ON CONTROL OF PROBABILITY FLOWS WITH INCOMPLETE INFORMATION Текст научной статьи по специальности «Математика»

V- l™|■■■■ О

Серия «Математика»

2022. Т. 42. С. 27—42

Онлайн-доступ к журналу: http://mathizv.isu.ru

Research article

УДК 517.977.5

MSC 49N30, 49K15, 34A60

DOI https://doi.org/10.26516/1997-7670.2022.42.27

On Control of Probability Flows with Incomplete Information

Dmitry V. Khlopin1®

1 N. N. Krasovskii Institute of Mathematics and Mechanics UB RAS, Yekaterinburg, Russian Federation И [email protected]

Abstract. The mean-field type control problems with incomplete information are considered. There are several points of view that can be adopted to study the dynamics in probability space. Eulerian framework describes probability flows by the continuity equation. Kantorovich formulation describes each probability flows in terms of a single distribution on the set of admissible trajectories. The superposition principle connects these frameworks for uncontrolled dynamics. In this article, a probability flow in the both frameworks must be generated by a control that based on incomplete information about state and/or the probability at every time instance. This article presents some links between these frameworks in the case of incomplete information. In particular, besides the convexity condition, the assumptions are founded that guarantees the equivalence between the Kantorovich and Eulerian framework. This expands [6, Theorem 1] to mean-field type control problem with incomplete information.

Keywords: probability flow, continuity equation, incomplete information, mean-field optimal control

Acknowledgements: The study was performed as a part of research carried out in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (agreement number 075-02-2022-874).

For citation: KhlopinD. V. On Control of Probability Flows with Incomplete Information. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 42, pp. 27-42.

https://doi.org/10.26516/1997-7670.2022.42.27

Научная статья

Об управлении вероятностными потоками в условиях неопределенности

Д.В.Хлопин1 И

1 Институт математики и механики им. Н. Н. Красовского УрО РАН, Екатеринбург, Российская Федерация И [email protected]

Аннотация. Рассматриваются задачи управления средним полем в случае неполной информации. Имеется несколько подходов к описанию динамической системы в пространстве вероятностных мер. Подход, восходящий к Эйлеру, описывает поток заданных вероятностных мер как решение некоторого уравнения неразрывности. Подход, названный в [6] именем Канторовича, задает такой поток как поток образов одной и той же меры, заданной на множестве всех допустимых траекторий. Хорошо известный принцип суперпозиции связывает эти два подхода в случае отсутствия управления. В работе предполагается, что и в той, и в другой формулировке поток вероятностных мер должен быть порожден управлением, соблюдающим все ограничения, включая информационные. При этом неполной может оказаться как информация о позиции, так и информация о реализовавшейся вероятностной мере. Для таких задач управления средним полем исследуются взаимосвязи между указанными выше подходами, в частности найдены условия, помимо предположения выпуклости, гарантирующие эквивалентность этих подходов. Это развивает результат, показанный в [6, Theorem 1], в том числе для случая неполной информации.

Ключевые слова: потоки вероятностных мер, уравнение неразрывности, неполная информация, управление средним полем

Благодарности: Работа выполнена в рамках исследований, проводимых в Уральском математическом центре при финансовой поддержке Министерства науки и высшего образования Российской Федерации (номер соглашения 075-02-2022-874).

Ссылка для цитирования: KhlopinD. V. On Control of Probability Flows with Incomplete Information // Известия Иркутского государственного университета. Серия Математика. 2022. Т. 42. C. 27-42. https://doi.org/10.26516/1997-7670.2022.42.27

The article deals with a control of a dynamic system in a space of probabilistic measures in the case of incomplete information. The dynamics in a space of probability measures can be described in various ways [1], [6]. In this paper, the Eulerian and Kantorovich frameworks [6] are considered. Eulerian framework suggests that each velocity field sets the continuity equation, and a flow of probability measures is only its distributional solution. In Kantorovich framework we track all trajectories generated all velocity fields; then, we suggest that a flow at every time instance is the push-forward of some distribution on such trajectories by the evaluation map at this time instance. In this paper, based on the superposition principle [1], we investigate the conditions that guarantee the equivalence

of these frameworks if the control is determined by a given observation function 2 and, in addition, the corresponding total resource g is assumed be finite. This result expands the result [6, Theorem 1] to mean-field type control system [4] with information constraints.

The rest of the paper is organised as follows. First, in Section 1, we introduce some general notations. The next two sections are devoted to the assumptions on dynamics f, total resource g, and observation z (Section 2) and its various interpretations (Section 3). The basic definitions of Eulerian and Kantorovich frameworks are given in Section 4. The links between them are investigated in Section 5 (the general case) and Section 6 (the convex case). The applications to the mean-field optimal control are considered in Section 7.

1. Preliminaries

Let 1 be a Polish space, i.e., a separable completely metrizable topological space. By B(1) denote the set of all Borel subsets of 1, the a-algebra generated by all opens subsets of 1. Then, by V(1) denote the space of all Borel probabilities over 1. We also endow this probability space with the topology of narrow convergence [1]. Now, for every interval K, denote by C(K, V(1)) the set of all narrowly continuous functions from K to V(1).

By B(11, 12) denote the set of all Borel measurable maps from a Polish space 1i to a Polish space 12. For a Borel map 0 e B(11, 12) and a probability v e V(11) the pushforward measure e V(12) is defined by the rule:

(№)(A) = v{x e 1i | $(x) e A} VA e Bfa).

Further, for every Polish space 1, an interval K, and a time instance t e K denote by et the evaluation map B(K, 1) 3 y ^ et(y) = y(t) e 1.

Let consider a real p > 1. Denote by VP(Y) the set of all Borel measures over a Banach space Y with finite p-th moment. This space is endowed with the metric Wp defined by the rule: for all m', m" e VP(Y)

W*(m',m")= inf If \\y' - y"r v (y',y") L JYxY

m', m" are marginal measures of v e V(Y.

Let A be a set and let B be its subset. The symbol ib denotes the indicator function of the subset B. This function from A to {0, has value 0 on B and elsewhere.

2. The dynamics, observation, and resource

Let an Euclidean space Rd and a time interval [0; T] be given. Let also

Rd and the space r △ C([0; T], Rd) are equipped the usual norms.

Let a control set U as well as a set of observation Z be given. Assume that U and Z are Polish spaces.

Denote by U the set of all LB-measurable maps from [0; T] x Z to U. Recall that the LB-measurability is the measurability with respect to the ^-algebra generated by the products of Lebesgue measurable subsets in [0; T] and Borel subsets in Z. Every u £ U is called an admissible control. Notice that every u £ U also is an admissible control.

Let a general dynamics function f : [0; T] x Rd x V(Rd) x U ^ Rd be given; assume that this function is Borel.

Let an observation function z be a given Borel mapping from Rd x V(Rd) to Z.

Notice that each admissible control u £ U generates the function fu : [0;T] x x V(Rd) xU by the following rule:

fu(t, x, v) = f (t, x, v, u(t, z(x, v)))

whenever (t,x,u). All these functions are also LB-measurable.

Let a total resource g be a tuple (g\,g2,... ,gr), here every function gk : [0; T] x x V(Rd) x U ^ R U is Borel.

3. Cases of z and g

Now we will consider different information and resource constraints. These lists are not claiming to be full.

Let's list several classes for observation z.

complete information: z(x, v) = (x, v), here Z △ x V(Rd);

no information: z(x,u) = 0, here Z = {0}; the programmed controls, corresponding to this case, is considered in [8].

only state z(x, v) = x with Z =

only distribution z(x, v) = v, here Z △ V(Rd);

only the support: z(x, v) = supp v, here the Polish space Z is the set of all non-empty closed subsets of Rd, equipped Wijsman convergence [5];

only barycenter: z(x,v) = fRd у и(dy), here Z =

only some average: z(x, v) = fRd <fi(y) u(dy) for a given Borel function 0 from Rd to a Banach space Z;

only some pushforward measure: z(x, v) = for a given Borel function 0 from Rd to a Polish space 1 and for Z = V(1);

only deviation of state: z(x, v) = x — fRd yv(dy), here Z = Rd;

only one observable: z(x,u) = (<fi(x),<fi$v) if a given Borel function 0 from Rd to a Polish space 1 is observable with Z = 1 x V(1); e.g.

— 1 = R and $(x) = x\ is the first coordinate of x; in this case the probability coincides with the first marginal distribution X\\v;

— 1 = Rd and $(x) is a nearest to x point of a given e-net of Rd.

Recall that the total resource is g = (g\,... ,gr). A such map

(t, x, v, u) ^ gk(t, x, v, u)

may be

phase constraints: ic(t)(x,u) for a given multi-valued mapping G from [0; T] to Rd x V(Rd) that is nonempty-valued and measurable;

control constraints: %G(t)(u) for a given multi-valued mapping G from [0; T] to U that is nonempty-valued and measurable;

velocity constraints: ic(t,v)(f(t,x,u,u)) for a given multi-valued mapping G from [0; T] x V(Rd) to Rd that is measurable;

mixed constraints: iG(t) (x, v, u) for a given multi-valued mapping G from [0; T] to Rd x V(Rd) x U that is nonempty-valued and measurable; e.g., the proximal normal cone condition in [3];

running cost: a lower semicontinuous function from [0; T] x RdxV(Rd) xU to R;

energy condition: either ||«||p, or \\x\\p, or \\f (t,x,v,u)\\p, or an interaction potential [7, (2.11)], or some its sums.

Thereinafter, we prescribe that g\ is a runnung cost.

4. Eulerian and Kantorovich frameworks

In this section we formulate deterministic mean-field dynamics within Eulerian and Kantorovich frameworks. The Eulerian approach describes a probability flow as the evolution of distribution of agents by the controlled continuity equation. The Kantorovich approach identifies each agent by its trajectory in r = C([0; T], Rd) and allows to consider the probability flow as some distribution of a random process on r.

Definition 1. For a certain Borel velocity field v : [0; T] x Rd ^ Rd we say that a probability flow y e C([0; T], V(Rd)) is a distributional solution of the continuity equation

dtp(t) + div(v(t, x)p(t)) = 0 (4.1)

if, for every smooth function $ in C^°((0; T) x Rd), one has

I f [dtj>(t) + V^(t,x)v(t,x)] v(t,dx) dt = 0. (4.2)

J 0 JRd

Here, Cc°°((0; T) x Rd) is the set of all smooth compactly supported functions from (0; T) x Rd to R.

Definition 2. We say that a pair (y,u) e C([0; T], V(Rd)) is an Eulerian pair iff the flow ^ is a distributional solution of continuity equation

(4-1) coupled with the velocity field (t,x) ^ v(t,x) = fu(t, x,^(t)) that satisfies the resource constraints:

i-T r

10

g{t,x,^(t),u{t,z(x,^,(t)))) y(t,dx) dt is finite. (4.3)

Definition 3. We say that a pair (v,ur) G V(r) x 5([0; T] x r, U) is a Kantorovich pair iff the probability is concentrated on the set of absolutely continuous curves, 'q-a.e. curves 7 G r satisfying the differential equation

^ = f (t,1(t),et^n,uT(t,1)) a.e. on [0; T];

furthermore,

i i 9(t,l(t),e4ri,ur(t,j)) rj(dj) dt is finite, (4.4)

J0 Jr

and one has the following implication: for r/-almost all curves 7'

(z(j(t),etM= z(i(t),etM a.e. on [0;T])

^ (ur(t,i) = uT(t,i) a.e. on [0;T]). (4.5)

This implication guarantees that the coincidence of the observation on [0; T] gives the coincidence of the controls on this interval.

Notice that resource conditions (4.3) as well as (4.4) may be phase, control and/or velocity constraints.

5. Links between Euler and Kantorovich pairs

The following statement generalizes [6, Proposition 7.4].

Proposition 1. Let (y,,u) be an Eulerian pair and the flow ^ lies in (AC )P([0; T ], Vp (Rd)).

Then, there exists a Kantorovich pair (*q, ur) that satisfies et#"q = ^(t) for all t G [0; T] and

ur(t,j)= u(t,z(-y(t),»(t))) (5.1)

for all 7 G r and almost all t G [0; T\; furthermore, one has

! f g(t,7(t),e4v,ur(t,j)) v(dj) dt J o J r

= / g(t,x, y(t),u(t, z(x, y(t)))) ^(t,dx) dt. (5.2) J0 JRd

In addition,

f f \\fu(t,x,v(t))f Kt,dx) dt< (5.3)

J0 JRd

holds true.

Proof. Let a pair u) be Eulerian. Define the velocity field [0; T] x Rd 9

(t,x) ^ one has

(t,x) ^ v(t,x) = fu(t,x,ß(t)). Now, for almost all t and ^(i)-almost all x,

v(t, x) = f{t, x, ^(t),u(t, z(x, y(t)))) (5.4)

and the flow ^ is a solution of continuity equation (4.1) with this field v.

Since ^ G (AC)p([0; T], Vp(Rd)), due to the superposition principle [1, Theorem 8.2.1], there exists a probability measure ^ G V(r) that is concentrated on the set of all curves 7 G (AC)p([0;T], Rd) solving

^ = v(t,7(t)) a.e. on [0; T] and et#-n = ^(t) for all t G [0; T].

Define the control ur by the rule (5.1). Then, ц is concentrated on the set of absolutely continuous curves, ^-a.e. curves 7 e Г satisfying the differential equation = f (t,j(t),et^,rg,ur(t,j)). Furthermore, the control ur satisfies implication (4.5) by the definition. Finally, by the Tonelli-Fubini theorem, equality (5.2) as well as condition (4.4) are the direct consequences of (4.3). □

Remark 1. The requirement у e (AC)p([0; T], Vp(Rd)) for a given Eule-rian pair (ц,, и) is satisfied if, one find an A e R that one has

max(||/(t, x, v, и) ||p, ||ж||p) < 1 + lgi(t, x, v, u)| +

+ lg2(t,x,u,u)l + ... + lgr(t,x,u,u)^j. (5.5)

for all (t,x,v,u) e [0;T] x Rd x V(Rd) x U.

Indeed, in this case, from (4.3) and (5.5), it follows that y(t) lies in Vp(Rd) for almost all t e [0; Т]. In particular, one find t' e [0; T] such that y(t') lies in Vp(Rd). Then, similar to the proof of [6, (6.7)], one has

I ЦхЦр ^(t',dx) < / ЦхЦр p(f,dx)

JRd JRd

/•max(t',t") f

+ p / llv(t,x)ll ■ ЦхЦр-1 fl(t,dx) dt

J min(t',t") JRd

</ ||ж||рv(f,dx)+ Ap(1+ j f \\g1(t,x,^(t),u)\

JRd J0 JRd L

+ ... + \ gr (t,x,^(t),v) \ ^(t,dx) dt.

So, all the ^(t) lie in Vp(Rd); furthermore, p e С([0; T], Vp(Rd)). Now, since from (5.5) it follows (5.3), the solution ц, to the corresponding continuity equation (4.1) lies in (AC)p([0; T], Vp(Rd)) due to [1, Theorem 8.3.1].

Notice that similar to (5.5) inequalities are typical assumptions for continuity equation. See [6, (3.3)], [3, (4)], and [2, (H3)]. Thereinafter, for simplicity's sake, we assume that (5.5) holds

The following proposition shows that if a Kantorovich pair applies only one admissible control, then one must generate some Eulerian pair.

Proposition 2. Assume that (5.5) holds true. Let a pair (^,ur) be a Kantorovich pair satisfying

(U) one finds a Borel subset Г' e #(Г) with ^(Г') = 1 such that the images

V(U) △ [ur(t,j) | 37 e Г' z(j(t),e4v) = С} (5.6)

are singletons for all ( e {( e Z | З7 e Г' z(j(t), et^) = (} and almost all t e [0; T].

Then, there exists an Eulerian pair (y, u) that satisfies y(t) = et#"q for all t £ [0; T] and (5.1) for ^-almost all 7 £ r and almost all t £ [0; T\. Furthermore, (5.2) holds true.

Proof. Let ('q, ur) be a Kantorovich pair. Define the flow ^ : [0; T] ^ V(Rd) by the following rule: y(t) = et#"q for all t £ [0; T].

Decreasing r' if it is needed, we can propose that (4.5) holds for all

l,i £ r'.

Since the images V(t, () are singletons and ur is Borel map, there exists a LB-measurable control u : [0; T] x Z ^ U satisfying

{u(t, z(1(t),eiir,))} = [ur(t, 7') | Vi £ r' z(i(t)Mrj) = z(i(t)Mrj)}

for all 7 £ r' and almost all t £ [0; T]. It means that (5.1) holds true and u is admissible.

Define the velocity field v by the rule (5.4). Further, by the Tonelli-Fubini theorem, condition (4.3) with equality (5.2) are the direct consequences of (4.4). Now, from (5.5) it follows that y(t) £ Vp(Rd) for almost

all t £ [0; Tj; further, the map

f T

supp v 3 R(j) = \\v(t, 7(i)) II J 0

dt

(5.7)

is ^-summable, in particular, this map is finite ^-a.e.

Notice that ^ is concentrated on the set of all solutions to =

v(t,j(t)), therefore, for all 0 £ ((0; T) x Rd) and for ^-almost all 7 £ r, one has

rT

0 = 0(T-,7(T)) - 0(O+,7(T))= dt0(t,7(t)) + V0(t,7(t))

dt

dt.

Since the norms of all derivatives of 0 are bounded by a number N, we obtain

rT

dt^(t,1(t))+ V0(t,7 (t))

dj (t)

dt

dt < N(1 + R(j))(d + 1).

Hence, since R is ^-summable, the Tonelli-Fubini theorem yields

T

It J0

i !

/0 JT

dt^(t,1(t))+ V0(t,7(t))

d-y(t) dt

dt ^(d'y )

dt<j>(t, j(t)) + V^(t, j(t))v(t, 7(t)) v(dj) dt

for all $ £ C£°((0; T) x Rd). By the definition, we obtain that the flow y is the distributional solution of (4.1) coupled with velocity field v.

0

0

0

Finally, fix a sequence of tn e [0; T] converging to a time instance t. Consider also a continuous and bounded real functions 0 on Rd. Then, 0(7(tn)) converges to ^(7(t)) for all 7 e r. Since 0 is bounded, due to the Lebesgue dominated convergence theorem, we obtain that

/ ^(tn,x)= ^(7(in)) ^ 0(7(t)) = V(t,x)

jRd Jr Jr JRd

for all continuous and bounded real function 0 on Rd. Then, by the definition, the sequence of y(tn) narrowly converges to lim tn) for every

converging sequence of tn e [0; T]. It means that ^ is a narrowly continuous function,

i.e. p e C([0; T], V(Rd)). Thus, (p,u) is an Eulerian pair. □

6. Convex case

In this section we assume that (C1) the set U is convex;

(C2) the maps U 3 u^ fu(t, x, v) are affine for all x e v e V(Rd) and almost all t e [0; T];

(C3) the maps U 3 u^ g(t,x,v,u) are convex for all x e v e V(Rd) and almost all t e [0; T].

These assumptions are similar to [6, Assumption 3.4] and [2, (C1)-(C3)]. We also assume that

(Z) the set [x e | z(x, v) = (} is a singleton for (z(-, ^))ft^-almost all ( and all v e V(Rd).

This condition guarantees that state x may be reconstructed by z(x,v)

and v. In particular, this condition is satisfied if either z(x,u) = (x,v), or

z(x,v) = x, or z(x,v) = x — fRd y v(dy).

The following proposition expands [6, Proposition 7.5].

Proposition 3. Let inequality (5.5), conditions (C 1)-(C3) and (Z) hold.

Then, for every Kantorovich pair (q, ur) there exists an Eulerian pair (y, u) that satisfies y(t) = et#"q for all t e [0; T] and

! f g(t,7(t),e4v,ur(t,j)) v(dj) dt J0 Jr

< / g(t,x, y(t),u(t, z(x, y(t)))) ^(t,dx) dt, (6.1) J0 JRd

Proof. Let (~q, ur) be a Kantorovich pair. Define the flow ^ : [0; T] ^ V(Rd) by the following rule: y(t) = et#"q for all t £ [0; T].

Recall that the map z is Borel map from Polish space Rd x V(Rd) to

Z; it follows that the map Rd 3 x ^ zt(x) = z(x,^(t)) is also Borel. By assumption (Z), there exists a Borel function X : [0; T] x Z ^ Rd that satisfies {X(t,()} = [x £ Rd | z(x,^(t)) = (} for zt^(i)-almost all ( and almost all t £ [0; T].

By the disintegration theorem [1, Theorem 5.3.1], for almost all t £ [0; T], one find a Borel measurable family of probability measures ^t,x £ V(r), x £ Rd, that satisfies ^t,x{7 | ^(t) = x} = 1 for all ^(i)-a.e. x £ Rd and

/ $(x) y,t,x(dj) (e4rj)(dx) = $(x) ^(t,dx)

JRd Jr JRd

for all Borel bounded functions 0 : Rd ^ R. Define

u(t,() = J Ur(t,j) ßt,x(t,c)(dj)

for all ( £ Z and almost all t £ [0; T]. Since U, as U, is convex, this control is well-defined; furthermore, by zt(X(t,()) = ( for (z(-,v))^-almost all (, this control is admissible and

u(t, zt(x)) = u(t, z(x, ¡¿(t))) = J ur(t, 7) Vt,x(dj)

for ^(i)-almost all x.

Define the velocity field v by the rule (5.4). Since, by the convexity of g, from the Jensen's inequality it follows

g(t,x,e4rj,u(t,zt(x))) < j g(t,x,e4-q,ur(t)) ^t,x(dl)

for ^(i)-almost all x and almost all t £ [0; T], the relations (4.4) and (6.1) are the direct consequences of (4.3). Now, (5.5) entails that the map R (5.7) is ^-summable.

Let's consider a function 0 £ C£°((0; T) x Rd). The support of 0 is a compact subset of (0; T) x Rd, therefore dt<fi and V<fi are bounded. Hence, the function (t, 7) ^ dt^(t,j(t)) + (t)) is A ® ^-summable.

Integrating the identity

rT

0 = HT-, J(T)) - 0(0+, 7(0))= / dtHt,j(t))+ VHt,j(t))

>0

d-y (t)

dt

dt

and using Fubini-Tonelli theorem, we have

d7(t)

0

T

IT J0

rT ,

дгф(t,7 (t)) + W(t,-y(t))

дгф(1П (t)) + W(t,i(t))

dt dl(t) dt

dt 'q(d'y) ) dt

'0 Jt

Î [ \df^(t,x)+ I Уф&ч(t)) VtAdl) Kt,dx) dt. Iо JRd 1 Jt dt

On the other hand, by | l(t) = x} = 1 and (C2), we also obtain

that

Vt,x(dj) = J Уф(^х)/(t,x,et^v,UT(t,j)) Vt,x(dj) = Уф^, x)f(t, x, e4v, J^ UT(t, 7) ^t,x(dj)^j = Уф^, x)f(t, x, e^v, u(t, z(x, ¡J,(t)))j

= Уф(t,x)fu(t,x,et^^) for ^(i)-almost all x. Thus, we obtain

0

/7

10 JRd

дгф^, x) + Уф(t, x)fu(t, x, et^rj) ¡j,(t, dx) dt

for all 0 e Cc°°((0; T) x Rd). By the definition, it means that the flow ^ is a distributional solution of (4.1) coupled with the velocity field (5.4).

Repeating the last paragraph in the proof of Proposition 2 word-forword, we obtain V e C([0; T], V(Rd)) and (p,u) is an Eulerian pair. □

The following example will show that assumption (Z) in Proposition 3 is essential and can not be omitted.

л

л

л

Example 1. Put Rd = R 2, [0; T] = [0;2], U = R, Z = R. Define

л

л

fi(t, xl,x2, v, u) = — 33x1/2, f2(t, xl, x2, v, и) = u,

A

A

z(x1,x2, v) = x\, g(t, x1,x2, v, u) = x2)\\p + lv]p.

for all (t,x1,x2,u,u) e [0;2] x R2 x V(R2) x R. Notice that conditions (C 1)-(C3) hold.

For all t e [0; 2] and 7 = (a, e C([0; 2], R2) consider also

UT(t,j) =

0 t e [0; 1),

5a(t — 1)/2, t e [l; 2].

(6.2)

Now, for every c £ [0; 1] define the solution = (ac, fic) to the system

dj (t)

= (fi, f2)(t, 7 (t),v, ur(t, 7)) = (-3 3~a(t)/2, 5a(t - 1)/2) (6.3)

dt

that satisfies ac(0) = c3/2 and ¡3c(0) = 0. It's easy to calculate that

(c - t)3/2, t £ [0; c),

ac(Î) = ï0, t e [c; 2];

0, t e [0; 1),

ßc(t) = { c5/2 - (c + 1 -1)5/2, t e [1; 1 + c),

c5/2, t e [1 + c; 2].

In particular, (ac(1),fîc(1)) = (0,0) and z(ac(t),fic(t),v) = ac(t) = 0 for all t > 1 and c G [0; 1].

Let C be a random variable with the uniform (rectangular) distribution on [0;1]. Let be the distribution of the random process 7c(■) = (ac, fîc)(■). We claim that {q,ur) is a Kantorovich pair. Indeed, the probability is concentrated on solutions of (6.3), the total resource is bounded; finally, since ur is a function depend on the time instance and the map t ^ 2(7(t),et$rf), we obtain (4.5). So, (rq,ur) is a Kantorovich pair.

Define the flow y : [0; T] ^ V(Rd) by the following rule: ^(t) = et#rj for all t G [0; 2]. Then, for all t G [1; 2] the probability y(t) is concentrated on {0} x [0; 1 - (2 -1)5/2]. This yields that xi = 0, and z(x,p(t)) = 0 for ^(i)-almost all x = (x1,x2) G R2 if t G [1;2]. Now, z(x,^(t)) = 0 entails that the map x ^ j2(t,x,^(t),u(t,z(x,^,(t))) = u(t,z(x,^,(t)) is constant on suppy(t) for almost all t G [1;2] for all admissible controls u GU.

Let us prove that there no Eulerian pair (y,,u). Suppose it is false, there could be an Eulerian pair u) for some admissible control u G U. Then, by Proposition 1, there exists a Kantorovich pair (rf, u'r) that satisfies y(t) = et#rf and (5.1) for all t G [0; 2]. On the one hand, from suppy(t) C {0} x R for t G [1;2], it follows that al[1;2] = 0 for rf-almost all 7 = (a,fî).

Then, ^^ = 0 for almost all t G [1;2] and rf -almost all 7 = (a,fî ). On the other hand, the map x ^ j2(t,x,^(t),u(t,z(x,^,(t))) is constant on suppy(t), therefore, by (5.1), (a, fî) ^ ^^ is constant map rf-a.e. on r

for almost all t G [1;2]. Thus, the map [1;2] x supp rf 3 (t,j) ^ is independent of 7. However, since e1 \rf = ^(1) is concentrated on {(0,0)}, the probabilities et\rf = ^(t), t G [1;2], would be atomic. This would contradict supp ^(2) = {0} x [0; 1]. Thus, there are no Eulerian pairs (y,u).

7. Application to mean-field type optimal control problem

Recall that gl is a running cost.

Fix an endpoint cost a : V(Rd) x V(Rd) ^ R U

Definition 4. We say that an Eulerian pair (ß, u) is an Eulerian mini-mizer (with cost (gl,a)) iff a(ß(0), ß(T)) is finite and every Eulerian pair (ß,u) satisfies

/ / gi (t, x, ß(t), u(t, z(x, ß(t)))) ß(t,dx) dt + a(ß(0),ß(T)) Jo JRd

— / 9^,x, ß(t), u{t, z(x, ß(t)))) ß(t,dx) dt + a(ß(0), ß(T)).

J0 JRd

Let's consider the following mean-field type optimal control problem: minimize / / gi (t,x, ß(t),u(t, z(x, ß(t)))) ß(t,dx) dt + a(ß(0),ß(T))

Jo JRd

subject to dtß(t) + div (/(t, x, ß(t),u(t, z(x, ß(t))))ß(t)) =0, u eU,

/ / g(t,x, ß(t),u(t, z(x, ß(t)))) ß(t,dx) dt is finite. Jo JRd

Then, every Eulerian minimizer (ß,u) (with cost (gi,a)) is its optimal solution.

Definition 5. We say that a Kantorovich pair (rj,ûr) is a Kantorovich minimizer (with cost (gi,a)) iff (e0$rj, erfä) is finite and every Kantorovich pair (q, ur) satisfies

[ [ 9i(t,j(t),eéri,ûr(t,j)) v(dj) dt + a(eo^v,^T№) Jo Jr

— / 9i(t,l(t),etirj,uv(t,7)) rj(d^) dt + a(eo^rj,eTflrç). Jo Jr

So, a Kantorovich minimizer (-q, û7) (with (gl,a)) is the optimal solution of the following mean-field type optimal control problem:

minimize / gi(t,j(t),et^^,ur(t,j)) 'q(d^) dt + a(eo^v,^TM Jo Jr

dMt)

subject to = f (t,x,et$r),ur(t,7)) for rç-a.a. 7 e r, V eV (r), ur e 5([0; T] x r, Rd),

/ / 9(t,j(t),et§V,ur(t,l)) v(dj) dt is finite and (4.5) holds. Jo Jr

The following corollary is the direct consequence of Propositions 1-3 and Remark 1.

Corollary 1. Assume that (5.5) holds true. Then the following statements also hold true:

(t) Let a Kantorovich pair (rq,ur) be a Kantorovich minimizer (with cost (gi ,a)) and

— either U, f, and g satisfy convexity assumptions (C 1)-(C3) and z satisfies assumption (Z);

— or ur satisfies the condition (U).

Then, there exists an Eulerian minimizer (^.,u) (with (gi,a)) that satisfies ef.^'q = ^(t) for all t £ [0; T].

(tt) Conversely, for an Eulerian minimizer (^, u) (with (gi,a)), there exists a Kantorovich minimizer ur) (with to (gi, a)) that satisfies e^^ = ^(t) for all t £ [0; T] and (5.1) for r/-almost all 7 £ r and almost all t £ [0;T].

8. Conclusion

Corollary 1 demonstrates the conditions guaranteeing the equivalence between the Kantorovich and Eulerian frameworks in mean-field type optimal control problems. In the case of complete information this result expands [6] by the condition (U). Example 1 does not allow to transfer directly the conditions [6, Theorem 7.3] for probability flows with incomplete information. Is it due to non-Markovian strategy (6.2)? Is it possible to weaken the assumptions (Z) and (U)? This is another question the author does not know the answer to.

References

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4. Bensoussan A., Frehse J., Yam P. Mean field games and mean field type control theory. NY, Springer, 2013. https://doi.org/10.1007/978-1-4614-8508-7

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Об авторах

Хлопин Дмитрий Валерьевич,

канд. физ.-мат. наук, Институт математики и механики им. Н. Н. Красовского УрО РАН, Российская Федерация, 620108, г. Екатеринбург, [email protected], https://orcid.org/0000-0002-8942-6520

About the authors Dmitry V. Khlopin,

Cand. Sci. (Phys.-Math.), N. N. Krasovskii Institute of Mathematics and Mechanics UB RAS, Yekaterinburg, 620108, Russian Federation, [email protected], https://orcid.org/0000-0002-8942-6520

Поступила в 'редакцию / Received 01.09.2022 Поступила после рецензирования / Revised 11.10.2022 Принята к публикации / Accepted 17.10.2022