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Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
2024. Volume 64. Pp. 3-16
MSC2020: 46E27, 46G05, 82C21, 37A10 © Yu. Averboukh
NAGUMO-TYPE VIABILITY THEOREM FOR NONLOCAL BALANCE EQUATION
The main object of the paper is a nonlocal balance equation that describes an evolution of a system of infinitely many identical particles those move according to a vector field and can also disappear or give a spring. For such system we examine the viability property that means that the systems starting inside a given set of measures does not leave this set. We prove an analog of the Nagumo-type viability theorem that gives the equivalent form of the viability property in the terms of the tangent cone.
Keywords: balance equation, viability theorem, tangent cone, space of nonnegative measures. DOI: 10.35634/2226-3594-2024-64-01
Introduction
The main object of the paper is the dynamical system in the space of (nonnnegative) measures governed by the balance equation:
dtmt + div(f (x,mt)mt) = g (x,mt )mt. (0-1)
We study the viability property of a subset of the space of nonnegative measures G that means that, for each initial measure in G, a motion generated by the balance equation from this measure entirely lies at G.
The balance equation describes an evolution of a system of particles those can either move along the vector field f or disappear/give a spring according to the probability rate described by the function g. It appears within the study of opinion dynamics [16,20]. Apparently, the balance equation can be also used to study behavior of herds of animals, groups of drones, etc. It should be noticed that, if g = 0, then the balance equation turns out to be a continuity equation. We refer [12,17,18] for the current state in this area.
Notice that the natural space for the continuity equation is the space of probability measures endowed with the seminal Wasserstein (Kantorovich) distance [26]. At the same time balance equation (0.1) does not preserve the total mass, whilst the measure is always nonnegative. A modification of the Wasserstein measure for the case of nonnegative measures was proposed by Piccoli and Rossi [21-24]. The key idea of this approach is to remove some extra masses and find a normalized distance between the residuals of the measures those have the same mass. The removing procedure can be interpreted as a jumps to/from a remote point (see [4] for details). Thus, the modified Wasserstein metric can be computed using a normalization of the probability mass on the space augmented by the remote point.
The existence and uniqueness theorems for the nonlocal balance equation are derived in [21, 24,25]. Additionally, paper [4] gives the superposition principle for equation (0.1), i.e., there it is shown that a solution of the nonlocal balance equation can be computed using a specific instantaneous evaluation of a measure on the space of curves those almost everywhere satisfies a system of ODEs.
The key results in the viability theory are viability theorems those reformulates the viability property in the terms of nonsmooth analysis. The first viability theorem was proved by Nagumo in seminal paper [19]. It dealt with ODEs and uses the tangent cone. The case of finite dimensional differential inclusions is described in [2]. The viability theorem that involves the proximal
tangents was proved by Clarke also for finite-dimensional case (see [11]). There are several extension of these classical viability theorems. The case of stochastic differential equations was examined in [8,13-15], the case of dynamical systems in Hilbert spaces was considered in [10], whilst paper [9] examines the case of Banach space. The viability theorem for the morphological dynamics in an abstract metric space was recently discussed in [7].
As it was mentioned above, the balance equation is a natural generalization of the continuity equation. The viability theorems for various settings in this field were developed in [1,3,5,6]. Notice that the case of continuity equation governed by an external force was studied in [1,6], whereas papers [3,5] deal with mean field type control systems where it is assumed that agents are intellectual and choose their own controls keeping in the mind a common purpose.
The paper provides a first step in the viability theory for the dynamics governed by a nonlocal balance equation. We consider an uncontrolled system and prove an analog of the classical Nagumo theorem. The main contribution of the paper is the form of the tangent cone to a set in the space of nonnegative measures that allows to obtain the Nagumo-type viability theorem.
The paper is organized as follows. Section 1 provides the general notation. Additionally, here we discuss a metric on the space of nonnegative measures and definitions of a solution of the nonlocal balance equation. The main result that is the viability theorem is formulated in Section 2. Its proof relies on auxiliary statements presented in Section 3. Finally, Section 4 contains the proof of the viability theorem.
§ 1. Preliminaries §1.1. General notation
• If n is a natural number, X1}..., Xn are sets, i e {1,..., n}, then pj denotes the natural projection of Xi x ... x Xn onto Xj defined by the rule ,..., xn) = x.
• If (Q, F), (Q', F') are measurable spaces, h: Q ^ Q' is a F/F'-measurable mapping, whilst m is a measure on F, then a push-forward measure h%m is a measure on F' defined by the rule: for Y eF',
(htfm)(T) = m(h-1(Y)).
• If (X, pX) is a Polish space, then Cb(X) denotes the set of all bounded continuous functions from X to R.
• Elements of Rd are assumed to be column-vectors. At the same time, Rd'* stands for the space of row-vectors. If 0 is a differentiable function at x, then we regard its gradient V0(x) as a row-vector.
• If (X, pX) is a Polish space, then M(X) is the set of (nonnegative) Borel measures on X. Furthermore, P(X) denotes the set of Borel probabilities on X, i. e.,
p(X) 4 {m e M(X): m(X) = 1}.
On the space M(X) it is natural to consider the topology of the narrow convergence. This means that {mnC M(X) converges narrowly to a measure m if, for each 0 e Cb(X), one has that
/ 0(x)mn(dx) ^ 0(x)m(dx).
XX
If m e M(X), then ||m|| stands for the total mass of m, i.e., ||m|| = m(X). If, additionally, m' also is a measure on X, then, we write m' ^ m when, for each Borel set Y, m'(Y) < m(Y).
P1 (X) denotes the set of Borel probabilities on (X, px) with the finite first moment, i. e., a probabi point x* 6 X.
i.e., a probability m lies in P 1(X) provided that fX p(x,x*)m(dx) < to for some fixed
• If T > 0, then rT denotes the set of all continuous functions from [0, T] to Rd x [0, Additionally, if C > 0, we use r^ to designate the set of continuous functions from [0,T] to Rd x [0, C].
§ 1.2. Metrics on spaces of measures
First, let us recall the definition of the Wasserstein (Kantorovich) metric. Let m1, m2 6 P 1(X), then
W1(m1 ,m2) = min< / pX(x1, x2)n(d(x1; x2)): n 6 n(m1;m2)
yJx xX
Hereinafter, n(m1; m2) denotes the set of plans between m1, m2, i. e., a measure n 6 M(X x X) lies in n(m1; m2) provided that p ftn = m,, i = 1, 2.
The concept of Wasserstein distance was extended to the space of measures by Piccoli and Rossi. The corresponding metric is defined for some positive parameter b by the rule:
if m1, m2 6 M(X), then
(m1,m2) = min< b||m1-m1|| + b||m2 - mm1| + px(x1, x2)w(d(xb x2)):
Xx X
rn1 ^ m1; m2 ^ m2, = ||m21|, w 6 n(rn1,rm2)
Below, we call a triple (rh 1,rn2, w) such that m 1 ^ m1, rm,2 ^ m2, ||m 1| = ||rn21|, w 6 n(mh 1; mh2) and
W1,b(m1 ,m2) = b|m1 - ?h 11 + b|m2 - mm2! + px(x1, x2)w(d(x1, x2))
Xx X
an optimal triple for m1 and m2.
It is shown in [4] that the metric W1)b can be rewritten using a usual Wasserstein distance on the augmented space. The latter appears, when we add to the space X a remote point * such that a distance from this point to every element of X is b. Thus, we define the distance on X U {*} by the rule:
{0, x = y =
b, x = y 6 X or x 6 X, y =
px(x, y) A b, x,y 6 X.
Furthermore, if m 6 M(X), R > ||m||, then we denote by (mt>R) the probability on X U {*} defined for each Borel set Y c X U {*} by the rule
(m>R)(X) = R-1 |"m(Y n X) + (R - ||m|)1T(*)
Here 1Y denotes the indicator function for the set Y. It is shown (see [4]) that, if
R > ||m11| V||m2||,
W1,b(m1,m2) = RW1((m1 >r), (m2>fl)). (1.1)
Using this equality, one can reformulate the compactness condition on M(X). Below we say that a set G c M(X) is bounded if there exists R > 0 such that
||m|| < R
for each m G G. Additionally, G is called tight if for each e > 0, there exists a compact K such that
m(X \ K) < e
whenever m G G.
Proposition 1.1. A set G C M(X ) is compact if and only if it is closed, tight and bounded.
This statement directly follows from (1.1) and the Prokhorov theorem. § 1.3. Weak solution of the balance equation and the superposition principle
We assume the following conditions on the functions f and g:
(A1) the functions f and g are Lipschitz continuous w.r. t. x and m with constants CLf and CLg respectively;
(A2) the functions f and g are bounded by constants Cg and Cf respectively.
D e f i n i t i o n 1.1. Let T > 0, we say that a flow of measures [0, T] 9 t ^ mt G M(Rd) is a weak solution of (0.1) provided that, for each 0 G C1([0, T] x Rd),
/ 0(T,x)mT (dx) — / 0(0, x)m0(dx) =
/0
[dt0(t, x) + V0(t, x)f (x, mt) + 0(t, x)g(x, mt)]mt(dx) dt.
Here Cc1[0,T] x Rd denotes the space of continuously differentiate functions with compact support.
The equivalent (and more convenient) form of this definition is given by the superposition principle. The latter works with distributions over the set rT. Below we interpret an element y(-) e rt as a pair (x(-), w(-)) as a curve with time varying weight regarding x(t) as a state of a particle at time t, and w(t) as its relative weight. Furthermore, if n e P 1(rT), then the instantaneous distribution of particles at time t is given by the measure |_nJt defined by the rule: for 0 e Cb(Rd),
/ 0(x)LnJt = [ 0(x(t))w(t)n(d(x(-),w(-))).
Rd ./rr
Definition 1.2. We say that a probability n e P 1(rT) is an equilibrium distribution of curves with time varying weights if n-a. e. curves (x(-), w(-)) e rT satisfy
d
dt (1.2)
= IvUMt), w(0) = Wo.
It is shown (see [4, Proposition 4]) that if n1; n2 e P 1(rC), then
W1>6(Ln1jt, Ln2jt) < (C vb)W1 (m,n2). (1.3)
Assume that we are given with a measure v e M(Rd) such that fRd |x|v(dx) < to and T > 0. Additionally, let C > |m0|eCgT. In [4, Theorems 6, 7] it is proved that
• there exists a unique equilibrium distribution of weighted curves n e P 1(rC) such that LnJo = v;
• a flow of probabilities m. is a weak solution of balance equation (0.1) with the initial condition m0 = v if and only if there exists an equilibrium distribution of the weighted curves n £ P 1(rC) such that
mt = LnJt;
• the weak solution of nonlocal balance equation (0.1) exists and is unique.
§ 2. Formulation of the main result
We start this section with the definition of the set tangent elements to a set of measures on Rd. First, given a measure v £ M(Rd), a function (C, Z) £ L^(Rd, v; RdxR) and a number t > 0, we define a measure 6 V [C, Z] by the rule: for 0 £ Cb(Rd),
i 0(x)6V[C,Z](dx) — / 0(z + tC(z))exp [tZ(z)]v(dz).
Rd Rd
The operation 6T can be clearly rewritten using a dynamical system. We consider the following system of ODEs:
x(t) = C (z), x(0) = z; ww(t) = Z(z)w(t), x(0) = a.
This system produces the mapping that assigns to each z £ Rd a trajectory on [0, t] (x(-), w(-)) £ rT. We denote this mapping by TT[C, Z]. Given v £ M(Rd), we have that
6V [C,Z ] = №1 [C,Z ]«(|v ||-1v )JT.
Definition 2.1. Let G be a closed subset of M(Rd), v £ M(Rd), we say that a pair of functions (C, Z) £ (Rd, v; Rd x R) is tangent to G provided that
liminf [t-1 dist(6V[C,Z], G)] =0.
The set of all tangent elements is denoted by Tg (v).
Here, we assume that dist(m, G) stands for the distance from the measure m to the set G that is equal to
dist(m, G) — inf(W1;b(m,m/): m' £ G}.
Definition 2.2. We say that a set G C M(Rd) is viable w.r. t. nonlocal balance equation (0.1) if, for each v £ G there exist T > 0 and a flow of measures [0, T] 9 t ^ mt £ M(Rd) such that
• mo = v;
• m. is a weak solution of (0.1) on [0, T];
• mt £ G for each t £ [0,T].
The main result of the paper is the following Nagumo-type viability theorem.
Theorem 2.1. Let G C M(Rd) be a compact subset of M(Rd) such that fRd |x|v (dx) < to for every v £ G• Then, the set G is viable w. r. t. nonlocal balance equation (0.1) if and only if, for every v £ G, one has that
(f(-,v),g(.,v)) £Tg(v).
The proof of this theorem is given below in Section 4. It relies on the auxiliary results proved in the next section.
§ 3. Properties of the motion and shift operator
The first statement evaluates the distance of shifts.
Proposition 3.1. Let
• Vi, v2 G M(Rd) be such that fRd |x|^(dx) < œ;
• £2 be continuous functions from Rd to itself bounded by Cx;
• (b Z2 be real-valued continuous functions defined on Rd bounded by Cu
• a triple (//i, //2, w) be optimal for v1 and v2. Then, for each t G [0,T],
Wi,6(©Vi [Ci,Ci], ©V2 [6,(2])
< (1 + CiT)Wi,b(Vi, V2)
+ C2W (|^1(Z1) - &(*2)| + |Z1(Z1) - <2(Z2)|)w(d(z1,z2)) + Car
JRdxRd
where C1; C2, C3 are constants dependent only on b, Cx, Cw and T. Proof. First, denote
= e^[6,6], ^2 = ©v2[&,&].
Further, define a measure c on Rd x Rd by the rule: for 0 6 Cb(Rd x Rd),
2
/ 0(xi,x2)ç (d(Xi,X2))
JRdxRd
= 0(z1 + T6(z1),z2 + T^2(z2)) </RdxRd
Additionally, set
gTCi(zl) A eTC2(z2)
w(d(zi,z2)).
/ = p1 fc, /2 = p2 fc.
2
Notice that, for each 0 G C6(Rd), / 0(x)(^i-/ii) (dx)
0(z + T£i(z))erÇi(z)(V -^i) (dz)
+
0(zi + T^i (Zi))
erCi(zi) __ eTCi(zi) a eTCi(zi)
w(d(Zi,Z2 )).
Thus, we have that
Wi,b (©Vi [Ci,Ci], ©V2 [6,(2])
< b||/i - //ill + &II/2 - //2 I + = b||Vi - />i|| + b||V2 - /2II
|xi - X2 |ç(d(Xi,X2))
+ b
xR
erZi(zi) V eTZ2(z2)) - (erÇi(zi) A eTZ2(z2))] w(d(zi,z2))
w(d(zi, Z2)).
+ / |(zi + T^i(zi)) - (z2 + T^2(z2))|
jRdxRd
erCi(zi) A erz2(z2)
X
X
Furthermore,
ЖД©; [6,Ci], ©V2 [6,(2])
< b||vi - />i|| + 6||V2 - />2! +
|zi - Z2|^(d(Zi,Z2))
+ b
+
+ T
(VCl(zi) V eTZ2(z2)) - (eTil(zi) Л етСз(z2))] ет(фь;г2))
|zi - Z2|
eTZl(zi) Л eTi2(z2^ - 1
w(d(zi,z2))
|6(zi) - ^2(Z2)|
gtcl (z1) л etz2(z2 )
Since due to the choice of (/>i, >2, ет)
Wi,b(vi, V2) = b|vi - >i|| + b|v2 - />2! +
|zi - Z2|^(d(Zi,Z2)),
using the assumptions that the functions are bounded by Cx, while the functions Z are bounded by Cw, we derive the conclusion of the proposition. □
Now, let us evaluate the distance between measures along a solution of the balance equation.
Proposition 3.2. If [0,T] Э t M mt satisfies the balance equation, then, for each s,r G [0,T],
Wi>b(ms, mr) < C4|s - r|. Here C4 is a constant depending on the functions f, g and the constant T.
Proof. Without loss of generality, we assume that 0 = s < r. Let n G P(Гт) be such that for n-a. e. (x(-), w(-)) equation (1.2) holds true, while mt = |_nJt. Let ет be such that, for each ф G Cb(Rd x Rd),
f ф(жьж2)^(d(xi,x2)) = / ф(х(0), x(r)) [ew(r) Л|то||]n(d(x(-),w(-))).
Further, we set
We have that
^ A i м ^Aru
m0 = p дет, mr = p дет.
Wi)b(m0,mr) < ||m0 - m0|| + ||mr - mr|| + / |xi - )).
./rT
We have that
m0 - m> 0|
/гт
m0| - (ew(r) Л ||m0||) n№(-),w(-))) < c0r
Here ci is a constant. Similarly,
mr - mr |
Гт
ew(r) - (ew(r) Л ||m0||)l n№(-),w(-))) < cir.
Further, using the boundness of f and g, we have that
(3.1)
(3.2)
(3.3)
/ |xi - X2Md(xi,X2)) = |x(r) - x(0)| (ew(r) Л М)n№(-),4))) < c0r.
' Гт J Гт
(3.4)
X
X
X
X
Here c° is a constant dependent on f, g and T.
Summarizing estimates (3.1)-(3.4), we obtain the conclusion of the proposition. □
The next statement evaluates the distance between a flow of probabilities generated by a vector field and a shift along constant vector field.
Proposition 3.3. Let
• v be such that fRd |x| v(dx) < oo;
• let m. be a weak solution of (0.1) satisfying the initial condition m0 = v;
• £(x) = f (xv);
• C(x) - g(xv)-
Then,
Wi>6(mr, ev]) < C5T2.
As above C5 can depend on T.
Proof. Let n satisfy (1.2) and be such that Lnjt = mt. Further, let
n' = t: [£,z MM-1 v)
for a = ||v||. The key idea of the proof is to evaluate the distance between n and n'. To this end, we first introduce the operator that assigns to each initial state z the trajectory (x(-),w(-)) on [0, t] that satisfies
d
jfx(t) = f(x(t),[V\t), x(0) = z; d
—w(t) = g(x(t),[r]\t)w(t), w(0) = a. dt
We denote this operator by XJ. Due to Proposition 3.2,
Wi,6(Lnjt, Lnjo) < C4t. This gives that, if (x(-),w(-)) = X||"vy(z), then
|x(t) — z| < c1t, |w(t) — ||v||| < c2t.
Here the constants c1; c2 are determined only by the constant b together with the functions f, g. Further, if (x(-),w(-)) = XTvy (z), (x'(-),w'(-)) = 7jf y [£,Z](z) we have that
|x(t) — x'(t)| |f (x(s), Lnjs) — f (z, v)|ds < cat2. (3.5)
Jo
Here, c3 is a constant. Simultaneously,
|w(t) — w'(t)| < / |g(x(s), Lnjs)w(s) — g(z, v)w'(s)| ds Jo
< |g(z, v)| / |w(s) — w'(s)| ds + c4t2. Jo
Using the Gronwall's inequality, we conclude that
|w(t) — w'(t)| < c5t2. (3.6)
Here, as above, c5 is a constant.
Now, we define that following plan on rT x rT:
= №n> HI-1 v).
From (3.5) and (3.6), we have that
Wi< C6T2.
This together with (1.3) gives the statement of the proposition. □
§ 4. Proof of the viability theorem §4.1. Necessity part
The proof of Necessity part of Theorem 2.1 directly follows from the viability property (see Definition 2.2), and Proposition 3.3.
§ 4.2. Sufficiency part
Let m* e G. We choose T > 0 and consider a flow of probabilities [0, T] 9 t m mt. The aim of the proof of the sufficiency part of the theorem is to show that
m e G for each t e [0,T].
To this end, we fix e > 0 and consider the set
T = {t e [0, T]: dist(mt, G) < eC6*Cret}
where
C6 4 Ci + C2(CL/ + CLg)(1 + Cg), (4.1)
C7 = (Ca + C5 + 1), Cg 4 sup{|MI: veG}. (4.2)
while dist as above stands for the distance function. Recall that due to Proposition 1.1, we have Cg < to. First, we notice that 0 e T.
L e m m a 4.1. The set T£ is closed.
Proof. Assume that a sequence {tn} C T£ converges to t. Due to compactness of G, we have that, for each n, there exists e G such that
Wi)b(min ) < eC6tnCVetra. From this and Proposition 3.2, we have that
Wi,6(mt,^n) < eC6tnCVetra + C4|t - tra|. This implies that dist(mt, G) < eC6tCVet. □
Lemma 4.2. Given t G T£ and 5 < e, there exists r > (t, t + 5] a/so lying in T£.
Proof. We choose v e G such that
Wi,&(mt, v) = dist(mt, G).
Since t e T£,
Wi,6(mt, v) < eC6iC7£t.
Recall that the pair of functions (£, Z) such that £(x) = f (x, v), Z = g(x, v) lies in 7g( v). This means that one can find a measure ^ e G and a positive number t < 5 satisfying
wMev[£,ZU) < £t. (4.3)
Further, we consider a pair (£Zsuch that £^(x) = f (x, mt), Z^(x) = g(x, mt). Proposition 3.1 gives that
Wi,*(em Me[£,Z]) < (1 + Cit)Wi>6(mt, v)
+ C2T(|£^(zi) — £(z2)| + |Zh(zi) — Z(z2)hw(d(zi, z2)) + CaT
(4.4)
11 (zi) — Z(z2)^W(d(zi,z2)) + C3T ,
where ( z^, z, w) is an optimal triple for mt and v.
Taking into account the choice of (£, Z) and (£Zwe have that
|£h(zi) — £(z2)| < CL/|zi — z2| + CL/Wi,6(m(t), v), (4.5)
|Z*(zi) — Z(z2)| < CLg|zi — z2| + CLgWi,6(m(t), v). (4.6)
Recall that ( Z, zz, w) is an optimal triple for mt and v. Therefore,
/ |zi — z2|w(d(zi, z2)) < Wi,6(mt, v). (4.7)
JRdxRd
Additionally, taking into account the inclusion v e G, we have that
w(Rd x Rd) < ||v|| < Cg. (4.8)
Here the constant Cg is defined by (4.2).
Combining estimates (4.4)-(4.8), we arrive at the following inequality:
Wi,*(em[£",Z>], ev[£,Z]) < (1 + c6T)Wi,6(mt, v) + C3T2.
Here we use the definition of C6 (see (4.1)). This and (4.3) yield that
Wi,6(emt [£*,ZV) < (1 + C6T)Wi,6(mt, v) + C3T2 + £T, where ^ e G. Now we apply Proposition 3.3 and deduce that
Wi,*(mt+r ,ju) < (1 + C6T)Wi,6(mt, v) + (C3 + C5)t2 + £t. The choice of the measures ^ and v gives that
dist(mt+r, G) < (1 + C6t) dist(mt, G) + (C3 + C5)t2 + £t.
Therefore,
dist (mt+r, G) < (1 + C6T)eC61(C3 + C5 + 1)et + (C3 + C5)t2 + £t. This gives that r = t + t e T. □
Proof of Theorem 2.1. Sufficiency part.
Let t e [0, T], if t / T£, due to the closedness of T£, we have that the time instant
1 4 max{s e T£: s < t}
is strictly less than t and lies in T£. Simultaneously, by Lemma 4.2, there exists r > (£, t] such
that r e T£. This contradicts with the choice of 1 Therefore, each t e [0,T] belongs to T£. In
other words, for every t e [0, T], one has that
dist(mt, G) < C6eiCzet.
Passing to the limit when e m 0 and using the compactness of G, we obtain that
m e G.
This proves the sufficient part of the theorem. □
Funding. This work was funded by the Russian Science Foundation, project number 24-2100373, https://rscf.ru/project/24-21-00373/.
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26. Villani C. Optimal transport. Old and new, Berlin-Heidelberg: Springer, 2009. https://doi.org/10.1007/978-3-540-71050-9
Received 22.08.2024 Accepted 27.09.2024
Yurii Vladimirovich Averboukh, Doctor of Mathematical Sciences, Leading Researcher, Department of
Differential Equations, Krasovskii Institute of Mathematics and Mechanics, ul. S. Kovalevskoi, 16, Yekaterinburg, 620108, Russia.
ORCID: https://orcid.org/0000-0002-6541-8470
E-mail: [email protected]
Citation: Yu. Averboukh. Nagumo-type viability theorem for nonlocal balance equation, Izvestiya Instituta
Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2024, vol. 64, pp. 3-16.
Ю. В. Авербух
Теорема о выживаемости типа Нагумо для нелокального уравнения баланса
Ключевые слова: уравнение баланса, теорема о выживаемости, касательный конус, пространство неотрицательных мер.
УДК: 517.986.7
DOI: 10.35634/2226-3594-2024-64-01
Основным объектом работы является нелокальное уравнение баланса, описывающее эволюцию системы из бесконечно большого числа идентичных частиц, которые движутся в соответствии с векторным полем, а также могут исчезать или давать потомка. Для такой системы мы исследуем свойство выживаемости, которое означает, что система, начинающаяся внутри заданного множества мер, не покидает это множество. Мы приводим аналог теоремы о выживаемости типа Нагумо, которая дает эквивалентные формы свойства выживаемости в терминах касательного конуса.
Финансирование. Исследование выполнено за счет гранта Российского научного фонда № 24-2100373, https://rscf.ru/project/24-21-00373/.
СПИСОК ЛИТЕРАТУРЫ
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6. Badreddine Z., Frankowska H. Viability and invariance of systems on metric spaces // Nonlinear Analysis. 2022. Vol. 225. Article number: 113133. https://doi.org/10.1016/j.na.2022.113133
7. Bonnet B., Frankowska H. Viability and exponentially stable trajectories for differential inclusions in Wasserstein spaces // 2022 IEEE 61st Conference on Decision and Control (CDC). IEEE, 2022. P. 5086-5091. https://doi.org/10.1109/CDC51059.2022.9992888
8. Buckdahn R., Peng Shige, Quincampoix M., Rainer C. Existence of stochastic control under state constraints // Comptes Rendus de l'Academie des Sciences - Series I - Mathematics. 1998. Vol. 327. Issue 1. P. 17-22. https://doi.org/10.1016/S0764-4442(98)80096-7
9. Carja O., Necula M., Vrabie 1.1. Viability, invariance and applications. Elsevier, 2007.
10. Clarke F. H., Ledyaev Yu. S., Radulescu M. L. Approximate invariance and differential inclusions in Hilbert spaces // Journal of Dynamical and Control Systems. 1997. Vol. 3. No. 4. P. 493-518. https://doi.org/10.1023/A:1021873607769
11. Clarke F. H., Ledyaev Yu. S., Stern R. J., Wolenski P. R. Nonsmooth analysis and control theory. New York: Springer, 1998. https://doi.org/10.1007/b97650
12. Crippa G., Lecureux-Mercier M. Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow // Nonlinear Differential Equations and Applications NoDEA. 2013. Vol. 20. No. 3. P. 523-537. https://doi.org/10.1007/s00030-012-0164-3
13. Da Prato G., Frankowska H. Invariance of stochastic control systems with deterministic arguments // Journal of Differential Equations. 2004. Vol. 200. Issue 1. P. 18-52. https://doi.org/10.1016/j.jde.2004.01.007
14. Frankowska H., Da Prato G. Invariance of closed sets under stochastic control systems // Control and boundary analysis. CRC Press, 2005. P. 218-229. https://doi.org/10.1201/9781420027426.ch17
15. Da Prato G., Frankowska H. Stochastic viability of convex sets // Journal of Mathematical Analysis and Applications. 2007. Vol. 333. Issue 1. P. 151-163. https://doi.org/10.1016/j.jmaa.2006.08.057
16. Duteil N.P. Mean-field limit of collective dynamics with time-varying weights // Networks and Heterogeneous Media. 2022. Vol. 17. Issue 2. P. 129-161. https://doi.org/10.3934/nhm.2022001
17. Keimer A., Pflug L. Chapter 6 - nonlocal balance laws - an overview over recent results // Handbook of Numerical Analysis. 2023. Vol. 24. P. 183-216. https://doi.org/10.1016/bs.hna.2022.11.001
18. Kolokoltsov V. Differential equations on measures and functional spaces. Cham: Birkhauser, 2019. https://doi.org/10.1007/978-3-030-03377-4
19. Nagumo M. Uber die Lage der Integralkurven gewohnlicher Differentialgleichungen // Proceedings of the Physico-Mathematical Society of Japan. 3rd Series. 1942. Vol. 24. P. 551-559. https://doi.org/10.11429/PPMSJ1919.24.0_551
20. Piccoli B., Duteil N. P. Control of collective dynamics with time-varying weights // Recent Advances in Kinetic Equations and Applications. Cham: Springer, 2021. P. 289-308. https://doi.org/10.1007/978-3-030-82946-9_12
21. Piccoli B., Rossi F. Generalized Wasserstein distance and its application to transport equations with source // Archive for Rational Mechanics and Analysis. 2014. Vol. 211. No. 1. P. 335-358. https://doi.org/10.1007/s00205-013-0669-x
22. Piccoli B., Rossi F. On properties of the generalized Wasserstein distance // Archive for Rational Mechanics and Analysis. 2016. Vol. 222. No. 3. P. 1339-1365. https://doi.org/10.1007/s00205-016-1026-7
23. Piccoli B., Rossi F. Measure-theoretic models for crowd dynamics // Crowd dynamics. Vol. 1. Cham: Birkhauser, 2018. P. 137-165. https://doi.org/10.1007/978-3-030-05129-7_6
24. Piccoli B., Rossi F., Tournus M. A Wasserstein norm for signed measures, with application to nonlocal transport equation with source term // Communications in Mathematical Sciences. 2023. Vol. 21. No. 5. P. 1279-1301. https://doi.org/10.4310/CMS.2023.v21.n5.a4
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26. Villani C. Optimal transport. Old and new. Berlin-Heidelberg: Springer, 2009. https://doi.org/10.1007/978-3-540-71050-9
Поступила в редакцию 22.08.2024
Принята к публикации 27.09.2024
Авербух Юрий Владимирович, д. мат. н., ведущий научный сотрудник, отдел дифференциальных уравнений, Институт математики и механики им. Н. Н. Красовского УрО РАН, 620108, Россия, г. Екатеринбург, ул. С. Ковалевской, 16. ORCID: https://orcid.org/0000-0002-6541-8470 E-mail: [email protected]
Цитирование: Ю. В. Авербух. Теорема о выживаемости типа Нагумо для нелокального уравнения баланса // Известия Института математики и информатики Удмуртского государственного университета. 2024. Т. 64. С. 3-16.
