{LT}(\|x_{1}-x_{2}\|+C|t_{1}-t_{2}|)+\tilde{L}\|m_{1}(\cdot)-m_{2}(\cdot)\|_{H^{1}({\mathbb{R}}^{d})}\right)\\\
+T\omega_{\ell}\left(e^{LT}(\|x_{1}-x_{2}\|+C|t_{1}-t_{2}|)+\tilde{L}\|m_{1}(\cdot)-m_{2}(\cdot)\|_{H^{1}({\mathbb{R}}^{d})}\right),$

where the last inequality holds due to (14) in §2.1. Therefore, as
$|t_{1}-t_{2}|+\|z_{1}-z_{2}\|+\|m_{1}(\cdot)-m_{2}(\cdot)\|_{H^{1}(\mathbb{R}^{d})}$
goes to zero, (23) is infinitesimal.

Now we assume $t_{2}\leq t_{1}$. Then

$\Bigg{|}\int_{t_{1}}^{T}\ell(y_{(x_{1},t_{1})}(s),m(\cdot,s;\beta,t_{1},m_{1}),s,\gamma_{2}^{1}[\beta_{1}],\beta_{1}(\cdot,s))ds+\psi(y_{(x_{1},t_{1})}(T),m(\cdot,T;\beta,t_{1},m_{1}))\\\
-\int_{t_{2}}^{T}\ell(y_{(x_{2},t_{2})}(s),m(\cdot,s;\beta,t_{2},m_{2}),s,\gamma_{2}[\beta_{1}^{2}](s),\beta_{1}^{2}(\cdot,s))ds\\\
-\psi(y_{(x_{2},t_{2})}(T),m(\cdot,T;\beta,t_{2},m_{2}))\Bigg{|}\\\
=\Bigg{|}\int_{t_{1}}^{T}\ell(y_{(x_{1},t_{1})}(s),m(\cdot,s;\beta,t_{1},m_{1}),s,\gamma_{2}[\beta_{1}](s),\beta_{1}(\cdot,s))ds\\\
+\psi(y_{(x_{1},t_{1})}(T),m(\cdot,T;\beta,t_{1},m_{1}))\\\
-\int_{t_{2}}^{t_{1}}\ell(y_{(x_{2},t_{2})}(s),m(\cdot,s;\beta,t_{2},m_{2}),s,\gamma_{2}[\beta_{1}^{2}](s),\beta_{1}^{2}(\cdot,s))ds\\\
-\int_{t_{1}}^{T}\ell(y_{(x_{2},t_{2})}(s),m(\cdot,s;\beta,t_{2},m_{2}),s,\gamma_{2}[\beta_{1}](s),\beta_{1}(\cdot,s))ds-\psi(y_{(x_{2},t_{2})}(T),m(\cdot,T;\beta,t_{2},m_{2}))\Bigg{|}\\\
\leq
G_{1}|t_{1}-t_{2}|+|\psi(y_{(x_{1},t_{1})}(T),m(\cdot,T;\beta,t_{1},m_{1}))-\psi(y_{(x_{2},t_{2})}(T),m(\cdot,T;\beta,t_{2},m_{2}))|\\\
+\int_{t_{1}}^{T}\left|\ell(y_{(x_{1},t_{1})}(s),m(\cdot,s;\beta,t_{1},m_{1}),s,\gamma_{2}[\beta_{1}](s),\beta_{1}(\cdot,s))\right.\\\
\left.-\ell(y_{(x_{2},t_{2})}(s),m(\cdot,s;\beta,t_{2},m_{2}),s,\gamma_{2}[\beta_{1}](s),\beta_{1}(\cdot,s))\right|ds\\\
\leq G_{1}|t_{1}-t_{2}|\\\
+\omega_{\psi}\left(e^{LT}(\|x_{1}-x_{2}\|+C|t_{1}-t_{2}|)+\|m(\cdot,T;\beta,t_{1},m_{1})-m(\cdot,T;\beta,t_{2},m_{2})\|_{H^{1}(\mathbb{R}^{d})}\right)\\\
+T\omega_{\ell}\left(e^{LT}(\|x_{1}-x_{2}\|+C|t_{1}-t_{2}|)+\|m(\cdot,s;\beta,t_{1},m_{1})-m(\cdot,s;\beta,t_{2},m_{2})\|_{H^{1}(\mathbb{R}^{d})}\right)\\\
\leq
G_{1}|t_{1}-t_{2}|+\omega_{\psi}\left(e^{LT}(\|x_{1}-x_{2}\|+C|t_{1}-t_{2}|)+\tilde{L}\|m_{1}(\cdot)-m_{2}(\cdot)\|_{H^{1}({\mathbb{R}}^{d})}\right)\\\
+T\omega_{\ell}\left(e^{LT}(\|x_{1}-x_{2}\|+C|t_{1}-t_{2}|)+\tilde{L}\|m_{1}(\cdot)-m_{2}(\cdot)\|_{H^{1}({\mathbb{R}}^{d})}\right),$

where in the last inequality we have used again (14). Letting
$|t_{1}-t_{2}|+\|z_{1}-z_{2}\|+\|m_{1}(\cdot)-m_{2}(\cdot)\|_{H^{1}(\mathbb{R}^{d})}$
go to zero, we conclude. ∎

##  4 The Hamilton-Jacobi-Isaacs equation for $\underline{V}$

The value function $\underline{V}$ is a function of
$(x,m,t)\in{\mathbb{R}}^{d}\times H^{1}({\mathbb{R}}^{d})\times[0,T]$. Here we
want to derive the corresponding Hamilton-Jacobi-Isaacs equation and consider
it in a suitable set $\tilde{X}$ for the variables $(x,m,t)$, which is compact
in ${\mathbb{R}}^{d}\times L^{2}({\mathbb{R}}^{d})\times[0,T]$ and invariant
for the controlled evolutions $y_{(x,t)}(\cdot;{\alpha})$,
$m(\cdot,s;\beta,t,\bar{m})$. In order to determine such a possible set
$\tilde{X}$, we first fix an open bounded subset
$\Omega\subset{\mathbb{R}}^{d}$ which is going to contain the supports of all
the admissible initial distribution $m_{0}$, at time $t=0$, for the mass.
Moreover, we fix a constant $K>0$ such that all the admissible initial
distributions of the mass belong to

${\cal M}=\left\\{m_{0}\in
W^{1,\infty}(\Omega):\|m_{0}\|_{W^{1,\infty}(\Omega)}\leq K,\
\mathrm{supp}\;m_{0}\subset\Omega\right\\}.$

Referring to (15), we set $B:=\max\\{1+\tilde{M},L_{\varphi^{-1}}\\}K$ and for
all $m_{0}\in{\cal M}$, $\beta\in{\mathcal{B}}(0)$ and $s\in[0,T]$ we get

$\|m(\cdot,s;\beta,0,m_{0})\|_{W^{1,\infty}(\Omega)}\leq Be^{MT}.$ (24)

Now, denoting by $M$ a bound for $\|\beta(\cdot,t)\|_{\infty}$, as in (7),
i.e. the maximal possible velocity of the mass’ agents, we define, for all
$t\in[0,T]$ and for all $s\in[t,T]$, the bounded sets

$\Omega_{1}(t)=\overline{B}(\Omega,Mt),\ \
\Omega_{1}(s,t)=\overline{B}\left(\Omega_{1}(t),M(s-t)\right),$

being, as example,
$\overline{B}(\Omega,Mt)=\\{x\in{\mathbb{R}}^{d}:\text{dist}(x,\Omega)\leq
Mt\\}$. Note that $\Omega_{1}(t)$ represents the maximal set that can be
invaded, up to the time $t$, by agents starting from $\overline{\Omega}$ at
the time $t_{0}=0$, whereas $\Omega_{1}(s,t)$ represents the maximal set that
can be invaded, at the time $s$, by agents starting from $\Omega_{1}(t)$ at
the time $t$. Of course, we have a sort of semigroup property:
$\Omega_{1}(0)=\overline{\Omega}$, $\Omega_{1}(s,t)=\Omega_{1}(s)$ for all
$t\in[0,s]$ and, in particular, $\Omega_{1}(T)=\Omega_{1}(T,t)$ is the maximal
set that can be invaded at the final time $T$ by a mass having support in
$\Omega_{1}(t)$ at the time $t$. That is, assuming $\bar{m}\in
W^{1,\infty}({\mathbb{R}}^{d})\cap H^{1}({\mathbb{R}}^{d})$ with
$\mathrm{supp}\;\bar{m}\subset\Omega_{1}(t):=\overline{B}(\Omega,r(t))$, we
have, independently of $\beta\in{\mathcal{B}}(t)$,

$\mathrm{supp}\;m(\cdot,s^{\prime};\beta,\bar{m})\subset\Omega_{1}(s^{\prime},t)\subset\Omega_{1}(s^{\prime\prime},t)\subset\Omega_{1}(T,t)=\Omega_{1}(T)\
\ \text{for all }t\leq s^{\prime}\leq s^{\prime\prime}\leq T.\\\ $

For all $t\in[0,T]$, we then define the following set (recall (24))

$X(t):=\left\\{\bar{m}\in W^{1,\infty}(\Omega_{1}(T)):\right.\\\
\left.\mathrm{supp}\;\bar{m}\subset\Omega_{1}(t),\
\|m(\cdot,s;\beta,t,\bar{m})\|_{W^{1,\infty}(\Omega_{1}(T))}\leq Be^{MT}\
\text{for all }s\in[t,T]\text{ and }\beta\in{\mathcal{B}}(t)\right\\}.$

In this way, the domain of the Hamiltonian we are going to consider (see (26)
below) concerning the pair $(m,t)$ is defined as the following set:

$X=\bigcup_{t\in[0,T]}(X(t)\times\\{t\\})=\left\\{(\bar{m},t)\in
W^{1,\infty}(\Omega_{1}(T))\times[0,T]:\right.\\\
\left.\mathrm{supp}\;\bar{m}\subset\Omega_{1}(t),\
\|m(\cdot,s;\beta,t,\bar{m})\|_{W^{1,\infty}(\Omega_{1}(T))}\leq Be^{MT}\
\text{for all }s\in[t,T]\text{ and }\beta\in{\mathcal{B}}(t)\right\\}.$

Such a set $X$ is bounded in $W^{1,\infty}(\Omega_{1}(T))\times[0,T]$
$\left(\|\bar{m}\|=\|m(\cdot,t;\beta,t,\bar{m})\|\leq Be^{MT}\right)$ and
hence relatively compact in $H^{1}(\Omega_{1}(T))\times[0,T]$ (and in
$L^{2}(\Omega_{1}(T))\times[0,T]$). Actually, it is indeed compact, because it
is closed in $H^{1}\times[0,T]$. Let us prove it.

Let $(m_{n},t_{n})\in X$ be converging in $H^{1}\times[0,T]$ to
$(\bar{m},\bar{t})$. We want to show that $(\bar{m},\bar{t})\in X$. Since
$(m_{n},t_{n})\in X$, the sequence $m_{n}$ is bounded in $W^{1,\infty}$ and
hence the convergence to $m$ is also uniform, from which we obtain
$\mathrm{supp}\;\bar{m}\subset\Omega_{1}(\bar{t})$. Now, take $s>\bar{t}$ and
$n$ sufficiently large such that $t_{n}<s$. By (13) and (14) (the modulus of
continuity of (13) is independent of $m_{n}\in X$), we have the convergence in
$H^{1}$ of $m(\cdot,s;\beta,t_{n},m_{n})$ to
$m(\cdot,s;\beta,\bar{t},\bar{m})$. The boundedness of the former in
$W^{1,\infty}$ implies the weak-star convergence, which implies

$\|m(\cdot,s;\beta,\bar{t},\bar{m})\|_{W^{1,\infty}}\leq\liminf\|m(\cdot,s;\beta,t_{n},m_{n})\|_{W^{1,\infty}}\leq
Be^{MT}.$

Due to compactness, the $H^{1}$ and $L^{2}$ norms are topologically equivalent
in $X$, that is there exists a modulus of continuity $\tilde{\omega}$ such
that for every $(m_{1},t_{1}),(m_{2},t_{2})\in X$,

$\|m_{1}-m_{2}\|_{H^{1}(\Omega_{1}(T))}\leq\tilde{\omega}\left(\|m_{1}-m_{2}\|_{L^{2}(\Omega_{1}(T))}\right).$
(25)

Moreover, $X$ contains all the possible trajectories
$s\longmapsto(m(\cdot,s;\beta,t,\bar{m}),s)=(m(s),s)$ (with field
$\beta\in{\mathcal{B}}(t)$) starting from any $(\bar{m},t)\in X$. Indeed, for
all $s\in[t,T]$ and for $\tau\in[s,T]$, it is

$\|m(\cdot,\tau;\beta,s,m(s))\|_{W^{1,\infty}(\Omega_{1}(T))}=\|m(\cdot,\tau;\beta,t,\bar{m})\|_{W^{1,\infty}(\Omega_{1}(T))}\leq
Be^{MT}$

and hence $(m(s),s)\in X$ because, by definition,
$\mathrm{supp}\;m(s)\subset\Omega_{1}(s)$ since
$\mathrm{supp}\;\bar{m}\subset\Omega_{1}(t)$. Hence $X$ is viable under the
evolution of the continuity equation with fields $\beta\in{\mathcal{B}}(t)$.
In particular, it contains all the trajectories $t\longmapsto(m(t),t)$
starting from $(m_{0},0)$ with $m_{0}\in{\mathcal{M}}$.

A similar construction is naturally done for the trajectories of the single
player of (18). More precisely, we define
$\Omega_{2}(t):=\overline{B}(\Omega,Ct)$, where $C$ is the bound for
$\|f(\cdot,a)\|$, and hence we have that
$y_{(x,t)}(\tau)\in\Omega_{2}(s,t):=\overline{B}(\Omega_{2}(t),C(s-t))$ for
every $\tau\in[t,s]$. We define then the set

$\tilde{X}:=\bigcup_{t\in[0,T]}(\Omega_{2}(t)\times X(t)\times\\{t\\})=\\\
\left\\{(x,m,t)\in\Omega_{2}(T)\times
W^{1,\infty}(\Omega_{1}(T))\times[0,T]:x\in\Omega_{2}(t),\ (m,t)\in
X\right\\}.$

Hence $\tilde{X}$ is a compact set in ${\mathbb{R}}^{d}\times
H^{1}\times[0,T]$, and is viable under all the admissible trajectories with
controls $\alpha\in{\mathcal{A}}(t)$ and $\beta\in{\mathcal{B}}(t)$. That is,
all our admissible trajectories in the time interval $[0,T]$, when starting
from points of $\tilde{X}$, can not exit from $\tilde{X}$ itself. From the
point of view of the Hamilton-Jacobi-Isaacs equation that we are going to
state and study on $\tilde{X}$, this means that, besides the time boundary at
$t=0$ and $t=T$, there will not be other spatial boundaries requiring boundary
conditions.

We then define the Hamiltonian $H:\tilde{X}\times{\mathbb{R}}^{d}\times
L^{2}({\mathbb{R}}^{d})\longrightarrow{\mathbb{R}}$ as

$H(x,m,t,p,q):=\min_{b\in\tilde{\mathcal{B}}}\max_{a\in A}\left\\{-f(x,a)\cdot
p+\langle
q,\operatorname{div}(bm)\rangle_{L^{2}({\mathbb{R}}^{d})}-\ell(x,m,t,a,b)\right\\},$
(26)

where $\langle\cdot,\cdot\rangle_{L^{2}({\mathbb{R}}^{d})}$ denotes the scalar
product in $L^{2}({\mathbb{R}}^{d})$. In the sequel, we denote by $D_{x}$ the
gradient $1\times d$, and we are going to consider the fields $\beta(\cdot,s)$
defined on the compact set $\Omega_{1}(T)$, i.e.,
$\beta:\Omega_{1}(T)\times[t,T]\longrightarrow{\mathbb{R}}^{d}$.

######  Theorem 4.1.

Under the hypotheses of §3, the lower value function $\underline{V}$ is a
viscosity solution of

$\begin{cases}-V_{t}+H(x,m,t,D_{x}V,D_{m}V)=0&\text{in
}\tilde{X}\times{\mathbb{R}}^{d}\times L^{2}({\mathbb{R}}^{d})\\\
V(x,m,T)=\psi(x,m),&(x,m)\in\Omega_{2}(T)\times X(T)\times\\{T\\}\end{cases},$
(27)

where as test functions for the definition of viscosity solution we are going
to take $\varphi\in C^{1}({\mathbb{R}}^{d}\times
L^{2}({\mathbb{R}}^{d})\times[0,T])$.

For the proof, we need the following

######  Lemma 1.

Assume the hypotheses of Theorem 4.1. Let $(x,m,t)\in\tilde{X}$ and
$\varphi\in C^{1}({\mathbb{R}}^{d}\times L^{2}({\mathbb{R}}^{d})\times[0,T])$
be such that

$-\varphi_{t}(x,m,t)+H(x,m,t,D_{x}\varphi(x,m,t),D_{m}\varphi(x,m,t))=\theta>0.$
(28)

Then there exists $\gamma^{*}\in\Gamma(t)$ such that for all
$\beta\in{\mathcal{B}}(t)$ and $\tau>t$ sufficiently close to $t$,

$\int_{t}^{\tau}\Big{\\{}\ell(y_{(x,t)},m(\cdot,s;\beta,t,\bar{m}),s,\gamma^{*}[\beta],\beta(\cdot,s))+f(y_{(x,t)},\gamma^{*}[\beta])\cdot
D_{x}\varphi(y_{(x,t)},m(\cdot,s;\beta,t,\bar{m}),s)\\\ -\langle
D_{m}\varphi(y_{(x,t)},m(\cdot,s;\beta,t,\bar{m}),s),\operatorname{div}(\beta(\cdot,s)m(\cdot,s;\beta,t,\bar{m}))\rangle_{L^{2}(\Omega_{1}(T))}\\\
+\varphi_{t}(y_{(x,t)},m(\cdot,s;\beta,t,\bar{m}),s)\Big{\\}}ds\leq-\frac{\theta(\tau-t)}{4},$
(29)

where $y_{(x,t)}=y_{(x,t)}(\cdot,\gamma^{*}[\beta])$.

###### Proof.

We define

$\Lambda(z,m,t,a,b):=-\varphi_{t}(z,m,t)-f(z,a)\cdot D_{x}\varphi(z,m,t)\\\
+\langle
D_{m}\varphi(z,m,t),\operatorname{div}(bm)\rangle_{L^{2}(\Omega_{1}(T))}-\ell(z,m,t,a,b).$

By definition of $H$ and (28), we have

$\min_{b\in\tilde{\mathcal{B}}}\max_{a\in A}\Lambda(x,m,t,a,b)=\theta.$

Then, for any $b\in\tilde{\mathcal{B}}$ there is $a=a(b)\in A$ such that

$\Lambda(x,m,t,a,b)\geq\theta.$

Observe that $\Lambda(x,m,t,a,\cdot)$ is uniformly continuous in
$\overline{\tilde{\mathcal{B}}}$ w.r.t. the $L^{2}$ topology, where we recall
that $\overline{\tilde{\mathcal{B}}}$ is the closure of $\tilde{\mathcal{B}}$
(see Remark 1). This gives that

$\Lambda(z,m,t,a,\zeta)\geq\frac{3\theta}{4}\ \ \text{for all}\ \zeta\in
B(b,r)\cap\overline{\tilde{\mathcal{B}}}$

for some $r=r(b)>0$. Since $\overline{\tilde{\mathcal{B}}}$ is compact in
$L^{2}(\Omega_{1}(T),\mathbb{R}^{d})$ (see Remark 1 again), there exist
finitely many points $b_{1},\ldots,b_{n}\in\overline{\tilde{\mathcal{B}}}$ and
$r_{1},\ldots,r_{n}>0$ such that

$\overline{\tilde{\mathcal{B}}}\subset\bigcup_{i=1}^{n}B(b_{i},r_{i})\cap\overline{\tilde{\mathcal{B}}},$
(30)

and, for $a_{i}=a(b_{i})$,

$\Lambda(x,m,t,a_{i},\zeta)\geq\frac{3\theta}{4}\ \ \text{for all}\ \zeta\in
B(b_{i},r_{i})\cap\overline{\tilde{\mathcal{B}}}.$

Since $\tilde{\mathcal{B}}$ is dense in $\overline{\tilde{\mathcal{B}}}$, we
can assume, without loss of generality, that (30) holds, for suitable
$r_{1},\ldots,r_{n}>0$, with the centers $b_{i}\in\tilde{\mathcal{B}}$ for
every $i=1,\ldots,n$. Now define $\Psi:\tilde{\mathcal{B}}\longrightarrow A$
by setting

$\Psi(b)=a_{k}\ \ \text{if}\ b\in
B(b^{\prime}_{k},r^{\prime}_{k})\setminus\bigcup_{i=1}^{k-1}B(b^{\prime}_{i},r^{\prime}_{i}).$

Observe that for any $\beta\in{\mathcal{B}}(t)$ and $s\in[t,T]$, we have that
$\Psi(\beta(\cdot,s))$ is measurable as the composition of two measurable
functions, and then we can define $\gamma^{*}\in\Gamma(t)$ by setting

$\gamma^{*}[\beta](s):=\Psi(\beta(\cdot,s)).$

By definition of $\Psi$, we have

$\Lambda(x,m,t,\Psi(b),b)\geq\frac{3\theta}{4}\quad\text{for all}\
b\in\tilde{\mathcal{B}},$

and by the continuity of $\Lambda(x,m,t,a,\cdot)$, by general estimates on the
trajectory $y_{(x,t)}$ and by (13), there exists $\tau>t$ such that

$\Lambda(y_{(x,t)}(s),m(\cdot,s;\beta,t,\bar{m}),s,\gamma^{*}[\beta](s),\beta(\cdot,s))\geq\frac{\theta}{2}\quad\text{for}\
t\leq s<\tau\ \text{and for each }\beta\in{\mathcal{B}}(t).$ (31)

By integrating both sides of (31) from $t$ to $\tau$, we obtain (29) for
$\tau$ sufficiently close to $t$. ∎

###### Proof of Theorem 4.1.

At first we prove that $\underline{V}$ is a subsolution of (27). Let
$\varphi\in C^{1}({\mathbb{R}}^{d}\times L^{2}({\mathbb{R}}^{d})\times[0,T])$
and $(\bar{x},\bar{m},\bar{t})$ be a local maximum point for
$\underline{V}-\varphi$, and
$\underline{V}(\bar{x},\bar{m},\bar{t})=\varphi(\bar{x},\bar{m},\bar{t})$. We
assume by contradiction that (28) holds and then, by Lemma 1, there exists
$\gamma^{*}\in\Gamma(\bar{t})$ such that, for all
$\beta\in{\mathcal{B}}(\bar{t})$ and all $\tau$ sufficiently close to
$\bar{t}$,

$\int_{\bar{t}}^{\tau}\ell(y_{(\bar{x},\bar{t})},m(\cdot,s;\beta,\bar{t},\bar{m}),s,\gamma^{*}[\beta],\beta(\cdot,s))ds\\\
+\varphi(y_{(\bar{x},\bar{t})}(\tau),m(\cdot,\tau;\beta,\bar{t},\bar{m}),\tau)-\varphi(\bar{x},\bar{m},\bar{t})\leq-\frac{\theta(\tau-\bar{t})}{4},$
(32)

where $y_{(\bar{x},\bar{t})}=y_{(\bar{x},\bar{t})}(\cdot,\gamma^{*}[\beta])$.
Since $\underline{V}-\varphi$ has a local maximum at
$(\bar{x},\bar{m},\bar{t})$ and
$\underline{V}(\bar{x},\bar{m},\bar{t})=\varphi(\bar{x},\bar{m},\bar{t})$, by
classical estimates on the trajectory $y_{(\bar{x},\bar{t})}$ and by (13), we
have

$\varphi(y_{(\bar{x},\bar{t})}(\tau),m(\cdot,\tau;\beta,\bar{t},\bar{m}),\tau)-\varphi(\bar{x},\bar{m},\bar{t})\geq\underline{V}(y_{(\bar{x},\bar{t})}(\tau),m(\cdot,\tau;\beta,\bar{t},\bar{m}),\tau)-\varphi(\bar{x},\bar{m},\bar{t})$

for $\tau$ sufficiently close to $\bar{t}$. Plugging this into (32), we obtain

$\inf_{\gamma\in\Gamma(\bar{t})}\sup_{\beta\in{\mathcal{B}}(\bar{t})}\left\\{\int_{\bar{t}}^{\tau}\ell(y_{(\bar{x},\bar{t})},m(\cdot,s;\beta,\bar{t},\bar{m}),s,\gamma^{*}[\beta],\beta)ds+\underline{V}(y_{(\bar{x},\bar{t})}(\tau),m(\cdot,\tau;\beta,\bar{t},\bar{m}),\tau)\right\\}\\\
-\underline{V}(\bar{x},\bar{m},\bar{t})\leq-\frac{\theta(\tau-\bar{t})}{4}<0,$

which contradicts the inequality “$\leq$” in (21). Then $\underline{V}$ is a
subsolution of (27).

Next we show that $\underline{V}$ is a supersolution of (27). Let $\varphi\in
C^{1}({\mathbb{R}}^{d}\times L^{2}({\mathbb{R}}^{d})\times[0,T])$ and
$(\bar{x},\bar{m},\bar{t})$ be a local minimum point for
$\underline{V}-\varphi$, and
$\underline{V}(\bar{x},\bar{m},\bar{t})=\varphi(\bar{x},\bar{m},\bar{t})$.
Suppose by contradiction that

$-\varphi_{t}(\bar{x},\bar{m},\bar{t})+H(\bar{x},\bar{m},\bar{t},D_{x}\underline{V},D_{m}\underline{V})=-\theta<0.$

By the definition of $H$, there exists $b^{*}\in\tilde{\mathcal{B}}$ such that

$-\varphi_{t}(\bar{x},\bar{m},\bar{t})-f(\bar{x},a)\cdot
D_{x}\varphi(\bar{x},\bar{m},\bar{t})+\langle
D_{m}\varphi(\bar{x},\bar{m},\bar{t}),\operatorname{div}(b^{*}\bar{m})\rangle_{L^{2}(\Omega_{1}(T))}-\ell(\bar{x},\bar{m},\bar{t},a,b^{*})\leq-\theta$

for all $a\in A$. For $\tau$ sufficiently close to $\bar{t}$ and any
$\gamma\in\Gamma(\bar{t})$, we have

$-\varphi_{t}(y_{(\bar{x},\bar{t})}(s),m(\cdot,s;b^{*},\bar{t},\bar{m}),s)-f(y_{(\bar{x},\bar{t})}(s),\gamma[b^{*}](s))\cdot
D_{x}\varphi(y_{(\bar{x},\bar{t})}(s),m(\cdot,s;b^{*},\bar{t},\bar{m}),s)\\\
+\langle
D_{m}\varphi(y_{(\bar{x},\bar{t})}(s),m(\cdot,s;b^{*},\bar{t},\bar{m}),s),\operatorname{div}(b^{*}m(\cdot,s;b^{*},\bar{t},\bar{m}))\rangle_{L^{2}(\Omega_{1}(T))}\\\
-\ell(y_{(\bar{x},\bar{t})}(s),m(\cdot,s;b^{*},\bar{t},\bar{m}),s,\gamma[b^{*}](s),b^{*})\leq-\frac{\theta}{2}$

for every $\bar{t}\leq s<\tau$, where
$y_{(\bar{x},\bar{t})}(s)=y_{(\bar{x},\bar{t})}(s;\gamma[b^{*}])$. By
integrating from $\bar{t}$ to $\tau$, and by (18) and (19), we get

$\varphi(\bar{x},\bar{m},\bar{t})-\varphi(y_{(\bar{x},\bar{t})}(\tau),m(\cdot,\tau;b^{*},\bar{t},\bar{m}),\tau)-\int_{\bar{t}}^{\tau}\ell(y_{(\bar{x},\bar{t})}(s),m(\cdot,s;b^{*},\bar{t},\bar{m}),s,\gamma[b^{*}](s),b^{*})ds\\\
\leq-\frac{\theta(\tau-\bar{t})}{4},$

for $\tau$ sufficiently close to $\bar{t}$. From

$\varphi(\bar{x},\bar{m},\bar{t})-\varphi(y_{(\bar{x},\bar{t})}(\tau),m(\cdot,\tau;b^{*},\bar{t},\bar{m}),\tau)\geq\underline{V}(\bar{x},\bar{m},\bar{t})-\underline{V}(y_{(\bar{x},\bar{t})}(\tau),m(\cdot,\tau;b^{*},\bar{t},\bar{m}),\tau),$

we obtain

$\underline{V}(y_{(\bar{x},\bar{t})}(\tau),m(\cdot,\tau;b^{*},\bar{t},\bar{m}),\tau)+\int_{\bar{t}}^{\tau}\ell(y_{(\bar{x},\bar{t})}(s),m(\cdot,s;b^{*},\bar{t},\bar{m}),s,\gamma[b^{*}](s),b^{*})ds\\\
\geq\frac{\theta(\tau-\bar{t})}{4}+\underline{V}(\bar{x},\bar{m},\bar{t})$

and thus

$\inf_{\gamma\in\Gamma(\bar{t})}\sup_{\beta\in{\mathcal{B}}(\bar{t})}\Bigg{\\{}\int_{\bar{t}}^{\tau}\ell(y_{(\bar{x},\bar{t})}(s),m(\cdot,s;\beta,\bar{t},\bar{m}),s,\gamma[\beta](s),\beta(\cdot,s))ds\\\
+\underline{V}(y_{(\bar{x},\bar{t})}(\tau;\gamma[\beta]),m(\cdot,\tau;\beta,\bar{t},\bar{m}),\tau)\Bigg{\\}}>\underline{V}(\bar{x},\bar{m},\bar{t}),$

which contradicts (21). This completes the proof that $\underline{V}$ is a
viscosity solution of (27). ∎

Now we want to characterize the lower value function of the problem as the
unique solution of the HJI equation (27).

######  Theorem 4.2.

Assume the hypotheses of Theorem 4.1. Let $u_{1},u_{2}$ be bounded and
uniformly continuous functions and, respectively, viscosity sub- and
supersolution of

$-V_{t}+H(x,m,t,D_{x}V,D_{m}V)=0\quad\text{in
}\tilde{X}\times{\mathbb{R}}^{d}\times L^{2}({\mathbb{R}}^{d}).$

Therefore if $u_{1}\leq u_{2}$ on $\Omega_{2}(T)\times X(T)\times\\{T\\}$,
then $u_{1}\leq u_{2}$ in $\tilde{X}$.

For the proof, we need the following lemmas.

######  Lemma 2.

For all $\xi>0$, $p\in{\mathbb{R}}^{d}$,
$(x_{1},m_{1},t_{1}),(x_{2},m_{2},t_{2})\in\tilde{X}$ and
$q=\frac{2(m_{1}-m_{2})}{\xi^{2}}$, we have

$\left|H\left(x_{1},m_{1},t_{1},p,\frac{2(m_{1}-m_{2})}{\xi^{2}}\right)-H\left(x_{2},m_{2},t_{2},p,\frac{2(m_{1}-m_{2})}{\xi^{2}}\right)\right|\\\
\leq
L\|x_{1}-x_{2}\|\|p\|+M\frac{\|m_{1}-m_{2}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}\\\
+\omega_{\ell}\left(\|x_{1}-x_{2}\|+\|m_{1}-m_{2}\|_{H^{1}(\Omega_{1}(T))}+|t_{1}-t_{2}|\right),$

where we recall that $L$ is the Lipschitz constant of $f(\cdot,a)$ and
$\omega_{\ell}$ is the modulus of continuity of the running cost $\ell$ (see
§3).

###### Proof.

Let us fix $b^{\prime}\in\tilde{\mathcal{B}}$ such that

$H\left(x_{1},m_{1},t_{1},p,\frac{2(m_{1}-m_{2})}{\xi^{2}}\right)\\\
\geq-f(x_{1},a)\cdot
p+\left\langle\frac{2(m_{1}-m_{2})}{\xi^{2}},\operatorname{div}(b^{\prime}m_{1})\right\rangle_{L^{2}(\Omega_{1}(T))}-\ell(x_{1},m_{1},t_{1},a,b^{\prime})\quad\text{for
every }a\in A.$

Then take $a^{\prime}\in A$ such that

$H\left(x_{2},m_{2},t_{2},p,\frac{2(m_{1}-m_{2})}{\xi^{2}}\right)\\\
\leq-f(x_{2},a^{\prime})\cdot
p+\left\langle\frac{2(m_{1}-m_{2})}{\xi^{2}},\operatorname{div}(b^{\prime}m_{2})\right\rangle_{L^{2}(\Omega_{1}(T))}-\ell(x_{2},m_{2},t_{2},a^{\prime},b^{\prime}).$

Therefore

$\left|H\left(x_{2},m_{2},t_{2},p,\frac{2(m_{1}-m_{2})}{\xi^{2}}\right)-H\left(x_{1},m_{1},t_{1},p,\frac{2(m_{1}-m_{2})}{\xi^{2}}\right)\right|\\\
\leq\left|-f(x_{2},a^{\prime})\cdot
p+\left\langle\frac{2(m_{1}-m_{2})}{\xi^{2}},\operatorname{div}(b^{\prime}m_{2})\right\rangle_{L^{2}(\Omega_{1}(T))}-\ell(x_{2},m_{2},t_{2},a^{\prime},b^{\prime})\right.\\\
\left.+f(x_{1},a^{\prime})\cdot
p-\left\langle\frac{2(m_{1}-m_{2})}{\xi^{2}},\operatorname{div}(b^{\prime}m_{1})\right\rangle_{L^{2}(\Omega_{1}(T))}+\ell(x_{1},m_{1},t_{1},a^{\prime},b^{\prime})\right|\\\
\leq\Bigg{|}(f(x_{1},a^{\prime})-f(x_{2},a^{\prime}))\cdot
p+\left\langle\frac{2(m_{1}-m_{2})}{\xi^{2}},\operatorname{div}(b^{\prime}(m_{2}-m_{1}))\right\rangle_{L^{2}(\Omega_{1}(T))}\\\
+\omega_{\ell}\left(\|x_{1}-x_{2}\|+\|m_{1}-m_{2}\|_{H^{1}(\Omega_{1}(T))}+|t_{1}-t_{2}|\right)\Bigg{|}\\\
\leq\|f(x_{1},a^{\prime})-f(x_{2},a^{\prime})\|\|p\|+\left|\frac{1}{\xi^{2}}\int_{\Omega_{1}(T)}\operatorname{div}(b^{\prime})(m_{1}-m_{2})^{2}dx\right|\\\
+\omega_{\ell}\left(\|x_{1}-x_{2}\|+\|m_{1}-m_{2}\|_{H^{1}(\Omega_{1}(T))}+|t_{1}-t_{2}|\right)\\\
\leq
L\|x_{1}-x_{2}\|\|p\|+M\frac{\|m_{1}-m_{2}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}\\\
+\omega_{\ell}\left(\|x_{1}-x_{2}\|+\|m_{1}-m_{2}\|_{H^{1}(\Omega_{1}(T))}+|t_{1}-t_{2}|\right),\\\
$

where in the second-to-last inequality we have integrated by parts and used
the fact that $m_{1},m_{2}$ vanish at $\partial\Omega_{1}(T)$. ∎

######  Lemma 3.

If $u_{1},u_{2}$ are respectively sub and supersolution of (27) in
$\tilde{X}$, they are also sub and supersolution in $\Omega_{2}(t)\times
X(t)\times[0,T[$. More precisely, the viscosity inequalities hold if the
maximum or minimum points are on $\Omega_{2}(0)\times X(0)\times\\{0\\}$.

###### Proof.

The proof goes similarly as the one of [6], Ch. II, §2, Lemma 2.10. ∎

Proof (of Theorem 4.2). The aim of the proof is to show that
$G=\sup_{\tilde{X}}(u_{1}-u_{2})$ is less or equal to $0$. We argue by
contradiction assuming that $G>0$.

In order to simplify the proof, we change $u_{1}$ with
$(u_{1}(x,m,t))_{\eta}:=u_{1}(x,m,t)-\eta t$ for some $\eta>0$ sufficiently
small, in such a way that, without loss of generality, we may assume that
$u_{1}$ is a strict subsolution of (27) since $(u_{1})_{\eta}$ is a
subsolution of

$-\frac{\partial}{\partial
t}(u_{1})_{\eta}+H(x,t,m,D_{x}(u_{1})_{\eta},D_{m}(u_{1})_{\eta})\leq-\eta<0\quad\text{in}\
\tilde{X}.$ (33)

To perform the proof, it is sufficient to show that $(u_{1})_{\eta}\leq u_{2}$
in $\tilde{X}$ for any $\eta$ and then to let $\eta$ tend to $0$. Notice also
that we still have $(u_{1})_{\eta}\leq u_{2}$ on $\Omega_{2}(T)\times
X(T)\times\\{T\\}$. To simplify the notation, and since the proof is clearly
reduced to compare $(u_{1})_{\eta}$ and $u_{2}$, we drop the $\eta$ and use
the notation $u_{1}$ instead of $(u_{1})_{\eta}$.

Next, we consider the difficulty with $\Omega_{2}(0)\times X(0)\times\\{0\\}$:
a priori, we do not know if $u_{1}\leq u_{2}$ on this part of the boundary and
a maximum point of $u_{1}-u_{2}$ can be located there. It is solved by Lemma
3.

Now, since $u_{1}$ and $u_{2}$ are not smooth, we need an argument in order to
be able to use the definition of viscosity solution. We introduce then the
test function

$\Psi_{\varepsilon,\xi,{\alpha}}(x^{\prime},m^{\prime},t^{\prime},x^{\prime\prime},m^{\prime\prime},t^{\prime\prime})=u_{1}(x^{\prime},m^{\prime},t^{\prime})-u_{2}(x^{\prime\prime},m^{\prime\prime},t^{\prime\prime})\\\
-\frac{\|x^{\prime}-x^{\prime\prime}\|^{2}}{\varepsilon^{2}}-\frac{\|m^{\prime}-m^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}-\frac{|t^{\prime}-t^{\prime\prime}|^{2}}{{\alpha}^{2}}.$

Since $\Psi_{\varepsilon,\xi,{\alpha}}$ is continuous on
$\tilde{X}\times\tilde{X}$, there exists a maximum point
$(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime},\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})$,
and we set
$\bar{G}:=\Psi_{\varepsilon,\xi,{\alpha}}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime},\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})$.
We prove that

  * $(1)$

when $\varepsilon,\xi,{\sigma}\to 0$, then $\bar{G}\to G$;

  * $(2)$

$u_{1}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})-u_{2}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})\to
G$ as $\varepsilon,\xi,{\alpha}\to 0$;

  * $(3)$

we have

$\frac{\|x^{\prime}-x^{\prime\prime}\|^{2}}{\varepsilon^{2}},\frac{\|m^{\prime}-m^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}},\frac{|t^{\prime}-t^{\prime\prime}|^{2}}{{\alpha}^{2}}\to
0\quad\text{as }\ \varepsilon,\xi,{\alpha}\to 0;$

  * $(4)$

$(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime}),(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})\in\tilde{X}$.

Since
$(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime},\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})$
is a maximum point of $\Psi_{\varepsilon,\xi,{\alpha}}$, for any
$(x^{\prime},m^{\prime},t^{\prime}),(x^{\prime\prime},m^{\prime\prime},t^{\prime\prime})\in\tilde{X}$,
we have

$u_{1}(x^{\prime},m^{\prime},t^{\prime})-u_{2}(x^{\prime\prime},m^{\prime\prime},t^{\prime\prime})-\frac{\|x^{\prime}-x^{\prime\prime}\|^{2}}{\varepsilon^{2}}-\frac{\|m^{\prime}-m^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}-\frac{|t^{\prime}-t^{\prime\prime}|^{2}}{{\alpha}^{2}}\\\
\leq
u_{1}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})-u_{2}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})-\frac{\|\tilde{x}^{\prime}-\tilde{x}^{\prime\prime}\|^{2}}{\varepsilon^{2}}-\frac{\|\tilde{m}^{\prime}-\tilde{m}^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}-\frac{|\tilde{t}^{\prime}-\tilde{t}^{\prime\prime}|^{2}}{{\alpha}^{2}}=\bar{G}.$

We choose $x^{\prime}=x^{\prime\prime}$, $m^{\prime}=m^{\prime\prime}$ and
$t^{\prime}=t^{\prime\prime}$ in the left-hand side to get

$u_{1}(x^{\prime},m^{\prime},t^{\prime})-u_{2}(x^{\prime},m^{\prime},t^{\prime})\leq\bar{G}\quad\text{for
all }(x^{\prime},m^{\prime},t^{\prime})\in\tilde{X},$

and, by taking the supremum over $(x^{\prime},m^{\prime},t^{\prime})$, we
obtain the inequality $G\leq\bar{G}$.

Since $u_{1}$ and $u_{2}$ are bounded, we can set
$R:=\max\\{\|u_{1}\|_{\infty},\|u_{2}\|_{\infty}\\}$ and we also have, by
arguing in an analogous way,

$G\leq
u_{1}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})-u_{2}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})-\frac{\|\tilde{x}^{\prime}-\tilde{x}^{\prime\prime}\|^{2}}{\varepsilon^{2}}-\frac{\|\tilde{m}^{\prime}-\tilde{m}^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}-\frac{|\tilde{t}^{\prime}-\tilde{t}^{\prime\prime}|^{2}}{{\alpha}^{2}}\\\
\leq
2R-\frac{\|\tilde{x}^{\prime}-\tilde{x}^{\prime\prime}\|^{2}}{\varepsilon^{2}}-\frac{\|\tilde{m}^{\prime}-\tilde{m}^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}-\frac{|\tilde{t}^{\prime}-\tilde{t}^{\prime\prime}|^{2}}{{\alpha}^{2}}.$

Recalling that $G>0$, we deduce

$\frac{\|\tilde{x}^{\prime}-\tilde{x}^{\prime\prime}\|^{2}}{\varepsilon^{2}}+\frac{\|\tilde{m}^{\prime}-\tilde{m}^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}+\frac{|\tilde{t}^{\prime}-\tilde{t}^{\prime\prime}|^{2}}{{\alpha}^{2}}\leq
2R.$ (34)

In particular,
$\|\tilde{x}^{\prime}-\tilde{x}^{\prime\prime}\|,\|\tilde{m}^{\prime}-\tilde{m}^{\prime\prime}\|_{L^{2}(\Omega_{1}(T))},|\tilde{t}^{\prime}-\tilde{t}^{\prime\prime}|\to
0$ as $\varepsilon,\xi,{\alpha}\to 0$.

Now, since $\tilde{X}$ is compact in ${\mathbb{R}}^{d}\times
L^{2}({\mathbb{R}}^{d})\times[0,T]$, we may assume, without loss of
generality, that the points
$(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime}),(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})$
converge to the same one: we deduce from this property that

$\liminf(u_{1}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})-u_{2}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime}))\leq\limsup(u_{1}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})-u_{2}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime}))\leq
G,$

and the inequality

$G\leq
u_{1}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})-u_{2}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})-\frac{\|\tilde{x}^{\prime}-\tilde{x}^{\prime\prime}\|^{2}}{\varepsilon^{2}}-\frac{\|\tilde{m}^{\prime}-\tilde{m}^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}-\frac{|\tilde{t}^{\prime}-\tilde{t}^{\prime\prime}|^{2}}{{\alpha}^{2}}\\\
\leq
u_{1}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})-u_{2}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})$
(35)

implies that necessarily
$\lim(u_{1}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})-u_{2}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime}))=G$.
Finally, because again of (35),

$\bar{G}=u_{1}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})-u_{2}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})-\frac{\|\tilde{x}^{\prime}-\tilde{x}^{\prime\prime}\|^{2}}{\varepsilon^{2}}-\frac{\|\tilde{m}^{\prime}-\tilde{m}^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}-\frac{|\tilde{t}^{\prime}-\tilde{t}^{\prime\prime}|^{2}}{{\alpha}^{2}}\to
G$

and, since
$u_{1}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})-u_{2}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})\to
G$, we immediately get

$-\frac{\|\tilde{x}^{\prime}-\tilde{x}^{\prime\prime}\|^{2}}{\varepsilon^{2}}-\frac{\|\tilde{m}^{\prime}-\tilde{m}^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}-\frac{|\tilde{t}^{\prime}-\tilde{t}^{\prime\prime}|^{2}}{{\alpha}^{2}}\to
0,$

and we have proved $(1)$, $(2)$ and $(3)$.

For $(4)$, it is enough to observe that, if the point
$(x^{\prime},m^{\prime},t^{\prime})$ is a limit of a subsequence of
$(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime}),(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})$,
then
$u_{1}(x^{\prime},m^{\prime},t^{\prime})-u_{2}(x^{\prime},m^{\prime},t^{\prime})=M>0$
and therefore $(x^{\prime},m^{\prime},t^{\prime})$ cannot be on
$\Omega_{2}(T)\times X(T)\times\\{T\\}$.

Now we assume that $\varepsilon,\xi,{\alpha}$ are sufficiently small in such a
way that $(4)$ holds. Since
$(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime},\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})$
is a maximum point of $\Psi_{\varepsilon,\xi,{\alpha}}$,
$(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})$ is a maximum
point of the function

$(x^{\prime},m^{\prime},t^{\prime})\longmapsto
u_{1}(x^{\prime},m^{\prime},t^{\prime})-\varphi^{1}(x^{\prime},m^{\prime},t^{\prime}),$

where

$\varphi^{1}(x^{\prime},m^{\prime},t^{\prime})=u_{2}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})+\frac{\|x^{\prime}-\tilde{x}^{\prime\prime}\|^{2}}{\varepsilon^{2}}+\frac{\|m^{\prime}-\tilde{m}^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}+\frac{|t^{\prime}-\tilde{t}^{\prime\prime}|^{2}}{{\alpha}^{2}},$

but $u_{1}$ is a viscosity subsolution of (33) and
$(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})\in\tilde{X}$.
Therefore

$-\varphi^{1}_{t}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})+H(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime},D_{x}\varphi^{1}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime}),D_{m}\varphi^{1}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime}))\\\
=\frac{2(\tilde{t}^{\prime\prime}-\tilde{t}^{\prime})}{{\alpha}^{2}}+H\left(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime},\frac{2(\tilde{x}^{\prime}-\tilde{x}^{\prime\prime})}{\varepsilon^{2}},\frac{2(\tilde{m}^{\prime}-\tilde{m}^{\prime\prime})}{\xi^{2}}\right)\leq-\eta.$

In the same way,
$(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})$
is a maximum point of the function

$(x^{\prime\prime},m^{\prime\prime},t^{\prime\prime})\longmapsto-
u_{2}(x^{\prime\prime},m^{\prime\prime},t^{\prime\prime})+\varphi^{2}(x^{\prime\prime},m^{\prime\prime},t^{\prime\prime}),$

where

$\varphi^{2}(x^{\prime\prime},m^{\prime\prime},t^{\prime\prime})=u_{1}(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime})-\frac{\|\tilde{x}^{\prime}-x^{\prime\prime}\|^{2}}{\varepsilon^{2}}-\frac{\|\tilde{m}^{\prime}-m^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}-\frac{|\tilde{t}^{\prime}-t^{\prime\prime}|^{2}}{{\alpha}^{2}},$

and hence
$(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})$
is a minimum point of the function $u_{2}-\varphi^{2}$. But $u_{2}$ is a
viscosity supersolution of (27) and
$(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})\in\tilde{X}$,
and therefore

$-\varphi^{2}_{t}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime})+H(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime},D_{x}\varphi^{2}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime}),D_{m}\varphi^{2}(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime}))\\\
=\frac{2(\tilde{t}^{\prime\prime}-\tilde{t}^{\prime})}{{\alpha}^{2}}+H\left(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime},\frac{2(\tilde{x}^{\prime}-\tilde{x}^{\prime\prime})}{\varepsilon^{2}},\frac{2(\tilde{m}^{\prime}-\tilde{m}^{\prime\prime})}{\xi^{2}}\right)\geq
0.$

Then we subtract the two viscosity inequalities to obtain

$H\left(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime},\frac{2(\tilde{x}^{\prime}-\tilde{x}^{\prime\prime})}{\varepsilon^{2}},\frac{2(\tilde{m}^{\prime}-\tilde{m}^{\prime\prime})}{\xi^{2}}\right)\\\
-H\left(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime},\frac{2(\tilde{x}^{\prime}-\tilde{x}^{\prime\prime})}{\varepsilon^{2}},\frac{2(\tilde{m}^{\prime}-\tilde{m}^{\prime\prime})}{\xi^{2}}\right)\leq-\eta.$

Now, by using Lemma 2, we get

$H\left(\tilde{x}^{\prime},\tilde{m}^{\prime},\tilde{t}^{\prime},\frac{2(\tilde{x}^{\prime}-\tilde{x}^{\prime\prime})}{\varepsilon^{2}},\frac{2(\tilde{m}^{\prime}-\tilde{m}^{\prime\prime})}{\xi^{2}}\right)-H\left(\tilde{x}^{\prime\prime},\tilde{m}^{\prime\prime},\tilde{t}^{\prime\prime},\frac{2(\tilde{x}^{\prime}-\tilde{x}^{\prime\prime})}{\varepsilon^{2}},\frac{2(\tilde{m}^{\prime}-\tilde{m}^{\prime\prime})}{\xi^{2}}\right)\\\
\leq
2L\frac{\|\tilde{x}^{\prime}-\tilde{x}^{\prime\prime}\|^{2}}{\varepsilon^{2}}+M\frac{\|\tilde{m}^{\prime}-\tilde{m}^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}\\\
+\omega_{\ell}\left(\|\tilde{x}^{\prime}-\tilde{x}^{\prime\prime}\|+\|\tilde{m}^{\prime}-\tilde{m}^{\prime\prime}\|_{H^{1}(\Omega_{1}(T))}+|\tilde{t}^{\prime}-\tilde{t}^{\prime\prime}|\right)\leq-\eta.$

But, on one hand,
$\|\tilde{x}^{\prime}-\tilde{x}^{\prime\prime}\|,\|\tilde{m}^{\prime}-\tilde{m}^{\prime\prime}\|_{L^{2}(\Omega_{1}(T))},|\tilde{t}^{\prime}-\tilde{t}^{\prime\prime}|\to
0$ as $\varepsilon,\xi,{\alpha}\to 0$ (see (34)) and hence also
$\|\tilde{m}^{\prime}-\tilde{m}^{\prime\prime}\|_{H^{1}(\Omega_{1}(T))}\to 0$
as $\xi\to 0$ since (25) holds. On the other hand

$2L\frac{\|\tilde{x}^{\prime}-\tilde{x}^{\prime\prime}\|^{2}}{\varepsilon^{2}}+M\frac{\|\tilde{m}^{\prime}-\tilde{m}^{\prime\prime}\|^{2}_{L^{2}(\Omega_{1}(T))}}{\xi^{2}}\to
0\quad\text{when }\varepsilon,\xi\to 0.$

The above inequality leads us to a contradiction. $\hfill\square$

##  5 Comments on a one-dimensional example

In this section, we exhibit a one-dimensional example ($d=1$) and we somehow
provide some comments, trying to have information on the problem from the
corresponding Isaacs equation. However, we have to say that our comments will
be not exhaustive and are to be considering as guess for the understanding of
the optimal behavior of the players. Indeed, even in the simple example we are
going to show, the differential game between a player and a mass, as studied
in the current paper, presents many non-trivial aspects and so many possible
admissible and optimal behaviors which are much sensible to the initial
conditions.

We consider the following one-dimensional example. The dynamics of the single
player is given by $x^{\prime}=\alpha$ with
$\alpha:[0,T]\longrightarrow[-c,c]=A$ the measurable control and $c>0$. The
control $\beta$ for the mass is given by $\beta:[0,T]\longrightarrow{\cal B}$
with

${\cal B}=\left\\{b:\mathbb{R}\longrightarrow\mathbb{R}:\|b\|_{\infty}\leq
c,\|b_{x}\|_{\infty}\leq c_{1}\right\\},$

where $c$ is as for the single player and $c_{1}$ is such that $b$ can assume
values from $c$ to $-c$, or vice-versa, in an interval of length $2r$. Here
$r>0$ is a datum of the problem, being the final cost defined as

$\Psi(x,m)=\int_{x-r}^{x+r}m(\xi)d\xi.$

The differential game is indeed of the Mayer type: we only have a final cost
and no running cost, $\ell\equiv 0$. The lower value function is then defined
as

$\underline{V}(x,m,t)=\inf_{\gamma}\sup_{\beta}\int_{x(T)-r}^{x(T)+r}m(\xi,T)d\xi,$
(36)

and the Hamilton-Jacobi-Isaacs problem is

$\begin{cases}-\underline{V}_{t}+\sup_{a\in[-c,c]}\left\\{-a\underline{V}_{x}\right\\}+\inf_{b\in{\cal
B}}\left\\{\langle D_{m}\underline{V},(bm)_{\xi}\rangle_{L^{2}}\right\\}=0,\\\
\underline{V}(x,m,T)=\Psi(x,m)\end{cases}.$ (37)

Note that the Isaacs equation is decoupled in the controls (the Isaacs
condition holds: the lower Isaacs equation and the upper Isaacs equation
coincide) and hence it is reasonable to expect that the lower value
$\underline{V}$ will turn out to be equal to the upper value function and so
to the min-max equilibrium of a possible differential game, where also the
mass uses non-anticipating strategies (see the comments in the Introduction).
In the sequel we then denote the lower value function (36) simply by V.

We now make the following assumption (whose validity and reasonableness are
discussed below): the value function is given by

$\text{V}(x,m,t)=\int_{x-r-h_{\ell}(t)}^{x+r+h_{r}(t)}m(\xi)d\xi,$ (38)

where the functions $h_{\ell},h_{r}$ ($\ell$ stands for “left” and $r$ for
“right”) are two time-dependent absolutely continuous functions satisfying the
backward ordinary differential equations

$\begin{cases}h^{\prime}_{r}(t)=-2c{\cal H}\Big{(}m(x-r-
h_{\ell}(t))-m(x+r+h_{r}(t))\Big{)},\\\ h_{r}(T)=0,\\\
h^{\prime}_{\ell}(t)=-2c{\cal H}\Big{(}m(x+r+h_{r}(t))-m(x-r-
h_{\ell}(t))\Big{)},\\\ h_{\ell}(T)=0,\\\
h^{\prime}_{\ell}(t)h^{\prime}_{r}(t)=0,\\\
h^{\prime}_{\ell}(t)+h^{\prime}_{r}(t)=-2c\end{cases},$ (39)

where the Heaviside function $\cal H$ is defined as ${\cal H}(\xi)=1$, when
$\xi>0$, and ${\cal H}(\xi)=0$, when $\xi<0$. In the case $\xi=0$ the
Heaviside function is non-defined or possibly multi-valued, but the last two
lines of (39) imply that, when $m(x+r+h_{r}(t))-m(x-r-h_{\ell}(t))=0$, the
quantities $h^{\prime}_{r}(t),h^{\prime}_{\ell}(t)$ are equal to zero and
equal to $-2c$, respectively.

Let us assume that a solution of (39) exists and prove that (38) gives a
solution of (37). The final condition is easily verified. We have

$\begin{array}[]{ll}\displaystyle\text{V}_{t}(x,m,t)=m(x+r+h_{r}(t))h^{\prime}_{r}(t)+m(x-r-
h_{\ell}(t))h^{\prime}_{\ell}(t),\\\
\displaystyle\text{V}_{x}(x,m,t)=m(x+r+h_{r}(t))-m(x-r-h_{\ell}(t)),\\\
\displaystyle D_{m}\text{V}(x,m,t)=\chi_{[x-r-
h_{\ell}(t),x+r+h_{r}(t)]},\end{array}$ (40)

where $\chi_{[x-r-h_{\ell}(t),x+r+h_{r}(t)]}$ is the characteristic function
of the interval $[x-r-h_{\ell}(t),x+r+h_{r}(t)]$. Plugging (40) into (37) we
get

$-m(x+r+h_{r}(t))h^{\prime}_{r}(t)-m(x-r-h_{\ell}(t))h^{\prime}_{\ell}(t)\\\
+c\Big{|}m(x+r-h_{r}(t))-m(x-r-h_{\ell}(t))\Big{|}\\\
-c\Big{(}m(x+r+h_{r}(t))+m(x-r-h_{\ell}(t))\Big{)},$ (41)

where in the last line we have used

$\inf_{b\in{\cal B}}\langle D_{m}V,(bm)_{\xi}\rangle=\inf_{b\in{\cal
B}}\int_{x-r-h_{\ell}(t)}^{x+r+h_{r}(t)}(bm)_{\xi}d\xi\\\ \inf_{b\in{\cal
B}}\Big{(}b(x+r+h_{r}(t))m(x+r+h_{r}(t))-b(x-r-h_{\ell}(t))m(x-r-
h_{\ell}(t))\Big{)}\\\ -c\Big{(}m(x+r+h_{r}(t))+m(x-r-h_{\ell}(t))\Big{)}$

because $m\geq 0$, $-c\leq b\leq c$ and $b$ can assume values from $c$ to
$-c$, or vice-versa, in any interval of length larger than $2r$ (as the
interval $[x-r-h_{\ell}(t),x+r+h_{r}(t)]$ certainly is). Since
$(h_{\ell},h_{r})$ satisfies (39), if $m(x+r-h_{r}(t))-m(x-r-h_{\ell}(t))>0$,
then (41) becomes

$2cm(x-r-h_{\ell}(t))+c\Big{(}m(x+r-h_{r}(t))-m(x-r-h_{\ell}(t))\Big{)}\\\
-c\Big{(}m(x+r+h_{r}(t))+m(x-r-h_{\ell}(t))\Big{)}=0.$

Similarly if $m(x+r-h_{r}(t))-m(x-r-h_{\ell}(t))<0$. If instead
$m(x+r-h_{r}(t))-m(x-r-h_{\ell}(t))=0$, for any possible choice of
$(h^{\prime}_{\ell}(t),h^{\prime}_{r}(t))=(-2c,0)$ or
$(h^{\prime}_{\ell}(t),h^{\prime}_{r}(t))=(0,-2c)$, we get (41) vanishing.

Modeling considerations. We look from the point of view of the mass, that is
looking for its optimal strategy. Let $t\longmapsto x(t)$ be the trajectory
chosen by the single player starting from $x_{0}$, say at $t=0$. Note that the
goal of the single player is to minimize the quantity of mass’ agents inside
the interval $[x(T)-r,x(T)+r]$ at the final time $T$, whereas the goal of the
mass is to maximize it. The agents of the mass that are already inside the
interval $[x_{0}-r,x_{0}+r]$ at $t=0$ will move jointly with the single
player. The other agents will try to enter the interval $[x(t)-r,x(t)+r]$ at a
certain time $t\leq T$ and then to stay in $[x(\tau)-r,x(\tau)+r]$ for all
remaining $t\leq\tau\leq T$. The single player cannot avoid the agents
entering its reference interval. However, having the same maximal velocity $c$
it can avoid the ones coming from one of the two sides: just moving in one of
the two directions at the maximal velocity. Note that, in this case, the
agents coming from the other side are entering with a relative velocity of
value $2c$, because they move at the maximal velocity $c$ towards $x(t)$. We
can adopt a “static point of observation” where the player is not moving and
the initial mass is not changing, but instead the interval of reference for
the player is growing with left and right extremals moving as $t\longmapsto
x_{0}-r-h_{\ell}(t)$ and $t\longmapsto x_{0}+r+h_{r}(t)$, for two suitably
chosen time dependent function $h_{\ell},h_{r}$. If such functions, from a
backward point of view, satisfy (39), then they are optimal and (38) gives the
value function. Moreover, an optimal strategy for the mass, still in this
static point of view, is, at time $t$,

$b(\xi)=\begin{cases}c&\mbox{if }\xi\leq x-r-h_{\ell}(t),\\\ linear&\mbox{if
}x-r-h_{\ell}(t)\leq\xi\leq x+r+h_{r}(t),\\\ -c&\mbox{if }\xi\geq
x+r+h_{r}(t),\end{cases}$ (42)

as indeed detected by the last line of (41) and the comments after it. Note
that the “static point of view”, as well as being a classical approach for
pursuit-evasion games ([21, 26], adopting the reference frame of one of the
two players), seems more convenient due to the feature of the game as
“accumulation of agents around the player”. However, from a non-static point
of view, (42) just means, for the single mass’ agent being in the position
$\xi\in\mathbb{R}$ at time $t$: if you are at the right of the interval
$[x(t)-r,x(t)+r]$, then move with maximal velocity $c$ towards the left; if
you are at the left of the interval $[x(t)-r,x(t)+r]$, then move with maximal
velocity $c$ towards the right; if you are already inside the interval
$[x(t)-r,x(t)+r]$, then move in order to stay inside the intervals
$[x(\tau)-r,x(\tau)+r]$ for all subsequent $\tau$.

Comments on the hypothesis (38) (and (39), existence of the solution). The
hypothesis (38), despite it enlightens some good points of the model, presents
some critical issues. The hypothesis that the functions $h_{\ell},h_{r}$ only
depend on $t$ is not generally satisfied. Indeed those functions satisfying
(39) are somehow an optimal feedback for the single player and then they
should depend on $x$ and $m$ too. In the equations (39) there should also be
the derivative of $h_{\ell},h_{r}$ with respect to $x$ and their Fréchet
differential with respect to $m$. This also reflects on the fact that a
solution $(h_{\ell},h_{r})$ of (39), as it is written above, may not exists,
for some profiles of $m$.

A simple possible situation with $h_{\ell},h_{r}$ depending only on $t$. We
end this section describing a simple favorable situation. Before that we
stress again that the goal of the present section is just to have some guess
on the optimal behavior which, even in this easy example, presents too many
non-trivial aspects to be evaluated with mere “hand-based” argument. If the
initial distribution $m$ is non-increasing in $[x-r-2cT,x+r+2cT]$ for some
$x\in\mathbb{R}$, then, for that $x$, (39) has a solution given by
$h_{\ell}\equiv 0$ and $h_{r}(t)=2c(T-t)$. The value function is then

$\text{V}(x,m,t)=\int_{x-r}^{x+r+2c(T-t)}m(\xi)d\xi,$

which means that, in a non-static point of view, the player, starting from
$x$, moves at the maximal velocity $c$ towards the right and the mass adopts
the strategy described above. Moreover, if we suppose that $m$ is strictly
decreasing in $[x-r-2cT-\sigma,x+r+2cT+\sigma]$ for some $\sigma>0$, then,
even if we perturb $x$ and $m$ a bit (remember that we are working in the set
$X$ as in Section 4 where the convergence of $m$ is also a uniform
convergence), the optimal behavior $(h_{\ell},h_{r})$ does not change, i.e. it
is not affected by such a perturbation. This means that their derivatives with
respect to $x$ and $m$ are null and that they are still solution of (39).

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