A stability property in mean field type differential games
Yurii Averboukh

	Towards a more complex description of chemical profiles in exoplanet retrievals: A 2-layer parameterisation
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    December 30, 2023
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	The paper is concerned with the feedback approach to the deterministic mean field type differential games. Previously, it was shown that suboptimal strategies in the mean field type differential game can constructed based on functions of time and probability satisfying the stability condition. This property realizes the dynamic programming principle for the constant control of one player. We present the infinitesimal form of this condition involving analogs of the directional derivatives. In particular, we obtain the  characterization of the value function of the deterministic mean field type differential game in the terms of directional derivatives and the set of  directions feasible by virtue of the dynamics of the game.
	
	93C30, 49L20, 46G05, 93A15
	




§ INTRODUCTION
 The theory of mean field type differential game studies  control systems consisting of a large number of similar agents with the mean field interaction between them governed by two players with opposite purposes. This problem is a natural extension of the theory of mean field type control systems dealing with the case of only one decision maker. Let us emphasize that the state space for the mean field type differential games and the mean field type control systems is the space of probability measures. This space is only metric. 

The theory of mean field type control systems started with  paper <cit.>. Nowadays, the mean field field type control theory is developed for the case when  the dynamics of each agent is given by SDE (see <cit.> and reference therein). For such type of systems the necessary optimality conditions based on forward-backward SDEs were obtained in  <cit.>, <cit.>, <cit.>. The existence result of the optimal control is also proved <cit.>. The dynamic programming principle for mean field type control systems was discussed in <cit.>, <cit.>, <cit.>, <cit.>.
Additionally, let us mention papers <cit.>, <cit.> concerning with the case of deterministic mean field type control systems.

The mean field type differential games previously were studied in <cit.>, <cit.>, <cit.>. Recall that as in the case of finite dimensional differential games one can formalize mean field type differential game using either nonanticipative or feedback strategies. 

A nonanticipative strategy is a mapping assigning to a control of a player a control of the other player satisfying the feasibility condition. Notice that this approach assumes that the players observe the control of each other. The nonanticipative strategies were introduced for the finite dimensional differential game for the mean field type differential games in <cit.>, <cit.>. This approach was developed in <cit.>. In that paper the existence result for the value function is proved and the dynamic programming is presented. 

The assumption that the players have information only about the current state leads to the feedback formalization. For the finite dimensional differential game feedback strategies were introduced by Krasovskii and Subbotin <cit.>. The extension of their approach to the mean field type differential is presented in <cit.>. The design of feedback strategies can be performed using so called u- and v-stable functions defined on the product of the time interval and the state space and taking  values in the set of real values. Given a u-stable (respectively, v-stable) function, the first (respectively, second) player can construct a suboptimal strategy guaranteeing the reward greater (respectively, smaller) than the value of the given function at the initial position <cit.> (see also <cit.> for the mean field type differential games). The u- (respectively v-) stability property means that the epigraph (respectively, hypograph) is viable with respect to the dynamics corresponding to the constant control of the second (respectively, first) player. Notice that the stability property realizes the dynamic programming principle for the frozen control of one player. It is proved that  the value function is simultaneously u- and v-stable <cit.>, <cit.>, <cit.>. 

The approaches based on nonaticipative and feedback strategies should be equivalent. This statement is proved for the case of finite dimensional differential games in <cit.>. Unfortunately, up to now this equivalence is not obtained for the mean field type differential games.

Recall that the dynamic programming reduces the original control problem to the Hamilton-Jacobi PDE,  which has no smooth solution in the general case. In particular, it was proved for the finite dimensional case that the u-stable (respectively, v-stable) function  is a supersolution (respectively, subsolution) of the corresponding Hamilton-Jacobi equation <cit.>.  For the finite dimensional case one can use two equivalent tools of nonsmooth analysis to define the viscosity solution of the Hamilton-Jacobi PDE <cit.>, <cit.>, <cit.>, <cit.>. First is based on sub- and superdifferentials, whereas the second involves directional derivatives. 

Nowadays, only the notions of sub- and superdifferentials are introduced for the functions of probability. Lions in <cit.> proposed the   extrinsic approach which is based on the lifting a probability measure to a random variable.  The intrinsic  definition of sub- and superdifferentials was introduced in <cit.>. The link between these two approaches is discussed in <cit.>. Solutions of Hamilton-Jacobi PDEs in the space of probabilities in the framework of the extrinsic approach were studied  in  <cit.>, <cit.>, <cit.>, <cit.>, <cit.>. The intrinsic sub- and superdifferentials were also used for this class of equations (see <cit.>, <cit.>, <cit.>). 



The paper aims to extend the approach involving  directional derivatives to the case of mean field type differential games. The main result of the paper  is the   infinitesimal forms of  u- and v-stability properties for the deterministic mean field type differential game expressed in the terms of directional derivatives. This statement is a modification of the famous viability theorem  <cit.> (see also <cit.> for the viability theorem for the mean field type control systems). In particular, we get the characterization of the value function in the terms of directional derivatives. 

The paper is organized as follows. In Section <ref> we introduce the general notation used in the paper. Furthermore,  in this section we present the feedback formalization of the deterministic mean field type differential game and recall the  link between the stability property and the value function of the game. The main result of the paper is presented in Section <ref>. Section <ref> contains the characterization of flows of propositions produced by distributions of constant controls. This reformulates the stability condition in the terms of differential inclusions. The properties of the shift operator in the space of probabilities used in the proof of the sufficiency part of the viability theorem are given in Section <ref>. Finally, the sufficiency and necessity parts of the main result are proved in Sections <ref>, <ref> respectively.



§ PRELIMINARIES

First, let us set down the notation for the paper.

	
  * If X_1,…, X_n are sets, i,j∈{1,…,n}, x=(x_1,…,x_n)∈ X_1×…× X_n, then 
    p^i(x)≜ x_i,  p^i,j(x)≜ (x_i,x_j).

	
  * If (Ω_1,Σ_1), (Ω_2,Σ_2) are measurable spaces, m is a probability on Σ_1, h:Ω_1→Ω_2 is measurable, then h_#m stands for the push-forward measure defined by the rule: for Υ∈Σ_2,
	
    (h_#m)(Υ)≜ m(h^-1(Υ)).

	
  * Given two metric spaces (X,ρ_X) and (Y,ρ_Y), we assume that the metric on X× Y is 
    ρ_X× Y((x',y'),(x”,y”))≜[(ρ_X(x',x”))^2+(ρ_Y(y',y”))^2]^1/2.

	
  * If (X,ρ_X) is a separable metric space, then 𝒫(X) stands the set of Borel probabilities on X. We endow 𝒫(X) with the topology of narrow convergence i.e. the sequence of probabilities {m_n}_n=1^∞ converges narrowly to m if, for any ϕ∈ C_b(X),
	
    ∫_Xϕ(x)m_n(dx)→∫_Xϕ(x)m(dx) as n→∞.

	
  * 𝒫^2(X) denotes the set of probabilities m∈𝒫(X) such that, for some (equivalently, any) x_*∈ X,
	
    ∫_X(ρ_X(x,x_*))^2 m(dx)<∞.

	
  * The 2-Wasserstein metric on 𝒫^2(X) is defined as follows: if m_1,m_2∈𝒫_2(X), then
	
    W_2(m_1,m_2)=[inf_π∈Π(m_1,m_2)∫_X× X(ρ_X(x_1,x_2))^2π(d(x_1,x_2))]^1/2,
 where Π(m_1,m_2) stands for the set of transport plans between m_1 and m_2 i.e. probabilities on X× X with the marginals equal to m_1 and m_2. Notice that if W_2(m_n,m)→ 0 as n→∞, then the sequence {m_n} converges to m narrowly. When X is compact, the converse is also true and 𝒫^2(X) is compact itself..
	
  * Denote by Π^0(m_1,m_2) the set of all plans π∈Π(m_1,m_2) minimizing the right-hand side in (<ref>).
	
  * If (X,ρ_X), (Y,ρ_Y) are separable metric spaces, m is a measure on X, then denote by Λ(X,m,Y) the set of measures on (X× Y) with the marginal on X equal to m. By the disintegration theorem, given measure α∈Λ(X,m,Y), there exists, a weakly measurable family of probabilities α(·|x)∈𝒫(Y) such that, for any ϕ∈ C_b(X× Y),
	
    ∫_X× Yϕ(x,y)α(d(x,y))=∫_X∫_Yϕ(x,y)α(dy|x)m(dx).
 If α'(dy|x), α”(dy|x) both satisfy (<ref>), then α'(·|x)=α”(·|x) for m-a.e. x∈ X. Conversely, given a weakly measurable family of probabilities α(dy|x), (<ref>) defines the unique measure α∈Λ(X,m,Y). Thus, we will identify a measure α∈Λ(X,m,Y) with the class of equivalence containing families of probabilities α(dy|x) satisfying (<ref>). 
	
  * Let m,m'∈𝒫(X), π∈Π(m',m), α∈Λ(X,m,Y).  Define the composition of π and α π*α∈Λ(X,m',Y) by the following rule: for ϕ∈ C_b(X,Y), 
	
    ∫_X× Yϕ(x',y)(π*α)(d(x',y))≜∫_X× X∫_Yϕ(x',y)α(dy|x)π(d(x',x)).
 Notice that if m,m' and α have the finite second moments, then π*α has also a finite second moment.
	
  * We assume that the state space for each agent is the d-dimensional torus ≜/ℤ^d. Elements of  are sets x={x'}⊂ such that if x',x”∈ x, then x'-x”∈ℤ^d. 
	
	
  * The distance on  is introduced as follows: for x,y∈, set
	
    x-y≜inf{x'-y':x'∈ x,y'∈ y}.

	
  * If x∈, v∈, then x+v is the set {x'+v:x'∈ x}.
	
  * 𝔹_c denotes  the ball in  of the radius c  centered at the origin.
	
  * For s,r∈ℝ, s<r, 𝒞_s,r stands for the set of continuous functions x(·) from [s,r] to .
	
  * If x(·)∈𝒞_s,r, t∈ [s,r], then e_t(x(·))≜ x(t). With some abuse of notation, we denote the distance between x(·),y(·)∈𝒞_s,r by x(·)-y(·). Recall that 
	
    x(·)-y(·)≜max_t∈ [s,r]x(t)-y(t).

	




 §.§ Mean field type differential game

We consider the mean field type differential game with the dynamics of each agent given by

    d/dtx(t)=f(   t,x(t),m(t),u(t),v(t)),
       t∈ [0,T],  m(t)∈𝒫^2(),  u(t)∈ U,  v(t)∈ V.
 Here m(t) is the distribution of all agents at time t; u(t) (respectively, v(t)) is the control of the first (respectively, second) player acting on the agent; U (respectively, V) is the control space for the first (respectively, second) player.


We assume that the players influence upon the dynamics of each player independently. The purpose of the first (respectively, second) player is to minimize (respectively, maximize) the functional

    g(m(T)).


We impose the following condition on the dynamics and the payoff function:

	
  * the sets U and V are metric compacts;
	
  * the functions f and g are continuous;
	
  * the function f is Lipschitz continuous w.r.t. x and m i.e. there exists L>0 such that, for any t∈ [0,T], x',x”∈, m',m”∈𝒫(), u∈ U, v∈ V,
	
    f(t,x',m',u,v)-f(t,x',m',u,v)≤ L(x'-x”+W_2(m',m”));

	
  * (Isaacs' condition) for any t∈ [0,T], x∈, m∈, u∈ U, v∈ V and w∈,
	
    min_u∈ Umax_v∈ V⟨ w,f(t,x,m,u,v)⟩=max_v∈ Vmin_u∈ U⟨ w,f(t,x,m,u,v)⟩.
 


Now, let us describe the dynamics of the representative agent produced by the players' controls. Denote by 𝒰^0 the set of measurable functions defined on [0,T] with values in U. Further, set 𝒰≜Λ([0,T],λ,U), where λ stands for the Lebesgue measure. Below we regard U, 𝒰^0 and 𝒰 as the sets of constant, measurable and relaxed controls of the first players respectively. Using the embedding provided by the Dirac measures one can assume that

    U⊂𝒰^0⊂𝒰.


Analogously, we define the set of measurable controls of the second player 𝒱^0≜{v(·):[0,T]→ V measurable} and the set of relaxed control of the second player 𝒱≜Λ([0,T],λ,V). As above, we have that

    V⊂𝒱^0⊂𝒱.


Now, assume that s,r∈ [0,T], y∈ is an initial position, m(·):[s,r]→𝒫^2() is a given flow of probabilities, ξ∈𝒰, ζ∈𝒱 are relaxed controls of the first and second player respectively. The corresponding motion of representative agent is a function defined on [s,r] with values in  solving the following initial value problem:

    d/dtx(t)=∫_U∫_V f(t,x(t),m(t),u,v)ξ(du|t)ζ(dv|t),  x(s)=y.
 Below we denote the solution of (<ref>) by x(·,s,y,m(·),ξ,ζ). Furthermore, let traj^s,r_m(·) stand for the operator which assigns to the triple (y,ξ,ζ) the  trajectory x(·,s,y,m(·),ξ,ζ).

Now, let us turn to the dynamics of the whole system. We start with the analogs of the open-loop controls. It is natural to assume that at each point x the player can share his/her control. Thus, we consider the distributions of controls. If m_*∈𝒫^2() is an initial distribution of agents, let 
	
  * 𝒜^c[m_*]≜Λ(,m_*,U) be the set of distributions of constant controls of the first player;
	
  * 𝒜^0[m_*]≜Λ(,m_*,𝒰^0) denote the set of distributions of measurable controls of the first player;
	
  * 𝒜[m_*]≜Λ(,m_*,𝒰) be the set of distributions of relaxed controls of the first player.
 Without loss of generality, we assume that

    𝒜^c⊂𝒜^0⊂𝒜.


Analogously, ℬ^c[m_*]≜Λ(,m_*,V), ℬ^0[m_*]≜Λ(,m_*,𝒱^0), ℬ[m_*]≜Λ(,m_*,𝒱) are the sets of distributions of constant, measurable and relaxed controls of the second player respectively. As above, we have that

    ℬ^c⊂ℬ^0⊂ℬ.


Set 𝒟[m_*]≜Λ(,m_*,𝒰×𝒱). In particular, this means that 𝒟[m_*]⊂𝒫(×𝒰×𝒱) and each element of  𝒟[m_*] is a distribution of pairs of controls. Finally, let us introduce the  consistent distributions of controls. If α∈𝒜[m_*], then let 𝒟_1[α] be the set of probabilities ϰ∈𝒟[m_*] such that its marginal distribution on ×𝒰 is equal to α i.e. p^1,2_#ϰ=α. Analogously, for β∈ℬ[m_*], we denote by 𝒟_2[β] the set of probabilities ϰ∈𝒟[m_*] such that   p^1,3_#ϰ=β.

In  the following, we call any function of time taking values in the space of probabilities a flow of probabilities.

 Let s,r∈ [0,T], s<r, m_*∈𝒫^2(), ϰ∈𝒟[m_*] we say that the flow of probabilities [s,r]∋ t↦ m(t)∈𝒫() is produced by s, m_* and distribution of pairs of controls ϰ if there exists χ∈𝒫_2(𝒞_s,r) such that
	
		
  * m(t)=e_t_#χ, m(s)=m_*;
		
  * χ=traj^s,r_m(·)_#ϰ.
	
 Below we denote the flow of probabilities produced by s, m_* and ϰ by m(·,s,m_*,ϰ).



 §.§ Feedback strategies and value function




	A feedback strategy of the first player is a function 𝔲:[0,T]×→𝒫(× U) satisfying the condition 
    𝔲[s,m]∈𝒜^c[m]
 for each s∈ [0,T] and m∈. Analogously, we call any function 𝔳:[0,T]×𝒫()→𝒫(× V) such that, for any s∈ [0,T], m∈𝒫(), 𝔳[s,m]∈𝒟^c[m_*] a feedback strategy of the second player.


We assume that the players form their controls stepwise. If t_0∈ [0,T] is an initial time, m_0∈ is an initial distribution of agents, 𝔲 is a strategy of the first player, Δ={t_k}_k=0^N is a partition of [t_0,T], then we say that the flow of probabilities [t_0,T]∋ t↦ m(t) is produced by t_0, m_0, 𝔲 and Δ if there exist  sequences of probabilities {α_k}⊂𝒫(× U) {ϰ_k}_k=0^N⊂𝒫(× U×𝒱) such that

	
  * α_k=𝔲[t_k,m_k], where m_k≜ m(t_k);
	
  * ϰ_k∈𝒟_1[α_k];
	
  * for k=0,…,N-1, t∈ [t_k,t_k+1], 
    m(t)=m(t,t_k,m_k,ϰ_k).

 Notice that t_k is the time of control correction.

We denote the set of flows of probabilities produced by t_0, m_0, 𝔲 and Δ by 𝒳_1(t_0,m_0,𝔲,Δ). 

Given an initial position (t_0,m_0)∈[0,T]×, a second player's control 𝔳 and a sequence of times of control correction Δ={t_k}_k=0^N, one can define the corresponding set of flows of probabilities in the similar way. We denote it by 𝒳_2(t_0,m_0,𝔳,Δ). 

The the players' outcome can be estimated as follows:

    Γ_1(t_0,m_0)≜inf_𝔲,Δsup_m(·)∈𝒳_1(t_0,m_0,𝔲,Δ)g(m(T));


    Γ_2(t_0,m_0)≜sup_𝔳,Δinf_m(·)∈𝒳_2(t_0,m_0,𝔳,Δ)g(m(T)).
 The function Γ_1 (respectively, Γ_2) is the upper (respectively, lower) value of the game. Under the imposed condition it is proved (see <cit.>) that there exists a value function of the game Γ=Γ_1=Γ_2. To characterize the value function we need the notions of u- and v-stability. 

 We say that a lower semicontinuous function ψ_1:[0,T]×→ℝ is u-stable if
	
		
  * for any m∈, g(m)≥ψ_1(T,m);
		
  * for any s,r∈[0,T], s≤ r, m_*∈, β∈ℬ^c[m_*], there exists ϰ∈𝒟_2[β] such that 
		
    ψ_1(s,m_*)≥ψ_1(r,m(r,s,m_*,ϰ)).

	


 An upper semicontinuous function ψ_2:[0,T]×→ℝ is said to be  v-stable if
	
		
  * for any m∈, g(m)≤ψ_2(T,m);
		
  * for any s,r∈[0,T], s≤ r, m_*∈, α∈𝒜^c[m_*], there exists ϰ∈𝒟_1[α] such that 
		
    ψ_2(s,m_*)≤ψ_2(r,m(r,s,m_*,ϰ)).

	



	If ψ_1 is u-stable, then Γ_1≤ψ_1. Analogously, if ψ_2 is v-stable, then Γ_2≥ψ_2.
	
	The value function exists and it is simultaneously u- and v-stable.

 Given a u-stable (respectively, v-stable) function one can construct the suboptimal strategy of the first (respectively, second) player. This is based on the variant of the extremal shift rule for the mean field type differential games (see <cit.> for details).




§ MAIN RESULT
 In this section we extend the notion of directional derivatives to functions of probability and formulate the infinitesimal variants of   u- and v-stability conditions using this notion.


For τ>0, let Θ^τ:×→ be defined by the rule:

    Θ^τ(x,u,w)≜ x+τ w.
 With some abuse of notation, we also denote by Θ^τ the operator from × to  acting by the rule:

    Θ^τ(x,v,w)≜ x+τ w.



	Let ψ:[0,T]×→ℝ, s∈ [0,T], c>0, η∈𝒫^2(×), β≜p^1,2_#η, m≜p^1_#β=p^1_#η. The value
	
    _cψ(s;η)≜lim inf_η'∈,  p^1,2_#η'=β
    τ↓ 0,  W_2(η',η)↓ 0 ψ(s+τ,Θ^τ_#η')-ψ(s,m)/τ
 is called a u-derivative of the function ψ at s and η for the radius c. 
	
	Analogously, if η∈𝒫^2(×), α≜p^1,2_#η, m≜p^1_#α=p^1_#η, c>0, the number
	
    _cψ(s;η)≜lim sup_η'∈,  p^1,2_#η'=α
    τ↓ 0,  W_2(η',η)↓ 0 ψ(s+τ,Θ^τ_#η')-ψ(s,m)/τ
 is  a v-derivative of the function ψ at s and η  for the radius c.


Notice that one can consider u-derivative  as a lower directional derivative of the extension of the function ψ to the space [0,T]×𝒫^2(). Analogously, v-derivative can be regarded to be an upper directional derivative of lifting of ψ to [0,T]×𝒫^2(). However, since there is no natural ways to define the set of tangent distribution to 𝒫^2() (respectively, 𝒫^2()) we use only shifts on  and introduce the special notions.

Now, we define analogs of the vectograms. 
First, for  s∈ [0,T], x∈, m∈, u∈ U, v∈ V, put 
    F_1(s,x,m,u)≜co{f(t,x,m,u,v):v∈ V},


    F_2(s,x,m,v)≜co{f(s,x,m,u,v):u∈ U}.
 The graphs of   F_1 and F_2 are introduced as follows. Set

    G_1(s,m)≜{(x,u,w):x∈,u∈ U,w∈ F_1(s,x,m,u)},


    G_2(s,m)≜{(x,v,w):x∈,v∈ V,w∈ F_2(s,x,m,v)}.
 Now, let s∈[0,T],
α∈𝒫(), m≜p^1_#α. Put 
    ℱ_1(s,α)≜{η:η∈𝒫^2(G_1(s,m)),  p^1,2_#η=α}.
 If β∈𝒫(), m≜p^1_#β, then set

    ℱ_2(s,β)≜{η:η∈𝒫^2(G_2(s,m)),  p^1,2_#η=β}.


The set ℱ_1(s,α) (respectively, ℱ_2(s,β)) plays the role the vectogram for the given distribution of the constant controls of the first (respectively, second) player.

 A lower semicontinuous function ψ_1:[0,T]→ is u-stable if and only if
	
		
  * for any m∈, g(m)≥ψ_1(T,m);
		
  * there exists c>0 such that, for any s∈ [0,T],  β∈𝒫(),
		
    inf{_cψ_1(s,η):η∈ℱ_2(s,β)}≤ 0.

		
	A upper semicontinuous function ψ_2:[0,T]→ is v-stable if and only if
	
		
  * for any m∈, g(m)≤ψ_2(T,m);
		
  * for any s∈ [0,T], α∈𝒫(),
		
    sup{_cψ_2(s,η):η∈ℱ_1(s,α)}≥ 0,
 where c is constant independent  of s and α.
	


This and Theorem <ref> immediately imply the following.

 A continuous function ψ:[0,T]×→ℝ is a value function of the mean field type differential game if and only if, for any m∈, g(m)=ψ(T,m) and one can find a constant c>0 satisfying the following condition: for each s∈ [0,T], α∈𝒫(), β∈𝒫(),
	
		
  * inf{_cψ(s,η):η∈ℱ_2(s,β)}≤ 0;
		
  * sup{_cψ(s,η):η∈ℱ_1(s,α)}≥ 0.
	




§ FLOWS PRODUCED BY DISTRIBUTION OF CONSTANT CONTROLS


Below we will consider only the v-stability condition. The case of the u-stability is studied in the similar way.

First, we replace the metric on U. Originally, we consider on U a metric ρ_U. Now, let ϖ be a modulus of continuity for f. In particular, for t∈ [0,T], x∈, m∈, u',u”∈ U, v∈ V,

    f(t,x,m,u',v)-f(t,x,m,u”,v)≤ϖ(ρ_U(u',u”)).
 Put

    ρ̂_U(u',u”)≜ϖ(ρ_U(u',u”))+ρ_U(u',u”).
 Obviously, ρ̂_U is a metric on U and the function f is Lipschitz continuous w.r.t. u in (U,ρ̂_U). However, we are to prove that the set of Borel probabilities does not change when we change the metric. To this end we prove the following.
 The topologies produced by ρ_U and ρ̂_U coincides.


	Assume that E is open within ρ_U. Let u∈ E and let ε> be such that {u'∈ U:ρ_U(u,u')<ε}⊂ E. Since ρ_U(u,u')≤ρ̂_U(u,u'), we have that 
    {u'∈ U:ρ̂_U(u,u')<ε}⊂{u'∈ U:ρ_U(u,u')<ε}⊂ E.
 
	
	Conversely, assume that E is open within ρ̂_U. Pick any u∈ E. There exists ε>0 such that 
	
    {u'∈ U:ρ̂_U(u,u')<ε}⊂ E.
 Since ϖ(δ)+δ→ 0 as δ→ 0, we have that there exists δ such that, for any δ'<δ, ϖ(δ')+δ'<ε. Using (<ref>), we get that if ρ_U(u,u')<δ, then ρ̂_U(u,u')<ε. Thus,
	
    {u'∈ U:ρ_U(u,u')<δ}⊂{u'∈ U:ρ̂_U(u,u')<ε}⊂ E.




Now, let us rewrite the v-stability condition using the probabilities on the set sr. To this end we need some additional designations. For s,r∈ [0,T], t∈ [s,r], a=(x(·),u), put

    t(a)≜ x(t),


    t(a)≜ (x(t),u).
 


	Let s,r∈ [0,T], s<r, m_*∈, α∈𝒜^c[m_*], a flow of probabilities [s,r]∋ t↦ m(t) is a motion produced by s, m_* and some ϰ∈𝒟_1[α] iff there exists ν∈sr) such that
	
		
  * s_#ν=α;
		
  * t_#ν=m(t) for every t∈ [0,T];
		
  * for any t',t”∈ [0,T],
		
    ∫_srdist(t”(a)-t'(a),∫_t'^t” F_1(t,t(a),m(t),p^2(a))dt)ν(da)=0;

	


	First, assume that m(·)=m(·,s,m_*,ϰ) for some ϰ∈𝒟_1[α]. This means that there exists χ∈𝒫(𝒞_0,T) such that m_*=m(s)=e_s_#χ and χ=traj_m(·)^s,r_#ϰ. Note that if (x,u,ζ)∈× U×𝒱, then x(·)=traj^s,r_m(·)(x,u,ζ) satisfies the differential inclusion
	
    d/dtx(t)∈ F_1(t,x(t),m(t) ,u).
 This implies that, for any t',t”∈ [s,r],
	
    dist(x(t”)-x(t'),∫_t'^t”F_1(t,x(t),m(t),u))=0.
 Introduce the mapping Traj^s,r_m(·):× U×𝒱→sr by the rule
	
    Traj^s,r_m(·)(x,u,ζ)≜ (traj^s,r_m(·)(x,u,ζ),u)
 and let ν≜Traj^s,r_m(·)_#ϰ. By construction we have that p^1_#ν=χ, s_#ν=α, t_#ν=m(t). Finally, using (<ref>) we get the third condition.  
	
	Conversely, assume that there exists a probability ν∈sr such that conditions 1–3 are fulfilled. Let SOL_m(·)^s,r denote the set of pairs (x(·),u)∈sr such that x(·) is absolutely continuous and (<ref>) holds for a.e. t∈ [s,r]. We shall prove that ν is concentrated on SOL_m(·)^s,r. Indeed, there exists a ν-null set 𝒩⊂sr such that, for any (x(·),u)∈(sr)∖𝒩 and any rational t', t”,
	
    x(t”)-x(t')=∫_t'^t”F_1(t,x(t),m(t),u).
 Passing to the limit, we get that (<ref>) for every (x(·),u)∈(sr)∖𝒩 and all t',t”∈ [s,r]. Therefore, (sr)∖𝒩⊂SOL_m(·)^s,r. Since ν(𝒩)=0, we can assume that ν  is itself concentrated on SOL_m(·)^s,r. 
	
	Further, for (x(·),u)∈SOL_m(·)^s,r, let 𝐛(x(·),u) be the set of triples (y,u,ζ)⊂×𝒱 such that y=x(s) and x(·)=traj_m(·)^s,r(y,u,ζ). By the continuity of traj we have that 𝐛 is upper semicontinuous. Moreover, 𝐛(x(·),u) is nonempty when (x(·),u)∈SOL_m(·)^s,r. Thus, by the Kuratowski–Ryll-Nardzewski selection theorem  <cit.> there exists a measurable function 𝐛_0:SOL_m(·)^s,r→×𝒱 such that 𝐛_0(x(·),u)∈𝐛(x(·),u). Put ϰ≜𝐛_0_#ν. By construction we have that
	
    p^1,2_#ϰ=s_#ν=α.
 This means that ϰ∈𝒟_1[α]. Further, put 
    χ≜traj^s,r_m(·)_#ϰ= p^1_#ν.
 Therefore, m(t)=e_t_#χ. Since m(s)=m_*, we get that m(·) is produced by s, m_* and ϰ∈𝒟_1[α].


Proposition <ref> and the definition of the v-stability (see Definition <ref>) imply the following.

 A upper semicontinuous functions ψ_2:[0,T]× is v-stable iff g(m)≤ψ_2(T,m) and, given s,r∈ [0,T], α_*∈𝒫(), there exists ν∈sr  such that
	
		
  * e_s_#ν=α_*;
		
  * for any t',t”∈ [s,r],
		
    ∫_srdist(t”(a)-t'(a),∫_t'^t” F_1(t,t(a),t_#ν,p^2(a))dt)ν(d(a(·)))=0;

		
  * ψ_2(s,s_#ν)≤ψ_2(r,r_#ν).
	




§ PROPERTIES OF THE SHIFT OPERATOR

Given τ>0, define the operator Ξ^τ:×→ by the rule:

    Ξ^τ(x,u,w)≜ (x+τ w,u).
 Notice that the operator Ξ^τ can be regarded as an extension of the operator Θ^τ defined above. This means that

    p^1(Ξ^τ(x,u,w))=Θ^τ(x,w).


The following lemmas are concerned with the transfer of distribution of direction determined by the composition operation  * defined by (<ref>). 


	Let c be a positive number, α,α'∈𝒫^2(), η∈𝒫^2() be such that p^1,2_#η=α, τ,θ>0, π∈Π^0(α',α) be an optimal plan between α' and α, then
	
    W_2(Ξ^τ_#η,Ξ^θ_#(π*η))≤ W_2(α,α')+|τ-θ| c.



	Let us consider the plan π̂ between Ξ^θ_#(π*η) and Ξ^τ_#η given by the rule: for ϕ∈ C_b(),
	
    ∫_×ϕ   (y',u',y,u)π̂(d(y',u',y,u))
       ≜∫_×∫_𝔹_cϕ(x'+θ w,u',x+τ w,u)η(dw|(x,u)π(d(x',x)).

	Hence, using the Minkowski inequality for the functions ϕ',ϕ”:××→ℝ^d+1, defined by the rule:
	
    ϕ'(y',u',y,u,w)≜ (y'-y,ρ̂_U(u',u)),  ϕ”(y',u',y,u,w)≜  ((τ-θ)w,0),

	we conclude that
	
    W_2(   Ξ^θ_#(π*η),Ξ^τ_#η)
       ≤[∫_× (y'-y^2+ρ̂^2_U(u',u))π̂(d(y',u',y,u))]^1/2
       =
    	[∫_×∫_𝔹_c (x'-x+(θ-τ)w^2+ρ̂^2_U(u',u))η(dw|(x,u))π(d(x',u',x,u))]^1/2
       =[∫_×∫_𝔹_cϕ'(y',u',y,u,w)+ϕ”(y',u',y,u,w)^2η(dw|(x,u))π(d(x',u',x,u))]^1/2
       ≤[∫_×∫_𝔹_cϕ'(y',u',y,u,w)^2η(dw|(x,u))π(d(x',u',x,u))]^1/2
       +[∫_×∫_𝔹_cϕ”(y',u',y,u,w)^2η(dw|(x,u))π(d(x',u',x,u))]^1/2
       =
    	[∫_×∫_𝔹_c (x'-x^2+ρ̂^2_U(u',u))η(dw|(x,u))π(d(x',u',x,u))]^1/2
       +
    	[∫_×∫_𝔹_c |θ-τ|^2w^2η(dw|(x,u))π(d(x',u',x,u))]^1/2.
 This implies the conclusion of the lemma.



Recall that L denotes the Lipschitz constant for the function f w.r.t. x and m, whereas ϖ is the modulus of continuity of f w.r.t. t, u and v. Assuming that L≥ 1 and using the definition of the metric ρ̂_U (see (<ref>)), we have that

    f(t,x',m',u',v)   -f(t,x”,m”,u”,v)
       ≤ϖ(t'-t”)+L(x'-x”+W_2(m',m”)+ρ̂_U(u',u”)).




	Let α,α'∈𝒫(), m≜p^1_#α, m'≜p^1_#α', η∈ be such that p^1,2_#η=α, π∈Π^0(α',α) be an optimal plan between α' and α, η'≜π*η. Then
	
    |∫_dist(w,F_1(t,x,m,u))η(d(x,u,w)   )
    - ∫_dist(w,F_1(t',x,m',u   ))η'(d(x,u,w))|  
       ≤ϖ(t'-t)+ 2LW_2(α',α).



	
	By definition of η' we have that
	
    |∫_dist(w,F_1(t,x,m,u))η(d(x,u,w)   )
    - ∫_dist(w,F_1(t',x,m',u   ))η'(d(x,u,w))| 
    ≤∫_×∫_𝔹_c|dist(w,F_1(t,x,m,u)   )- dist(w,F_1(t',x',m',u'))|
       η(dw|x,u)π(d(x',u',x,u)).
	
	To estimate |dist(w,F_1(t,x,m,u))- dist(w,F_1(t',x',m',u'))| recall that
	
    F_1(t,x,m,u)={∫_V f(t,x,m,u,v)ω_v(dv):ω_v∈𝒫(V) }.
 Pick ω_v^* such that
	
    dist(w,F_1(t,x,m,u))=min_ω_v∈𝒫(V)   w-∫_V f(t,x,m,u,v)ω_v(dv)
    =   w-∫_V f(t,x,m,u,v)ω_v^*(dv).
 We have that
	
    dist(w,F_1(t',x',m',u'))≤w-∫_V f(t',x',m',u',v)ω_v^*(dv).

	Thus, by (<ref>) we obtain that
	
    dist(w,F_1   (t,x,m,u))- dist(w,F_1(t',x',m',u')) 
       ≤w-∫_V f(t,x,m,u,v)ω_v^*(dv)-w-∫_V f(t',x',m',u',v)ω_v^*(dv)
       ≤ϖ(t'-t)+ L(x'-x+W_2(m',m)+ρ̂_U(u',u)).
 The opposite inequality is established in the same way. Hence, we get the estimate
	
    |dist(w,F_1(t,x,m,   u))- dist(w,F_1(t',x',m',u'))|
       ≤ϖ(t'-t)+ L(x'-x+W_2(m',m)+ρ̂_U(u',u)).
 This, (<ref>) and the Jensen's inequality yield that
	
    |∫_dist(w,F_1(t,x,m,u))η(d(x,u,w)   )
    - ∫_dist(w,F_1(t',x,m',u   ))η'(d(x,u,w))| 
    ≤ L ∫_× (x'-x+ρ̂_U(u',u))   π(d(x',u',x,u))
    +ϖ(t'-   t)+LW_2(m',m) 
    ≤
    	L[∫_× (x'-x^2+ρ̂_U(u',u   ))^2π(d(x',u',x,u))]^1/2
    +ϖ(t'-   t)+LW_2(α',α).

	Since π is an optimal plan between α' and α we get that the right-hand side of this inequality is equal to ϖ(t'-t)+2LW_2(α',α). This gives the conclusion of the lemma.




§ PROOF OF THEOREM <REF>. SUFFICIENCY


In this section we assume that the upper semicontinuous function ψ_2:[0,T]×→ℝ is such that 


  * for every m∈, ψ_2(T,m)≥ g(m), 

  * for every t∈ [0,T] and α∈𝒫(), 

    sup{_cψ_2(t,η):η∈ℱ_1(s,α)}≥ 0,
 where c is constant independent  of t and α.
 We aims to prove that in this case ψ_2 is v-stable. To this end given s,r∈ [0,T], s<r, α_*∈𝒫(), we construct ν∈sr satisfying conditions 1–4 of Corollary <ref>. 

Put m_*≜p^1_#α_*, z^*≜ψ_2(s,m_*). Let natural n be such that r-1/n>s.

The proof of sufficiency part is based on the following.  

 There exists ε_n∈ (0,1/n] satisfying the following property. For any (t,α) such that t∈ [s,r-1/n], ψ_2(t,p^1_#α)≥ z^*-(t-s), one can find t^+∈ (t+ε_n,t+1/n), α^+∈ and η∈ such that
	
		
  * ψ_2(t^+,p^1_#α^+)> ψ_2(t,p^1_#α)-(t^+-t)/n;
		
  * p^1,2_#η=α;
		
  * W_2(Ξ^t^+-t_#η,α^+)< (t^+-t)/n;
		
  * 
    ∫_dist(w,F_1(t,x,p^1_#α,u))η(d(x,u,w))<1/n.

	
	


	By assumption, for any (h,μ)∈ [s,r-1/n]×, there exists γ∈ {τ_k},{ϵ_k}⊂ (0,+∞), {γ_k}⊂ such that
	 τ_k,ϵ_k→ 0 as k→∞, p^1,2_#γ_k=p^1,2_#γ=μ, W_2(γ_k,γ)<ϵ_k, 
    ψ_2(t+2τ_k,Θ^2τ_k_#γ_k)> ψ(t,p^1_#μ)-2τ_kϵ_k
 and 
    ∫_dist(w,F_1(t,x,p^1_#μ),u)γ(d(x,u,w))=0.


	
	Choosing k sufficiently large and using Lemma <ref>, we can find τ_h,μ∈{τ_k}, γ_h,μ∈{γ_k} such that
	
	
  * p^1,2_#γ_h,μ=μ;
	
  * 3τ_h,μ<1/n;
	
  * 
    ψ_2(t+2τ_h,μ,Θ^2τ_h,μ_#γ_h,μ)> ψ_2(t,p^1_#μ)-2τ_h,μ/2n;

	
  * 
    ∫_dist(w,F_1(t,x,p^1_#μ),u)γ_h,μ(d(x,u,w))<1/2n.
 

	Denote h^+_h,μ≜ h+2τ_h,μ, μ^+_h,μ≜Ξ^2τ_h,μ_#γ_h,μ.
	
	Now, let 
    𝒦≜{(t,α,z)∈ [s,r-1/n]××ℝ:z∈ [z^*-(t-s),ψ_2(t,p^1_#α)]}.
 Since ψ_2 is upper semicontinuous, the set 𝒦 is compact. Given (h,μ)∈ [s,r-1/n]×, let ℰ(h,μ) be the set of triples (t,α,z)∈𝒦 such that for some η∈ the following inequalities are fulfilled:
	
		
  * |t-h|<τ_h,μ;
		
  * ψ^2(h^+_h,μ,p^1_#μ^+_h,μ)>z-(h^+_h,μ-t)/n;
		
  * W_2(Ξ^h^+_h,μ-t_#η,μ^+_h,μ)<(h^+_t,μ-h)/n;
		
  * 
    ∫_dist(w,F_1(t,x,p^1_#α),u)η(d(x,u,w))<1/n.
 
	
	The choice of h^+_h,μ, μ^+_h,μ and inequalities (<ref>), (<ref>) yield that 
    {(h,μ,z)∈ [s,r-1/n]××ℝ:z∈ [z^*-(h-s),ψ_2(h,p^1_#μ)]}⊂ℰ(h,μ).
 Thus, {ℰ(h,μ)}_h∈ [s,r-1/n],μ∈ is a cover of 𝒦. Let us prove that it is an open cover. To this end we are to show that each set ℰ(h,μ) is open. Let (t,α,z)∈ℰ(h,μ), ε be a positive number. Pick (t',α',z')∈𝒦  such that |t-t'|^2+W_2^2(α,α')+|z-z'|^2<ε^2. This implies that |t-t'|,  W_2(α,α'),  |z-z'|<ε. We shall show that, for sufficiently small ε, (t',α',z')∈ℰ(h,μ). Pick η such that conditions (E1)–(E4) are satisfied for (t,α,z) and η. Let π be an optimal plan between α' and α. Set η'≜π*η. 
	
	Condition (E1) holds for (t',α',z') and η' when ε<τ_h,μ-|t-h|.
	We have that if 
    ε(1+1/n)<ψ_2(h^+,p^1_#μ^+_h,μ)-z+(h^+_h,μ-t)/n
 then condition (E2) is fulfilled for (t',α',z') and η'. Further, by Lemma <ref>
	
    W_2(Ξ^h^+_h,μ-t'_#η',Ξ^h^+_h,μ-t_#η)≤ W_2(α',α)+|t'-t|c.
 Thus, when 
    ε(1+c)<(h^+_t,μ-h)/n-W_2(Ξ^h^+_h,μ-t_#η,μ^+_h,μ)
 condition (E3) is valid for (t',α',z') and η'. Finally, by Lemma <ref>
	
    |∫_dist(w,F_1(t,x,p^1_#α,u))η(d(x,u,w)   )
    - ∫_dist(w,F_1(t',x,p^1_#α',u   ))η'(d(x,u,w))|  
       ≤ϖ(t'-t)+ 2LW_2(α',α).
 Consequently, picking ε so small that
	
    ϖ(ε)+2Lε<1/n-∫_dist(w,F_1(t,x,p^1_#α),u)η(d(x,u,w)),
 we get condition (E4) for (t',α',z') and η'.
	
	
	Since {ℰ(h,μ)} is an open cover of 𝒦, we can find a finite number of pairs {(h_i,μ_i)}_i=1^I such that 
    𝒦=⋃_i=1^Iℰ(h_i,μ_i).
 Put
	
    ε_n≜min_i=1,… Iτ_h_i,μ_i.
 For (t,α)∈ [s,r-1/n]× and z=ψ_2(t,p^1_#α) there exists a number i∈{1,…,I} such that (t,α,z)∈ℰ(h_i,μ_i). Pick η satisfying conditions (E1)–(E4) for (t,α,z) and (h_i,μ_i). Set t^+≜ h^+_h_i,μ_i, α^+≜μ^+_h_i,μ_i.

 Notice that (t^+,α^+) constructed by Lemma <ref> is such that ψ_2(t^+,p^1_#α^+)≥ z^*-(t^+-s). 


Let us introduce operations used to prove the sufficiency part of Theorem <ref>. First, given t^1,t^2∈ [0,T], define the operator L^t^1,t^2:×→t^1t^2 by the rule:

    L^t^1,t^2(y,u,w)≜ (x(·),u),
 with x(t)≜ y+(t-t^1)w.

Now, let t^1<t^2<t^3, a_1=(x_1(·),u_1)∈t^1t^2, a_2=(x_2(·),u_2)∈t^2t^3. Assume that  x_1(t^2)=x_2(t^2), u_1=u_2. Concatenation of a_1, a_2 is a pair a_1⊙ a_2≜(x(·),u)∈t^1t^3 such that u=u_1=u_2, whereas

    x(t)≜{[       x_1(t),  t∈ [t^1,t^2];       x_2(t), t∈ [t^2,t^3]. ].


Further, let ν_1∈t^1t^2, ν_2∈t^2t^3 satisfy μ≜t^2_#ν_1=t^2_#ν_2. Let ν_2(·|y,u) be the disintegration of ν_2 along μ i.e. each ν_2(·|y,u_2) is concentrated on the set of pairs (x_2(·),u) such that x(t^2)=y, u=u_2 and, for any ϕ∈ C_b(t^2t^3),

    ∫_t^2t^3ϕ(a_2)ν_2(da_2)= ∫_∫_t^2t^3ϕ(a_2)ν_2(da_2|y,u)μ(d(y,u)).
 Define ν_1⊙ν_2 by the rule: for ϕ∈ C_b(t^1t^3),

    ∫_t^1t^3ϕ(a)(ν_1⊙ν_2)(da)≜∫_t^1t^2∫_t^2t^3ϕ(a_1⊙ a_2)ν_2(da_2|t^2(a_1))ν_1(da_1).

Notice that, for t∈ [t^1,t^2],

    t_#(ν_1⊙ν_2)≜t_#ν_1,
 whereas when t∈ [t^2,t^3],

    t_#(ν_1⊙ν_2)≜t_#ν_2.



	
	
	Given a sufficiently large natural number n, we construct a number J_n and sequences {t_n^j}_j=0^J_n⊂ [s,r], {α_n^j}_j=0^J_n,{μ_n^j}_j=0^J_n⊂, {η_n^j}_j=1^J_n, {γ_n^j}_j=1^J_n⊂ by the following rules.
	
		
  * Set t_n^0≜ s, μ_n^0=alpha_n^0≜
alpha_*;
		
  * If t_n^j<r-1/n, then  pick t^+, α^+ and η satisfying conditions of Lemma <ref> for t=t_n^j, α=α_n^j. Put t_n^j+1≜ t^+, α_n^j+1≜α^+, η_n^j+1≜η. Further, let π_n^j∈Π^0(μ_n^j,α_n^j). Set γ_n^j+1≜π_n^j*η_n^j+1, μ_n^j+1≜Ξ^t_n^j+1-t_n^j_#γ_n^j+1.
		
  * If t_n^j≥ r-1/n, then set J_n≜ j.
	 Notice that, since t_n^j+1-t_n^j∈ [ε_n,1/n] where ε_n is a positive number, this process is finite and t_n^J_n∈ [r-1/n,r). Furthermore, 
	
    ψ_2(t_n^j+1,p^1_#α_n^j+1)≥ψ_2(t_n^j,p^1_#α_n^j) - (t_n^j+1-t_n^j)/n.
 This gives that
	
    ψ_2(t_n^J_n,p^1_#α_n^J_n)≥ψ_2(s,p^1_#α_*)-(r-s)/n.

	
	The following inequality is fulfilled: 
    W_2(α_n^j,μ_n^j)≤ (t_n^j-s)/n.
 Indeed, (<ref>) is obviously fulfilled for j=0. If it holds for some j, then we have that
	
    W_2(α_n^j+1,μ_n^j+1)≤ W_2(Ξ^t_n^j+1-t_n^j_#γ_n^j+1,Ξ^t_n^j+1-t_n^j_#η_n^j+1)+W_2(Ξ^t_n^j+1-t_n^j_#η_n^j+1,μ_n^j+1).
 Since  γ_n^j+1≜π_n^j*η_n^j+1 and π_n^j∈Π^0(μ_n^j,α_n^j), Lemma <ref> gives that W_2(Ξ^t_n^j+1-t_n^j_#γ_n^j+1,Ξ^t_n^j+1-t_n^j_#η_n^j+1)≤ W_2(α_n^j,μ_n^j). Lemma <ref> states that W_2(Ξ^t_n^j+1-t_n^j_#η_n^j+1,μ_n^j+1)≤ (t_n^j+1-t_n^j)/n. Combining this estimates with the assumption, we obtain inequality (<ref>) for j+1.
	
	Further, we claim that 
	
    ∫_dist(w,F_1(t_n^j,x,p^1_#γ_n^j+1,u))γ_n^j+1(d(x,u,w))≤ (2L(t_n^j-s)+1)/n.
 Indeed, by Lemma <ref> we have that
	
    ∫_dist(w,F_1(t_n^j,x,p^1_#η_n^j+1,u))η_n^j+1(d(x,u,w))≤ 1/n.
 Since γ^j+1_n=π_n^j*η^j+1_n and π_n^j∈Π^0(μ_n^j,α_n^j), Lemma <ref> yields that 
	
    |∫_dist   (w,F_1(t_n^j,x,p^1_#γ_n^j+1,u))γ_n^j+1(d(x,u,w))
       - ∫_dist(w,F_1(t_n^j,x,p^1_#η_n^j+1,u))η_n^j+1(d(x,u,w))|
       ≤ 2LW_2(α_n^j,μ_n^j).
  This, (<ref>) and (<ref>) imply (<ref>).
	
	Let γ_n^J_n+1 be a probability on  such that p^1,2_#γ_n^J_n+1=μ_n^J_n and 
	
    ∫_dist(w,F_1(t_n^J_n,x,p^1_#γ_n^J_n+1,u))γ_n^J_n+1(d(x,u,w))≤ 1/n.
 Set t_n^J_n+1≜ r, μ_n^J_n+1≜Ξ^t_n^J_n+1-t_n^J_n_#γ_n^J_n+1. Since r-t^J_n_n≤ 1/n and the probability γ_n^J_n+1 is concentrated on , we have that W_2(μ_n^J_n+1,μ_n^J_n)≤ c/n. This and (<ref>) yield the estimate
	
    W_2(μ_n^J_n+1,α_n^J_n)≤ (r-s+c)/n.

	
	Now, define the probabilities on pairs consisting of motion and control. First, for j=0,…,J_n, set ν_n^j≜ L^t_n^j,t_n^j+1_#γ_n^j+1. Notice that  t_n^j+1_#ν_n^j=t_n^j+1_#ν_n^j+1 if j=0,…, J_n-1. Thus, the probability 
    ν_n≜ν_n^0⊙…⊙ν_n^J_n
 is well defined. Let us mention properties of ν_n. First, we have that 
    t_n^j_#ν_n=μ_n^j,  j=0,…,J_n+1.
 Thus,
	
    s_#ν_n=α_*.

	Inequality (<ref>) and equality (<ref>) imply that
	
    W_2(r_#ν_n,α_n^J_n)≤ (r-s+c)/n.
 
	
	Now, given t',t”∈ [s,r], let us evaluate the value
	
    ∫_srdist(x(t”)-x(t'),∫_t'^t”F_1(t,x(t),t_#ν_n,u)dt )ν_n(d(x(·),u)).
 Let I_n' and I_n” be such that t'∈ [t_n^I_n',t_n^I_n'+1], t”∈ [t_n^I_n”,t_n^I_n”+1]. Without loss of generality, we shall assume that I_n'<I_n”. Set τ_n^I_n'≜ t'. For j=I_n'+1,…, I_n”, let τ_n^j≜ t_n^j. Finally, put τ_n^I_n”+1≜ t”. We have that
	
    dist(   x(t”)-x(t'),∫_t'^t”F_1(t,x(t),t_#ν_n,u)dt ) 
       = dist(∑_j=I_n'^I_n”x(τ_n^j+1)-x(τ_n^j),∑_j=I_n'^I_n”∫_τ_n^j^τ_n^j+1F_1(t,x(t),t_#ν_n,u)dt) 
       ≤∑_j=I_n'^I_n”dist(x(τ_n^j+1)-x(τ_n^j), ∫_τ_n^j^τ_n^j+1F_1(t,x(t),t_#ν_n,u)dt).

	Recall that  ν_n is a concatenation of the probabilities ν_n^j (see (<ref>)), whereas each probability ν_n^j is concentrated on the set of pairs (x(·),u)∈t_n^jt_n^j+1, where x(·) is c-Lipschitz continuous. Furthermore, by (<ref>) we have that W_2(t_#ν_n,p^1_#μ_n^j)≤ c(t-t_n^j) when t∈ [τ_n^j,τ_n^j+1]. This implies that
	
    ∫_sr   dist(x(τ_n^j+1)-x(τ_n^j), ∫_τ_n^j^τ_n^j+1F_1(t,x(t),t_#ν_n,u)dt)ν_n(d(x(·),u)) 
       =∫_srdist(x(τ_n^j+1)-x(τ_n^j), ∫_τ_n^j^τ_n^j+1F_1(t,x(t),t_#ν_n,u)dt)ν_n^j(d(x(·),u)) 
       ≤∫_srdist(x(τ_n^j+1)-x(τ_n^j), ∫_τ_n^j^τ_n^j+1F_1(t_n^j,x(t_n^j),μ_n^j,u)dt)ν_n^j(d(x(·),u))
       +(τ_n^j+1-τ_n^j)(ϖ(1/n)+2Lc/n).
 Recall that ν_n^j≜ L^t_n^j,t_n^j+1_#γ_n^j+1. Therefore,
	
    ∫_sr   dist(x(τ_n^j+1)-x(τ_n^j), ∫_τ_n^j^τ_n^j+1F_1(t_n^j,x(t_n^j),μ_n^j,u)dt)ν_n^j(d(x(·),u)) 
       = (τ_n^j+1-τ_n^j)∫_dist(w,F_1(t_n^j,x,μ_n^j,u))γ_n^j+1(d(x,u,w)).
 This, (<ref>) and (<ref>) give that
	
    ∫_sr   dist(x(τ_n^j+1)-x(τ_n^j), ∫_τ_n^j^τ_n^j+1F_1(t_n^j,x(t_n^j),μ_n^j,u)dt)ν_n^j(d(x(·),u)) 
       ≤ (τ_n^j+1-τ_n^j)(2L(r-s)+1)/n.
 Applying this estimate for the right-hand side of (<ref>), we get
	
    ∫_sr   dist(x(τ_n^j+1)-x(τ_n^j), ∫_τ_n^j^τ_n^j+1F_1(t,x(t),t_#ν_n,u)dt)ν_n(d(x(·),u)) 
       ≤
    	(τ_n^j+1-τ_n^j)(ϖ(1/n)+2Lc/n+2L(r-s)/n+1/n).
 Combining this inequality with (<ref>), we conclude that
	
    ∫_sr   dist(x(t”)-x(t'),∫_t'^t”F_1(t,x(t),t_#ν_n,u) dt)ν_n(d(x(·),u))
       ≤ (r-s)(ϖ(1/n)+2Lc/n+2L(r-s)/n+1/n).

	
	
	By construction the probability ν_n is concentrated on the set of pairs (x(·),u) where x(·):[s,r]→ is c-Lipschitz continuous, u∈ U. Thus, the sequence {ν_n} is relatively compact. Pick a subsequence {ν_n_k} and a probability ν∈sr  such that
	
    W_2(ν_n_k,ν)→ 0 as k→∞.
 We shall prove that ν satisfy conditions of Corollary <ref>. This will imply v-stability of ψ_2. 
	
	First, we have that s_#ν=α_*. Thus, the first condition of Corollary <ref> is valid. Further, passing to the limit in (<ref>) we obtain the second condition. To show the third condition, notice that by construction of ν (see (<ref>)) and (<ref>)
	{(t_n_k^J_n_k,α_n_k^J_n_k)} converges to (r,r_#ν). Taking into account (<ref>) and upper semicontinuity of ψ_2, we get
	
    ψ_2(r,r_#ν)≥ψ_2(s,p^1_#α_*).
 This, the third condition is also fulfilled. 
	
	Since s, r, and α are chosen arbitrarily, using Corollary <ref>, we conclude that ψ_2 is v-stable. 





§ PROOF OF THEOREM <REF>. NECESSITY



Now we assume that ψ_2:[0,T]×→ℝ is v-stable.  We shall prove that, for any s∈ [0,T], m∈, and α∈,
    sup{_cψ_2(t,η):η∈ℱ_1(s,α)}≥ 0,
 where c=R.

For s,r∈ [0,T], s<r, let Δ^s,r be an operator from 𝒞_s,r× U to  × U× acting by the rule:

    Δ^s,r(x(·),u)≜(x(s),u,x(r)-x(s)/r-s).


Notice that if x(·) is c-Lipschitz continuous, then Δ^s,r(x(·),u)∈× U×𝔹_c.


	Since ψ_2 is v-stable, we have (see Corollary <ref>) that, for any s,r∈ [0,T], s<r, α∈, one can find ν∈sr  such that
	
		
  * e_t_#ν=α;
		
  * for any t',t”∈ [s,r],
		
    ∫_srdist(t”(a)-t'(a),∫_t'^t” F_1(t,t(a),t_#ν,p^2(a))dt)ν(d(a(·)))=0;

		
  * ψ_2(s,s_#ν)≤ψ_2(r,r_#ν).
	
	
	Put m≜p^1_#α.
	
	Notice that ν is concentrated on the set of pairs (x(·),u), where x(·) is R-Lipschitz continuous.
	
	Put η^s,r≜Δ^s,r_#ν. We have that each probability η^s,r is concentrated on the set × U×𝔹_R. Thus, the set {η^s,r}_r>s is relatively compact in [R].
	
	We have that there exist a sequence {r_n}_n=1^∞ and a probability η∈ such that  r_n→ s and W_2(η^s,r_n,η)→ 0 as n→∞. By construction p^1,2_#η^s,r=α for any r>s. Consequently, p^1,2_#η=α. Since η^s,r_n is concentrated on ×𝔹_R, Θ^r_n-s_#ν^s,r_n=r_n_#ν^s,r and ψ_2(s,m)≥ψ_2(r_n,r_n_#ν^s,r), we have that
	
    _Rψ_2(s,η)≥ 0.
 
	
	Further, we have that 
	
    0   =∫_srdist(x(r)-x(s),∫_s^r F_1(t,x(t),t_#ν,u)dt)ν(d(x(·),u)) 
       ≥∫_srdist(x(r)-x(s),(r-s) F_1(s,x(s),m,u))ν(d(x(·),u))
       -(α(r-s)+2LR(r-s))(r-s).

	
	Dividing both sides by (r-s) and taking into account the definition of η^s,r, we get
	
    ∫_×dist(w,F_1(s,x,m,u)dt)η^s,r(d(x,u,w))≤α(r-s)+2LR(r-s).
 Letting r=r_n and passing to the limit when n→∞, we conclude that
	η∈ℱ_1(s,α). This and (<ref>) imply the necessity part of the theorem.



