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

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.


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 R d , by the following controlled dynamics (respectively: ordinary differential equation for the single player and continuity partial differential equation for the mass): y ′ (t) = f (y(t), α(t)), m t (x, t) + div(β(x, t)m(x, t)) = 0, ( where the dynamics f : where the running cost ℓ and the final cost ψ are suitable functions whose dependence on m and β is given by their actual interpretations as state-dependent functions m(•, s) : R d −→ [0, +∞[ and β(•, s) : R d −→ 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 Being m X as given, the payoff of the game played by the single agent against the mass m Y is of the type (1.2).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(•, 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 ( [1,2]).In particular, we are concerned with the "inf sup" problem only, that is we study the lower value function V(x, m 0 , t) = inf γ sup β J(x, m 0 , t, γ[β], β).
Here, γ is any non-anticipating strategy for the single player, that is a function from the set of the controls β for the mass to the set of the controls α for the single player, such that Denoting by B a suitable set of functions from R d to itself (see (2.3)), and assuming that, for all controls β, it is β(•, t) ∈ B for almost every t, then, under suitable hypotheses, the Isaacs equation turns out to be where the unknown function u is a function of (x, m, t) ∈ R d × L 2 (R d ) × [0, T ], and ⟨•, •⟩ L 2 is the scalar product in L 2 (R d ).
In this paper, we prove that V is the unique viscosity solution u of (1.3) satisfying u(x, m, T ) = ψ(x, m).
Programming Principle and for the derivation of the corresponding Hamilton-Jacobi equation (for example, the possibility of gluing, at a fixed instant, different controls).On the contrary, the assumed regularity in space is a choice of the model, as a possible free choice of the agents.It does not affect the derivation and the validity of the equations, it may be closer to possible real applications and it allows to study the Hamilton-Jacobi-Isaacs equation in an Hilbert space, which is, as already said and due to its infinite dimensional feature, one of the major novelties of the paper.We remark that the two examples in §5 suggest the use, for the mass, of controls that are continuous in space.In some particular cases and under suitable hypotheses, the more regular problem can then be seen as an approximation of a less regular problem.
As far as the authors know, this is the first time that a similar approach is adopted.Compared to [11], 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 [14][15][16][17][18][19].We also refer the reader to [20][21][22] for further pursuit-evasion like games of multiple pursuers and one evader and to [23][24][25] for mean-field games with a major player.Possible applications are investigated in [26,27] as regards computer science and cyber-security, and in [28,29] as regards engineering and machine learning.

On the continuity equation for the mass
The evolution of a mass on R d can be represented by the evolution t −→ µ(t) of a measure on R d , where the quantity of mass in A ⊂ R d at time t is the measure of A at the same time: µ(t)(A).When the mass population is moving according to a given time-dependent vector field β : R d × [0, T ] −→ R d , then, at least formally, the evolution of the measure µ satisfies the following so-called continuity equation where T > 0 is the final time, μ is the value of the measure at the initial time t and, here and in the sequel, div means the spatial divergence.In (2.1) we have included the variable x in the notation of µ with the meaning that for any time s, it is a measure on R d whose elements are denoted by x.Equation (2.1) 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 [30] and [31][32][33].
The well-posedness of (2.1) is strongly related to well-posedness of the Cauchy problem In the sequel, we will denote the flow associated to (2.2) by Φ(x, t, s), which is defined as y(s) where y is the solution of (2.2) with the given (x, 2) has a unique solution for any initial data (x, t).For our future purposes, here we assume the stronger following hypothesis: the time-dependent controls β are functions from [0, T ] to the set B, which, for a given fixed M > 0, is defined as and moreover they belong to , Lipschitz continuous and C 1 , together with its inverse, with the Lipschitz constant independent on t and β ∈ B. We recall that, for every t ∈ [0, T ] and s ∈ [t, T ], the inverse of the flow Φ(•, t, s) is defined as Φ −1 (x, t, s) = y, where y is such that, starting from y at time t with the forward flow Φ, we arrive in x at time s.Still holding our hypotheses, in the sequel we will use the fact that Φ −1 (•, t, s) is Lipschitz continuous w.r.t.s independently on x, t, β.
Furthermore, if all the fields β have the support contained in the same bounded subset of R d , then the Proposition 2.2.Under hypothesis (2.4), for any Borel measure μ the solution of the continuity equation (2.1) in the sense of distribution is given by the push-forward where, for any Borel measurable set Proof.See [33], Proposition 4.
In the case when μ is absolutely continuous w.r.t. the Lebesgue measure L d , that is μ = mL d , where m : R d −→ R is the density, all the measures µ(•, s) are absolutely continuous w.r.t.L d too, and their density m(•, s) can be explicitly computed as (see [33]) where JΦ is the Jacobian matrix of Φ(•, t, s).Still referring to [33], and also using our hypothesis (2.4), we have the following inequalities Such estimates come from the fact that the function s −→ det JΦ(x, t, s) solves where, in the first line, (div β)(Φ(x, t, s), s) stands for the spatial divergence of β calculated in the point (Φ(x, t, s), s) ∈ R d ×]t, T ].In particular it is detJΦ(x, t, s) = e ( s t (div β)(Φ(x,t,τ ),τ )dτ ) .
In the following, by m(x, s; β, t, m) we denote the value at (x, s) ∈ R d ×]t, T ] of the density of the solution (2.5) of (2.1) with β ∈ B, where the initial datum at time t, μ, has density m.
From now on we assume (2.8)

Spatial and time estimates on the density m
In this subsection we prove that, under hypotheses (2.8) and (2.4), x −→ m(x, s; β, t, m) belongs to for every t ∈ [0, T ] and s ∈ [t, T ], where in (2.10) the modulus of continuity is independent of t, β and only depends on the norms of m.Moreover we prove that Hence, since Φ −1 (•, t, s) is a C 1 invertible function, by a known result (see for example [34]), (2.6) gives (2.9).So we prove that det JΦ(•, t, s) ∈ W 1,∞ (R d ): = ess sup function, and the (weak) derivative can be computed with the usual differentiation rules (see again [34]).
Furthermore, after similar calculations one can show that for the density m we have the following estimate where M > 0 is a constant enclosing the bounds for the spatial derivative of div β, Φ and Φ −1 , independent of t, β, and L Φ −1 is the Lipschitz constant of Φ where L 1 > 0 is the Lipschitz constant of e (•) and M 1 > 0 is the bound for ∂ x (div β)(Φ(x, t, τ ), τ )∂ x (Φ(x, t, τ )).Hence (2.13) holds with a Lipschitz constant independent of t, β.For proving (2.11), we observe that, similarly as the computations for (2.12), we have and then, by the linearity of the continuity equation for m, (2.14) gives the desired inequality.

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 where The pair (x, t) is the initial datum and the control is Given the control and the initial datum, we denote by y (x,t) (•; α) the unique solution of (3.1) in [t, T ], and, if no ambiguity arises, we do not display the dependence on the control α using the notation y (x,t) (•).
Coherently with Remark 2.3, the controlled equation for the evolution of the mass is where t ∈ [0, T ] and, using the same notation as in the previous section, m ) and the control β belongs to We consider a running cost 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 ω ℓ such that, for any fixed a, b it is Similarly we consider a final cost bounded and uniformly strongly continuous w.r.t.(x, m).
The corresponding cost functional J, for (x, m, t, α, β 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 where "non-anticipating" means, for all τ ∈ [t, T ], We then consider the lower value function  Proof.For all (ξ, u, τ Denote by w(x, m, t) the right-hand side of (3.4).Then we have to prove that .
Taken β ∈ B(t) and still denoting by β its restriction to [τ, T ], we have β ∈ B(τ ).We then define γ ∈ Γ(t) as where y (x,t) (τ ) is the position reached at time τ by the solution of (3.1) with control α = γ[β].The strategy γ is well-defined and non-anticipating, i.e., it belongs to Γ(t).We then have and we conclude.
We now address the regularity properties of V. Here, and in the following, we denote by ω ψ the modulus of continuity of ψ (being ω ℓ the one of ℓ, as above).We also set C := max{∥f (z, a)∥ : (z, a) ∈ R d × A}.Proposition 3.2.Under the hypotheses of Proposition 3.1, the lower value function V is bounded and uniformly continuous.
Proof.For the boundedness, for all (x, m, t) we get where G 1 > 0 and G 2 > 0 are the bounds for ℓ and ψ respectively.
We then consider ( We define γ 1 2 ∈ Γ(t 1 ) as We then take β 1 ∈ B(t 1 ) such that Finally, we define β 2 1 ∈ B(t 2 ) such that Hence we have where the last inequality holds due to (2.11) in Section 2.1.Therefore, as where in the last inequality we have used again (2.11).Letting to zero, we conclude.

The Hamilton-Jacobi-Isaacs equation for V
The value function V is a function of (x, m, t Here we want to derive the corresponding Hamilton-Jacobi-Isaacs equation and consider it in a suitable set X for the variables (x, m, t), which is compact in R d × L 2 (R d ) × [0, T ] and invariant for the controlled evolutions y (x,t) (•; α), m(•, s; β, t, m).In order to determine such a possible set X, we first fix an open bounded subset Ω ⊂ 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 Referring to (2.12), we set B := max{1 + M , L Φ −1 }K and for all m 0 ∈ M, β ∈ B(0) and s ∈ [0, T ] we get Now, denoting by M a bound for ∥β(•, t)∥ ∞ , as in (2.3), i.e. the maximal possible velocity of the mass' agents, we define, for all t ∈ [0, T ] and for all s ∈ [t, T ], the bounded sets being, as example, B(Ω, M t) = {x ∈ R d : dist(x, Ω) ≤ M t}.Note that Ω 1 (t) represents the maximal set that can be invaded, up to the time t, by agents starting from Ω at the time t 0 = 0, whereas Ω 1 (s, t) represents the maximal set that can be invaded, at the time s, by agents starting from Ω 1 (t) at the time t.Of course, we have a sort of semigroup property: Ω 1 (0) = Ω, Ω 1 (s, t) = Ω 1 (s) for all t ∈ [0, s] and, in particular, Ω 1 (T ) = Ω 1 (T, t) is the maximal set that can be invaded at the final time T by a mass having support in Ω 1 (t) at the time for all t ≤ s ′ ≤ s ′′ ≤ T .Note that all such functions m can be assumed as defined in the whole set Ω 1 (T ) putting them equal to 0 outside their support.For all t ∈ [0, T ], we then define the following set (recall (4.1)) In this way, the domain of the Hamiltonian we are going to consider (see (4.3) below) concerning the pair (m, t) is defined as the following set: We want to show that ( m, t) ∈ X.Since (m n , t n ) ∈ X, the sequence m n is bounded in W 1,∞ and hence the convergence to m is also uniform, from which we obtain supp m ⊂ Ω 1 ( t).Now, take s > t and n sufficiently large such that t n < s.By (2.10) and (2.11) (the modulus of continuity of (2.10) is independent of m n ∈ X), we have the convergence in H 1 of m(•, s; β, t n , m n ) to m(•, s; β, t, m).The boundedness of the former in W 1,∞ implies the weak-star convergence, which implies Due to compactness, the H 1 and L 2 norms are topologically equivalent in X, that is there exists a modulus of continuity ω such that for every (m 1 , t 1 ), (m 2 , t 2 ) ∈ X, Moreover, X contains all the possible trajectories s −→ (m(•, s; β, t, m), s) = (m(s), s) (with field β ∈ B(t)) starting from any ( m, t) ∈ X.Indeed, for all s ∈ [t, T ] and for τ ∈ [s, T ], it is and hence (m(s), s) ∈ X because, by definition, supp m(s) ⊂ Ω 1 (s) since supp m ⊂ Ω 1 (t).Hence X is viable under the evolution of the continuity equation with fields β ∈ B(t).In particular, it contains all the trajectories t −→ (m(t), t) starting from (m 0 , 0) with m 0 ∈ M. A similar construction is naturally done for the trajectories of the single player of (3.1).More precisely, we define Ω 2 (t) := B(Ω, Ct), where C is the bound for ∥f (•, a)∥, and hence we have that y (x,t) (τ ) ∈ Ω 2 (s, t) := B(Ω 2 (t), C(s − t)) for every τ ∈ [t, s].We define then the set and is viable under all the admissible trajectories with controls α ∈ A(t) and β ∈ B(t).That is, all our admissible trajectories in the time interval [0, T ], when starting from points of X, can not exit from X itself.From the point of view of the Hamilton-Jacobi-Isaacs equation that we are going to state and study on X, this means that, besides the time boundary at t = 0 and t = T , there will not be other spatial boundaries requiring boundary conditions.Indeed, from any point of X, each trajectory still remains inside X itself, then the Dynamic Programming Principle always holds and so definitely the Hamilton-Jacobi-Isaacs equation does, without spatial boundary conditions (see the proof of Thm.4.2).
We then define the Hamiltonian H : where ⟨•, •⟩ L 2 (R d ) denotes the scalar product in L 2 (R d ).In the sequel, we denote by D x the gradient 1 × d, and we are going to consider the fields β(•, s) defined on the compact set Ω 1 (T ), i.e., β : Ω Finally, u is a viscosity solution of (4.4) if it is simultaneously a viscosity sub-and supersolution.
We point out that the above local maximum/minimum point is w.r.t.X and not necessarily w.r.t.
where the test function is defined.This is not an issue because, as argued before, X is viable for our trajectories and moreover in X the convergences of m in H 1 (R d ) and in L 2 (R d ) are equivalent.
Proof.We define and, for Since B is dense in B, we can assume, without loss of generality, that (4.8) holds, for suitable r 1 , . . ., r n > 0, with the centers b i ∈ B for every i = 1, . . ., n.Now define Ψ : B −→ A by setting Observe that for any β ∈ B(t) and s ∈ [t, T ], we have that Ψ(β(•, s)) is measurable as the composition of two measurable functions, and then we can define γ * ∈ Γ(t) by setting By definition of Ψ, we have By integrating both sides of (4.9) from t to τ , we obtain (4.7) for τ sufficiently close to t.
Proof of Theorem 4.2.At first we prove that V is a subsolution (see Def. 4.1) of (4.5 ) and (x, m, t) be a local maximum point w.r.t.X for V − φ, and V(x, m, t) = φ(x, m, t).We stress again that any trajectory starting from (x, m, t) does not exit from X.We assume by contradiction that (4.6) holds and then, by Lemma 4.3, there exists γ * ∈ Γ( t) such that, for all β ∈ B( t) and all τ sufficiently close to t, where ).Since V − φ has a local maximum at (x, m, t) and V(x, m, t) = φ(x, m, t), by classical estimates on the trajectory y (x, t) and by (2.10), we have for τ sufficiently close to t. Plugging this into (4.10),we obtain inf which contradicts the inequality "≤" in (3.4).Then V is a subsolution (see Def. 4.1) of (4.5).
Next we show that V is a supersolution (see Def. 4.1) of (4.5 ) and (x, m, t) be a local minimum point for V − φ, and V(x, m, t) = φ(x, m, t).Suppose by contradiction that By the definition of H, there exists b * ∈ B such that for all a ∈ A. For τ sufficiently close to t and any γ ∈ Γ( t), we have for every t ≤ s < τ , where y (x, t) (s) = y (x, t) (s; γ[b * ]).By integrating from t to τ , and by (3.1) and (3.2), we get for τ sufficiently close to t.From and thus inf which contradicts (3.4).This completes the proof that V is a viscosity solution (see Def. For the proof, we need the following lemmas.Lemma 4.5.For all ξ > 0, p ∈ R d , (x 1 , m 1 , t 1 ), (x 2 , m 2 , t 2 ) ∈ X and q = 2(m1−m2) ξ 2 , we have where we recall that L is the Lipschitz constant of f (•, a) and ω ℓ is the modulus of continuity of the running cost ℓ (see Section 3).
Proof.Let us fix b ′ ∈ B such that Then take a ′ ∈ A such that where in the second-to-last inequality we have integrated by parts and used the fact that m 1 , m 2 vanish at ∂Ω 1 (T ).
Proof.The proof goes similarly as the one of [1], Ch.II, Section 2, Lemma 2.10.
Proof (of Thm.4.4).The aim of the proof is to show that G = sup X (u 1 − u 2 ) is less or equal to 0. We argue by contradiction assuming that G > 0.

Comments on two one-dimensional examples
In this section, we exhibit two one-dimensional examples (d = 1) and we somehow provide some comments on them, trying to have information on the problems from the corresponding Isaacs equations.However, we have to say that, in the first example, 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 simple situations, 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.In particular, this happens when the optimal behavior for the mass involves a change of the shape of the function m, that is it is not a rigid movement.The first example below concerns the case where the optimal behavior involves such a change.The second one concerns a case where such a change is not involved, the optimal behavior of the mass is a rigid movement and then it is easier to guess the Frechet differential w.r.t.m of the value.

First example
We consider the following one-dimensional example.The dynamics of the single player is given by x ′ = α with α : [0, T ] −→ [−c, c] = A the measurable control and c > 0. The control β for the mass is given by β 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 The differential game is indeed of the Mayer type: we only have a final cost and no running cost, ℓ ≡ 0. The lower value function is then defined as and the Hamilton-Jacobi-Isaacs problem is 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 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 (5.1) simply by V. We now make the following assumption (whose validity and reasonableness are discussed below): the value function is given by where the functions h ℓ , h r (ℓ stands for "left" and r for "right") are two time-dependent absolutely continuous functions satisfying the backward ordinary differential equations where the Heaviside function H is defined as H(ξ) = 1, when ξ > 0, and H(ξ) = 0, when ξ < 0. In the case ξ = 0 the Heaviside function is non-defined or possibly multi-valued, but the last two lines of (5.4) imply that, when m(x + r + h r (t)) − m(x − r − h ℓ (t)) = 0, the quantities h ′ r (t), h ′ ℓ (t) are equal to zero and equal to −2c, respectively.
Let us assume that a solution of (5.4) exists and prove that (5.3) gives a solution of (5.2).The final condition is easily verified.We have where χ [x−r−h ℓ (t),x+r+hr(t)] is the characteristic function of the interval [x − r − h ℓ (t), x + r + h r (t)].Plugging (5.5) into (5.2) we get (5.6)where in the last line we have used because m ≥ 0, −c ≤ b ≤ c and b can assume values from c to −c, or vice-versa, in any interval of length larger than 2r (as the interval ), we get (5.6)vanishing.Modeling considerations.We look from the point of view of the mass, that is looking for its optimal strategy.Let t −→ 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 ≤ T and then to stay in [x(τ ) − r, x(τ ) + r] for all remaining t ≤ τ ≤ 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 −→ x 0 − r − h ℓ (t) and t −→ x 0 + r + h r (t), for two suitably chosen time dependent function h ℓ , h r .If such functions, from a backward point of view, satisfy (5.4), then they are optimal and (5.3) gives the value function.Moreover, an optimal strategy for the mass, still in this static point of view, is, at time t, as indeed detected by the last line of (5.6) and the comments after it.Note that the "static point of view", as well as being a classical approach for pursuit-evasion games ( [36,37], 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, (5.7) just means, for the single mass' agent being in the position ξ ∈ 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(τ ) − r, x(τ ) + r] for all subsequent τ .Comments on the hypothesis (5.3) (and (5.4), existence of the solution).The hypothesis (5.3), despite it enlightens some good points of the model, presents some critical issues.The hypothesis that the functions h ℓ , h r only depend on t is not generally satisfied.Indeed those functions satisfying (5.4) are somehow an optimal feedback for the single player and then they should depend on x and m too.In the equations (5.4) there should also be the derivative of h ℓ , h r with respect to x and their Fréchet differential with respect to m.This also reflects on the fact that a solution (h ℓ , h r ) of (5.4), as it is written above, may not exists, for some profiles of m.The model studied in this section and the proposed solution are then just an ansatz of what the general situation should look like.Then, this simple model, which is representative of the overall problem and its difficulties, already deserves further future studies.In any case, whenever (5.4) has a solution for any x and m, then (5.3) solves (5.2) and, by uniqueness, is the value function.
A simple possible situation with h ℓ , h r depending only on t.We end this subsection describing a simple favorable situation.Before that we stress again that the goal of the present subsection 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 ∈ R, then, for that x, (5.4) has a solution given by h ℓ ≡ 0 and h r (t) = 2c(T − t).The value function is then V(x, m, t) = x+r+2c(T −t) x−r m(ξ)dξ, which indeed 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 − σ, x + r + 2cT + σ] for some σ > 0, then, even if we perturb x and m a bit (remember that we are working in the set X as in Sect. 4 where the convergence of m is also a uniform convergence), the optimal behavior (h ℓ , 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 (5.4).

Second example
Let us consider a one-dimensional Mayer problem, where the dynamics of the single player, its measurable controls and the set B of the controls of the mass are the same as the example in the subsection Section 5.1, and the only change is in the final cost which is now given by Besides the hypotheses on Section 4, we also consider only initial states for the mass such that R mdξ = 1 and m ≥ 0. Note that such properties are preserved by the solutions of the continuity equation (1.1) and are compatible with the arguments in Section 4. The single player x wants to minimize the cost J(x, m, t, α, β) = ψ(x(T ), m(T )) and the mass m wants to maximize it, that is the single player at the final time T wants to be close to the expectation of the distribution of the agents and the contrary for the mass.

Theorem 4 . 2 .
Under the hypotheses of Section 3, the lower value function V is a viscosity solution (see Def. 4.1) of

4 . 1 )
of (4.5).Now we want to characterize the lower value function of the problem as the unique solution of the HJI equation (4.5).Theorem 4.4.Assume the hypotheses of Theorem 4.2.Let u 1 , u 2 be bounded and uniformly continuous functions and, respectively, viscosity sub-and supersolution (see Def. 4.1) of Remark 2.1.As we have already said in the Introduction, one of our main goals is to study the problem in a Hilbert setting, and hence (2.3) and (2.4) guarantee a good regularity of the trajectories in Hilbert space.In particular, we have div β •, t) ∈ B for a.e.t}.(2.4) )for every t 1 , t 2 ∈ [0, T ], s ∈ [min{t 1 , t 2 },T ] and for some L independent of t 1 , t 2 and β.