Optimal control of the Navier—Stokes system with a space variable in a network-like domain Текст научной статьи по специальности «Математика»

UDC 517.929.4 Вестник СПбГУ. Прикладная математика. Информатика... 2023. Т. 19. Вып. 4 MSC 74G55

Optimal control of the Navier — Stokes system with a space variable in a network-like domain

A. P. Zhabko1, V. V. Provotorov2, S. M. Sergeev3

1 St. Petersburg State University, 7—9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation

2 Voronezh State University, 1, Universitetskaya pl., Voronezh, 394006, Russian Federation

3 Peter the Great St. Petersburg Polytechnic University, 29, Polytekhnicheskaya ul., St. Petersburg, 195251, Russian Federation

For citation: Zhabko A. P., Provotorov V. V., Sergeev S. M. Optimal control of the Navier — Stokes system with a space variable in a network-like domain. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2023, vol. 19, iss. 4, pp. 549-562. https://doi.org/10.21638/11701/spbu10.2023.411

The research of the problem of optimal control of the Navier — Stokes evolutionary differential system, considered in Sobolev spaces, the elements of which are functions with carriers in an те-dimensional network-like domain, is presented. Such domain consists of a finite number of subdomains, mutually adjacent to certain parts of the surfaces of their boundaries according to the graph type. For functions that are elements of these spaces, the conditions for the existence of traces on the surfaces of the joining are presented and the conditions of adjacency subdomains to which these functions satisfy are described. In applied questions of the analysis of the processes of transport of continuous media, the conditions of adjacency describe the regularities of the flow of fluid through the boundaries of the adjacent domains. The paper presents the results of following main research questions: 1) weak solvability of the initial boundary value problem for the Navier — Stokes system and obtaining the conditions for the existence of a weak solution to this problem; 2) the formation and solution of optimal control problems of various types of Navier — Stokes system. The fundamental approach to the analysis of the weak solvability of the initial boundary value problem is its reduction to the differential-difference problem (semi-digitization of the original system by a time variable) and subsequent use of a priori estimates for weak solutions of the obtained boundary value problems. The obtained a priori estimates are used to prove the theorem of the existence of a weak solution of the original differential system and indicate the way of the actual construction of this solution. A universal approach to solving the problems of optimal distributed and starting control of the Navier — Stokes evolutionary system is presented. The latter essentially expands the possibilities of analyzing non-stationary network-like processes of applied hydrodynamics (for example, processes of transporting various types of liquids through network or main line pipelines) and optimal control of these processes.

Keywords: differential-difference system, evolutionary Navier — Stokes system, network-like domain, solvability, optimal control.

1. Introduction. The paper be considered the question of the existence of a weak solution and the associated problems of optimal control of the Navier — Stokes evolutionary system, the spatial variable of which belongs to a network-like domain. The structure of such a domain is similar to the geometry of a connected graph (see [1-4] and the bibliography there). The work is a continuation of the studies presented in [5, 6], a essential difference from which was the use of a priori estimates of the differential-difference system

© St. Petersburg State University, 2023

to analyze the weak solvability of the Navier — Stokes differential system using a priori estimates. Namely, according to the obtained weak solutions of the differential-difference system, piecewise-constant approximations on the time variable are constructed, which form a weakly compact sequence of approximations of the solutions of the Navier — Stokes system. The obtained results are the basis for the analysis of the problems of optimal distributed and starting control of the Navier — Stokes evolutionary system, which have interesting analogies with the applied problems of optimization of multiphase hydrody-namic flows and composite polimers [7, 8].

2. Notations and concepts. Consider a bounded network-like domain 3 G 1" (n > 2), consisting of N subdomains (l G IN = {1, 2,..., N}), united in a certain way to each other by means of M, 1 < M < N — 1, nodal places wj (j G IM = {1, 2,..., M}):

N M ,

3 = 3U w, here 3 = |J 3w = |J wj, where 3; p| 3; = 0 (l = l ), wj p| w j = 0

i=i j=i

(j = j ), 3; p| wj = 0 [5]. In these nodal places 3; have common boundaries, which are the surfaces of the joining. If the index j G IM is fixed, then the nodal place is a totality of subdomains. Such a set consists of subdomain 3 j and subdomains 3,

lL G I m (j ) C I m , i = l,mj, from where follows the existence of the surface of the adjoining Sj (measSj > 0) of subdomain 3¡. to the subdomains its respective subsurfaces

Sj}U (measSju > 0), i = 1, mj : Sj = |J Sjl. In this case, Sj is part of the boundary d3,

' _ i=i °

and Sjyl (i = 1, mj) are the corresponding parts of the boundaries d3;;. The latter means that wj it is defined by the adjacency surface Sj, and the subsurfaces SjU are the surfaces of the joining to 3;;, lL G IM (j), i = 1,mj. The boundary d3 of a domain 3 is defined by

N M

the ratio d3 = |J d3i \ |J Sj. It is assumed that the surfaces Sj (j G IM) are smooth,

i=i j=i

subregions 3; (l G IN) are star-shaped relative to the ball fixed at each l (l G IN).

It should be noted that the structure of the network-like domain 3 is similar to the structure of the graph-tree [1, 2], we also note that any connected subdomain of the domain 3 also has a network-like structure.

Next, the initial boundary value problem for the evolutionary transfer equation is considered, which is a mathematical model of the transportation of a viscous fluid through pipeline networks.

3. Navier — Stokes evolutionary system. For functions Y(x, t) = {y1(x, t), y2(x, t),

...,y"(x,t)}, x,t G 3T = 3 x (0,T) (x = {x1,x2, ...,X"}, T < to), consider the system

"

% — vAY +£ Y§Y +grad p = f, (1)

i=i z

div Y = o(jC dY =0^ , (2)

however, for Y(x,t) in the nodal places wj (j G IM) there are conditions (conditions of adjacency the subdomain 3 j to 3, lL G IM, i = 1,mj)

Y (x,t)\xesjlCdQ= Y(x,t)\xesjlCdQ , 1 =l,mj, (3)

mj

I^ds + Ef d-jds = 00, (4)

Sj j 1=1 Sji j

on surfaces Sj, Sj1 (i = l, mj) at t e (0, T), where nj and nj1 are the external normals to Sj and Sjt, respectively, i = l,mj, j = 1,M. The relationships

Y(x, t) |t=o = Yo(x), x (5)

Y (x,t)lxea& = 0, (6)

describe the initial and boundary conditions. The set of relations (1)-(6) are the initial marginal boundary value problem (differential system (1)-(6)) for functions Y(x, t), p(x, t) in a closed domain 9T (9T = (^Ud9) x [0,T]).

In the mathematical description of the processes of transportation of viscous fluids, 9 it belongs to R3 and models the network (or main) hydraulic system, which is the carrier of the hydraulic flow. The function Y(x, t) describes the quantitative characteristics of the

n

velocities of the hydraulic flow, £ YijY are the convective change of the velocity vector.

i=i *

The ratios (1), (2) and (3), (4) define the Navier — Stokes system, which simulates the flow of a viscous fluid (viscosity is equal v) through the hydraulic system and forms the law of fluid flow at the branch places of the hydraulic system, (5) and (6) are initial and boundary conditions, respectively, p(x, t) is pressure in the hydraulic system.

Remark 1. You can use other conditions of adjacency (3), (4) depending on the goals pursued of an applied nature. It is necessary that the requirement of solvability of the problem (1)-(6) be satisfied (see work [5]).

To obtain the conditions of solvability of the differential system (1)-(6), a differential-difference system of the form

1 [Y(к) - Y(к - 1)] - vAY(к) + £ Yi(k)^ = fT(к) - gradр(к),

i=i г (7) div Y(k) = 0, к =1, 2,..., K, y(0) = Y0(x),

Y (k)\xea^ =0, к = 1, 2,...,K, (8)

is used, where т = T/K is step partition of segment [0, T] by points кт (к = 1,2, ...,K -1);

кт

Y (к) := Y (x; к); Y (к)г := 1 [Y (к) - Y (к - 1)]; fT (к) := fT (x; к) = if f (x,t)dt;

(k-i)

кт

Рт(к):= Рт (x; к) = if p(x,t)dt (к =1,2,..., K).

(k-i)

Let's introduce the necessary spaces, using the classical lebesgue and Sobolev spaces. Denote through L2(Q)n space, the elements of which are real Lebesgue measurable vector-functions u(x) = {ui(x,t),u2(x,t), ...,un(x,t)}, x = (xi, x2,..., xn) G Rn. The ratios

__N

(u,v) = f u(x)v(x)dx and ||u|| = \J(u, u) (here f ф(x)dx = £ f ф(x)dx) in L2(^)n define

9 9 l = i 9i

the scalar product and the norm, respectively. Let D(Q)n is set of infinitely differentiable finite functions ^>(x), for which div ф = 0: D(^)n = {ф : ф G D(^)n, div ф = 0}. The closure D(Q)n in L2(Q)n defines space Н(Щ, the elements of space rHi(Q) are functions ф(x) G H(^) with a generalized derivative dx G L2(^)n. The scalar product in Hi(Щ is defined by

the formula (ф,ф\ = (ф,ф) + (дф, Ц), IMIi = (М* + IIдФФЦ2)^. For the differential-difference system (7), (8) we introduce the state space ^o(^) as a closure in Hi(Q) the

mj

set of functions ф G D(^)n that satisfy the relations f дф^х^ds + £ f ддфпХds = 0.

Sj j l=i Sji

We will first analyze two differential forms:

PM= £ J dxj Txtdx' ß(u,v,w) = jr j uk§xkwdx

i,j=1 9 * * i,k = 1 9

for which the integrals f d-^Xtdx and /ukd^widx converge (here u(x) = {uo(x),

9 Xi Xi 9 xk

u2(x), ...,un(x)}, u(x) = {u>i(x),u2(x), ...,un(x)}, u(x) = {ui(x),u2(x), ...,un(x)}) (see also [9, pp. 79-81]).

Lemma 1. The form p(u,v) is continuous by u, v on ^o(^) x ^o(^); the form ß(u,v,w) is continuous by u, v, w on L4(3)n x Vo(3) x L4(3)n. Proof. For dx- and ^X- of form p(u, v) we get

I J dX-dXjdx\ < Jj(dX-)2dx.ß&dx < Wuj »1 \\vj Hi (9)

9 V 9 V 9

(where the Cauchy — Bunyakovskii inequality is used). Similar actions for ukwi and ^vk, and then for u\ and w2 of form a(u,v,w), reduce to the following inequalities:

\JukdXkwidxI f9(ukwi)2dxJI(dx)2dx ^

I-— n^-— I--(10)

^ M utdxM( dXi )2dx M wtdx < \\uk »¿4(9) \\vj\\i Wwi»L4 (9).

From inequality (9) follows continuity p(u,v) on Vq(3) x ^q(3), continuity g(u,v,w) on (VQ(3) n L4(3)") x VQ(3) x L4(3)" follows from (10).

Lemma 2. Let u, w are arbitrary elements of space Vq(3), then: 1) g(u,u,w) = —g(u,w,u); 2) g(u,w,w) = 0; 3) g(w,w,w)=0.

"

Proof. The first statement follows from the sum J uk-§%rwidx, when integrating

i,k=19 Xk

in parts all its integrals, the following statements follow from the first.

Lemma 3. From the weak convergence of sequences {um}m^0, {vm}m^0 in L2(3)" to the elements u and v follows the convergence of the sequence {umvm}m^0 in norm L2(3)" to the element uv.

Proof. Let's show convergence f umvmQdxdt ^ f uvQdxdt on any element of

m^oo

Z(x) G Vq(3). Since the sequences {um}m^0, {vm}m^0 converge weakly, the elements um,

vm are bounded in total and ||vm|| + ||v|| < c, \\um\\ + ||u|| < c. The sequence {vmZ}m^0 converges strongly in L2(3)" to the element vZ. Indeed, with arbitrarily small given e > 0, let's take as Z(x) a function Z(x), then

WvmZ — vZ Wl2(9)~ < Wvm — v||L2(9)n||Z W L2(9)n < e(lvm|i2(S)n + ||v||L2(9)n) < ec,

what means convergence {vmZ}m^0 to vZ in L2(3)". This and the estimates presented below

I / umvmZdx — f uvZdx\ = f \(umvm — uv)Z\dx <

< J (\\Um II + \\VmZ - VZ || + \M\\\umZ - UZII) dx

9

lead to the completion of the proof of the lemma statement.

The following approach for analyzing the weak solvability of the system (1)-(6) is based on the construction of a priori estimates of the solutions of the differential-difference system (7), (8) and use of the Galerkin method, which assume look for functions Y(k) e

Vg(9), k = l, 2, ...,K, in the form of expansions on a special basis of space Vq(9) —

n 2

system of generalized eigenfunctions of the operator AY = £ . Such a system forms

i=i x*

the basis in the spaces Vq(9) and L2(9)n (proof similar to the one given in the work [10]).

Let us turn to the issue of constructing a priori estimates of the weak solution of the system (7), (8). Let the input data Y0(x), f (x,t) of systems (1)-(6) satisfy the conditions Y0(x) e VQ (9), f (x,t) e L2Q (9t)n (the elements of space L2Q(9t)n belong to L1(9T)n

T

and have a finite norm ||w||2 q = f(f u(x,t)2dx)1/2dt). This means that for the system (7),

0 9

(8) the input data Yo(x) e VQ(9), fT(k) e L2(9)n.

Definition 1. The set of functions {Y (k) e Vq (9), k = 1, 2,..., K}, where for each k (k = 1, 2,..., K) the function Y(k) satisfies the relation

(Y(k)t, n) + vp(Y(k), V) + g(Y(k), Y(k), n) = (fT(k), n),

Y (0)= Yo (x), (11)

under any function n(x) e Vq(9), is called the weak solution of the differential-difference system (7), (8).

Since the set of generalized eigenfunctions [Ui(x)}i^1 is the basis in space Vq(9), then

m

illations y m(k) — C

i=1 the system

(Ym(k)t,Ui)+ Vp(Ym(k),Ui) + 6(Ym(k),Ym(k),Ui) = (fT (k),Ui), ( )

(12)

i = 1, 2,..,m, k = 1, 2,..,K,

Ym(0) = Yo m(x), (13)

m

where Yom(x) = J2 9imUi(x) (g°m — const), Yom(x) ^ Yo(x) in norm H1 (9).

i=i

Theorem 1. Let Yo (x) e VQ(9), fT (k) e L2(9)n (k = 1, 2,..,K). For functions Ym(k), k = 1, 2,..., K, valid estimates

IIYm(k)ll < IIYm(0)ll +2llfT (k)ll2ii, k =1, 2,..,K,

IIYm(k)f +2tv £ II ^Xf112 < C (\IYo I2 + (IIfT (k)l2,i)2) , k = 1, 2,..., K,

for approximations Ym(k) = £ gf mUi(x) of the functions Y(k), k = 1,2, ...,K, consider

i=i '

' k '

where the constant C is independent of t; IfT(k)I2 q = t £ IfT(k )I.

f=1

Proof. From the relation Y(k — 1) = Y(k) — tY(k)t follows the obvious relation 2t(Y(k), Y(k)t) = Y2(k)+ t2Y(k)2 — Y2(k — 1). The ratio (11), (12) multiplied by 2Tgfm and summed by i from 1 to m, we get

Ym (к) - Ym (к - 1) + т 2Ym (к)г + 2тив^т(к)^т(к)) = 2т (fT (к), Ym(^),

2Y2

к = 1, 2, ...,K,

(p(Ym(k),Ym(k),Ym(k)) =0 by virtue of statement 3 of Lemma 2), which follows

\\Ym(k)f - \\Ym(k - 1)||2 + т 2\\Ym(k)tf + 2tv у ^ II2 < 2t\\fT (k)\\\\Ym(k)l

k = 1, 2, ...,K.

and further

22

\\Ym(k)\\z -\\Ym(k - 1)\\2 < 2t\\/t(k)\\\\Ym(k)\\, k =1, 2,..., K. (15)

For inequality (15), consider two cases:

• \\Ym(k)\\ + \\Ym(k - 1)|| > 0. Since ||Y JY+k^m < 1 then, dividing (15) by \\Ym(k)\\ + \\Ym(k - 1)\\, we get

\\Ym(k)\\ - \\Ym(k - 1)\\ < 2t\\fT (k)\\, k = 1, 2,K; (16)

• \\Ym(k)\\ + \\Ym(k - 1)\\ = 0. From (15) follows 0 < 2t f (k)\\ \\Ym(k)\\ and \\Ym(k)\\2 - \\Ym(k - 1)\\2 < 2t\\fT(k)\\ \\Ym(k)\\, k =1,2, ...,K, and again we get (16).

If we sum (16) by k from 1 to k, we get the first estimate in the statements of the theorem:

k , , \\Ym(k)\\ < \\Ym(0)\\ + 2 £ t f (k )\\ = \\Yo\\ + 2\\fT(k)^, (17)

k = 1, 2,..., K,

k

\\fT (k)\\ 2,1 =12 t\\fT(k ^ (\\ • \\2,1 is analogue of the norm \\ • \\2,1 )-

k' = 1

If we sum (14) by k from 1 to k, using inequalities (17), we get a second estimate in the statements of the theorem:

k

\\Ym(k)\\2 + 2tv £ \\^^\\2 <

k =1

kk

t\\2 I 2TV X^ \\

<\Ym(k)\\2 + t2 £ \\Ym(k)t\\2 +2tv £ \\\2 < (18)

k' = 1 k' = 1 < С (\\Yo\\2 + (\\fT (k)^)2) , k = 1, 2,...,K,

where the constant С does not depend on т.

Consequence. The obtained a priori estimates make it possible to show a weak solvability of the differential-difference system (7), (8) (and hence (12), (13)), which is established similarly to the reasoning given in the work.

Remark 2. It is easy to show that from the estimates (17), (18) follows the continuous dependence of the weak solution {Y(k) e Vq(9), k = 1,2,...,K} of the differential-difference system (7), (8) on the input data Y0 (x), fT (k).

To analyze the differential system (1)-(6), we will introduce the necessary spaces.

Denote through W1,0(QT) the space with elements u(x,t), the generalized derivatives

1/2

of which belong to L2(9t)n, \\u\\w1,o(9t) = (\\uf + \\§U, and let W1 ($t) is the space with elements u(x,t) e L2(^T)n, for which dUgX't), au<(Xt't) belong to L2(^T)n, \\u\\W i(9t ) = (Н2 + \\^\\2 + \\^ Ц2)1/2. For the elements of space W 1(Qt ) the following 554 Вестник СПбГУ. Прикладная математика. Информатика... 2023. Т. 19. Вып. 4

properties are valid [11, p. 32]: elements are continuous according to t in norm L2(Q)n and traces of elements on sections QT by planes t = t0 (t0 & [0, T]) belong to L2 (Q)n.

Denote by Qq(Qt) C W1'0(Qt), ^(Qt) C W 1(Qt) sets whose elements with fixed t & (0, T) belong to the space Vq (Q). The closures Qt(QT) and Q2(QT) in spaces W 0'0(QT) and W0(QT) denote by W0'0(QT) and WQ(Qt), respectively [3]. Clearly, u(x,t)\dQ = 0 for u(x,t) & W0' (Qt) or u(x,t) & Wt(QT). The space W0' (QT) is the state space Y(x,t) of the Navier — Stokes system, W°(QT) is helper space. As above Y0(x) & Vg(Q), f(x,t) & L2 (Q)n, the scalar function p(x,t), characterizing quantitative changes of pressure, belong to the class C(QT).

Definition 2. A pair of functions

{Y(x,t), p(x,t) : Y(x,t) & W0'°(Qt), p(x,t) & C(Qt)}

is called a weak solution of a differential system (1)-(6), if the function Y(x,t) satisfies the integral identity

T T

-J Y (x, t ) ^(Tfl dxdT + v] p(Y, n)dr + j g(Y, Y, v)dr =

T 0 0 (19) = J Y0(x)n(x, 0)dx + f f (x,t)n(x,T)dxdT

at

for any n(x,t) & W°(QT) and n(x,T) = 0.

Remark 3. By virtue of Definition 2 for a function p(x,t) it is necessary that the relation (gradp(x,t), n(x,T)) =0 at any n(x,t) from W°(QT). The latter is possible, for example, when p(x,t) it belongs to the class C(QT). Note also that in many application problems of continuum transport, the function p(x, t) refers to the input data, therefore its existence not depend on the existence of the function Y (x,t) & wq0 (Qt ).

Further, using the obtained a priori estimates (17), (18) (statements of Theorem 1),

consider the issue of weak solvability of the differential system (1)-(6) [6] (see also [12, p. 191]). 1 n

Theorem 2. Let the conditions Y0(x) & Vg(Q), f(x,t) & L2,0(QT)n, then the initial boundary value problem (1)-(6) is weakly solvable.

Proof. Let's denote through YK(x,t) piecewise constant interpolations by t: YK(x,t) = Y(k), t & ((k - l)T,kT], k = 1, 2,..., K, YK(x, 0) = Y0(x). Here we proceed from the existence of the solution {Y (k) & V°(Q), k = 1,2,..., K} (consequence of Theorem 1). It is clear that uK(x,t) & W°'°(QT) and satisfies the a priori estimates (17) and (18), then for uK (x,t) a fair estimate

\\yK|| + || dJt II < C* (20)

with an independent of t the constant C* > 0. A similar representation is set for the function fK(x, t): fK(x, t) = f (x; k), t & ((k — \ )t, kT], k = 1, 2, ...,K. With an unlimited increase K to infinity, we get a sequence {YK(x,t)}, from which, given (20), we select the subsequence {YK(x,t)}, converging to Y(x, t) & w0'0(Qt). Let's assume that Y( x, t) is weak solution of the system (1)-(6). To do this, we will show what Y(x,t) satisfies the identity (19) with arbitrary n(x,t) & C0(QT+T)n, satisfying the conditions (3), (4) under any t & (0,T) and for which n\d&T = 0, n\te[T,T] = 0. By n(x,t) are defined n(k): n(k) = n(x, kT), k = 1, 2,..., K, at the same time n(k)t' = 0 [n(k + 1) — n(k)] (here n(k)t', n(k)t are right and left approximations dn at the point t = kT, respectively). By functions n(k) piecewise continuous approximations nK(x,t), dnKdXX't), are formed

for functions n(x,t), ,t), dngt,t) by analogy with YK(x,t), moreover nK(x,t), avKj£,t), amd\x,t) uniformly converge on 9T to n(x,t), dVgXx,t), dllgt,t) at K ^ to, respectively; vk(x,t) = 0, t G [T,T + t].

The identity (19) will be summed up by k from 1 to N, replaced n(x) by Tn(x):

N N

-tZJ Y (k)n(k)tdxdt -J Yon(1)dx + v £ Tp(Y (k), n(k)) +

k=19 9 k=1 (91)

N N y^1)

+ £ tq(y(k),Y(k),n(k)) = £ Tju(k)n(k) k=1 k=1 9

NN

(here t £ Y(k)tn(k) = -t £ Y(k)n(k)t - Y(0)n(1), n(N) = n(N +1) = 0). The relation k=1 k=1 (21), given the representation nK(x,t), takes the form

T

-J YK (x,t)nK (x,t)tdxdt -J Y0(x,t)nK (x,T )dx + vf p(YK ,nK )dt + 9t T 9 0 (22)

+ f q(Yk,Yk,nK)dt = J fK (x,t)nK (x, t)dxdt.

0 9T

Passing on to the limit in (22) by subsequence {YK(x,t)} (nK(x,t) replaced by rjK(x,t), which corresponds to {YK(x,t)}) taking into account Lemma 3, we obtain the identity (19) for Y(x,t). This proves the weak solvability of the initial boundary value problem (differential system) (1)-(6). It should also be noted, by virtue of comment 3, the continuous dependence Y(x,t) on the input data Y0(x), fT(k). The theorem is proven.

4. The problem of optimal control. For the Navier — Stokes system two types of optical control problems — distributed and start control are considered, which are most meeting in applications and do not reduce the community of analysis. Everywhere below, the control is indicated by the symbol u, the state of Y(x, t) of the Navier — Stokes system is indicated by the Y(x,t; u). In the case of distributed control u(x,t), the distributed effect operator (this operator determines the density of external forces) is present on the right side of equation (1) of the Navier — Stokes system:

n

dr - vAY +£ YidY + gradp = f + Bu, (23)

i=1 *

in the case of start control u(x), the external effect is realized out by means of the initial state of the Navier — Stokes system and determines the initial condition (5):

Y(x, t) 11=0 = u(x), x G9. (24)

Thus, for the Navier — Stokes system, the initial boundary value problem (23), (2)-(6) determines the problem of optimal distributed control, the initial boundary value problem (1)-(4), (24), (6) determines the problem of optimal start control. At the same time, in both cases, the state of the Navier — Stokes system is monitored both at the domain QT (distributed observation), and 9 at t0 G (0,T) (observation at a fixed point in time) or at t = T (final observation). Other types of observations are also possible, for example, at the boundary d9T or part of it (boundary observation). The physical task is to bring the dynamic characteristics of the viscous fluid (velocity, convective component values) to preassigned levels at an interval (0, T) or by a point in time t = t0 G (0, T].

The definition of a weak solution of the system (23), (2)-(6) or (1)-(4), (24), (6) exactly repeats the Definition 2 with the only difference that for the system (23), (2)-(6) the right part (19) changes to

fY0(x)n(x, 0)dx + f (f (x,T) + Bv(x,t))n(x,T)dxdT,

9 9T

and for the system (1)-(4), (24) the right part (19) changes to

fv(x)n(x, 0)dx + J f (x,T)n(x,T)dxdT.

9 9t

According to what has been said Y(x,t) replaced on Y(v) := Y(x,t; v) or on Yt (v) := Y(x,t; v) (YT(v) := Y(x,T; v)), the latter for the convenience of presenting the results.

To assess the state Y(x,t; v) of the Navier — Stokes system, we introduce the space of controls U and linear continuous observation operators Cq : L2(QT)n ^ H (q = 1,2), where H is the observation space: H = L2(QT)n in the case of distributed observation and in the case of final observation H = L2(Q)n. In addition C0Y(v) = DYT(v), here operator D : H ^ H is the linear bounded operator, C2Y(v) = YT(v), Yt(v) := Y(x,t; v) :=

± Yi(x, t; v) .

i=t

Let's set the functional J(v) on a closed convex set Ud C U:

J (v) = \\CiY (v) — $\\H + \\C2y (v) — *\\2h = Jt(v) + J2(v), (25)

where J0(v) = \\DYT(v) — $\\2H, J2(v) = \\YT(v) — ^\\H and ^ are preassigned functions: in the case of distributed observation $ := §(x,t), ^ := ^(x,t) & H = L2(QT)n, for the case of final observation $ := $(x), ^ := &(x) & H = L2(Q)n. In applications, the functional J0(v) establishes the difference between the characteristics of the velocity vector of the hydraulic flow from the defined $, the functional J2(v) characterizes the difference between the convective change in the velocity vector and the defined

Definition 3. The problem of optimal distributed or start control of the Navier— Stokes system (23), (2)-(5) or (1)-(4), (24) is to find inf J(v). Optimal control u & Ug of

veUg

the system will be called the minimizing element of the functional J(v): J(u) = inf J(v).

vEUg

In the future, we will assume the existence of optimal control of the Navier — Stokes system. We will prove the auxiliary statements beforehand.

Lemma 4. Let u,v & Uq and 6 & (0,1). The following relations are valid:

Yt(u)'(v — u) = Yt(v) — Yt(u), Yt(u)'(v — u)= Yt(v) — Yt(u), (26)

here the symbol "'" denotes Frechet derivative by control v of function Yt(v) and Yt(v).

P r o o f. Let's reason for the case of distributed control v(x, t), similar reasoning is true in the case of start control. Proceeding from the integral identity of the definition of the weak solution of the system (23), (2)-(6) and taking into account the relations

p(Yt(v),n) — p(Yt(u),n)= £ / ^Sdx — it J '-YXjdjdx =

i,j=1 9 i,j=1 9

= p(Yt(v) — Yt (u),n),

6(Yt(v),Yt(v),V) — 6(Yt(u),Yt(u),V)= ± J Yt(v)ka-Ygi Vidx — ± J Yt(u)ka-^ Vidx =

j,k = 1 9 j,k = 1 9

= 6(Y (v) — Y (u),V)

(here g(Y(ш),ц) = £ fYt(u)k dYX^ Vidx, ш = v or ш = u) for any u,v e Ud and

j,k=1 э k

arbitrary function n(x,t) e Wq(9t), we come to integral identities for arbitrary n(x,t) e W q0(9t ):

- J (Yt,(v) - Yt(u)) ЦХг1 dxdt + v ]p(Yt(v) - Yt(u),n)dt + ]g(Y(v) - Y(u),V)dt =

Эт 0 0

T

= J(B(v - u),n)dt,

0

T

-J (Yt(u + 0(v - u)) - Yt(u)) Щ^т1 dxdt + v J p(Yt(u + 6(v - u)) - Yt(u),n)dt + Эт 0

T _ _ T

+ J Y(Y(u + 6(v - u)) - Y(u), n)dt = 6 J(B(v - u), n)dt.

00

If there are limits lim Yt(u+e(v--u))-Yt(u), Y(u+e(u—u))-Y(u), then there are derivatives

e^0 в в^0 в

of Frechet Yt(u)', Y(u)', respectively. After dividing the second integral identity by 6 and calculating the limit at 6 ^ 0, we obtain the relations (26) after comparing the left parts of these identities. The lemma is proven.

Lemma 5. Let u e Uq be the minimizing element of the functional J(v) then

J(u)'(v - u) > 0 (27)

for any v e Ud.

Proof. Since u e Ud it is a minimizing element of the functional J(v), then J(u) = inf J(v). With any v e Ud and 6 e (0,1) element (1 - 6)u + 6v e Ud due to the

convexity of Ud, and therefore,

J(u) < J((1 - 6)u + 6v) = J(u + 6(v - u))

and then

j(u+e(v-u))-j(u) ^ 0 в ^0.

When 6 ^ 0 we get the inequality (27), the lemma is proven.

Next, consider the problems of optimal control of the Navier — Stokes system (1), (2). Distributed control. The Navier — Stokes system witch distributed control is of the form (23), (2)-(4). Its state {Y(v)(x,t),p(v)(x,t)} is determined by a weak solution of the initial boundary value problem (23), (2)-(6), for which the integral identity takes the form

T T

-J Y (x, t; v) Цд^ dxdt + vjp(Y,n)dt + j e(Y,Y,n)dt =

Эт 0 0 (28)

= J Y0(x)n(x, 0)dx + f (f (x,t) + Bv(x,t))n(x,t)dxdt.

Э Эт

Minimizing functional J(v) with distributed observation operators Cq, С2 (H = L2(QT)) has the form (25), control v e U = L2(^T).

Theorem 3. Let u(x,t) is the optimal control of the system (23), (2)-(4), then the state function Y(u)(x,t) of this system satisfies the identity (28) under v(x,t) = u(x,t) for any element n(x,t) e Wq(9;t), n(x,T) = 0; and the inequality

(DYt(u) - Ф,Б (Yt(v) - Yt(u)))H + (Yt(u) - Yt(v) - Yt(u))H > 0 (29)

for any u(x,t) € Ug; function p(x,t) is arbitrary element of class C(ST).

Proof. Since u(x, t) is the optimal control, it is a minimizing element of the functional J(u): J(u) = inf J(v). By virtue of the statement of Lemma 5,

v£Ug

inequality (27) is true. Based on (25), J(u) it is presented in the following form

(У (x,t; u)=± Yi(x,t; u) Щ^) :

J(u) = (DY(x,t; u) - Ф(x,t), DY(x,t; u) - <^(x,t))H + + (Y(x, t; u) - x, t), Y(x, t; u) - x, t))H

and, given the inequalities (26) of Lemma 4, we come to inequality

2 J(u)'(u - u)= (DYt(u) - Ф, D(Yt(u) - Yt(u)))H + + (Y(u) - 4,Yt(v) - Yt(u))H > 0

and get inequality (29). The relation (28) (at v(x,t) = u(x,t)) for any n(x,t) € ^¿(ST) (n(x,T) = 0) is the integral identity of the initial boundary value problem (23), (2)-(5). By virtue of Remark 4, the function p(x, t) is arbitrary element from C(ST). The theorem is proven.

It should be noted that the control effect on the Navier — Stokes system (1)-(4) in a finite number of fixed points of the domain S (point control) is a variant of distributed control [2]. For example, such points xj may belong to the surfaces Sj of the

M

nodal sites Uj, j = 1, 2,...,M. Then in the ratio (1) f (x,t) = E uj(t) ® S(x - xj),

j=i

u(t) = {u1(t),u2(t),...,uM (t)} € L2(0,T )m . In this case, the state Y (u) := Y (x,t; u) of the system (1)-(4) is an element ^¿(ST) and for Y(x,t; u) fair identity

T T

-J Y (x, t; u) dxdt + v] p(Y,n)dt + j e(Y,Y,n)dt =

9T 0 0

m T

= J Yo(x)n(x, 0)dx + J f (x,t)n(x,t)dxdt +E J uj(t)n(xj,t)dt

9 St j=1o

for any n(x,t) € ^¿(ST), n(x, T) = 0. Further reasoning is similar to the above.

Starting control. Consider the problem of optimal starting control of the Navier — Stokes system (1)-(4) with a control effect u(x) € U = L2(S)n, that determines the initial condition (24). The pair {Y(u)(x,t),p(x,t)} is a weak solution of the initial boundary value problem (1)-(4), (24), (6), p(x,t) € C(ST), the function Y(u)(x,t) satisfies the integral identity

T T

-J Y (x, t; u) Щ^Д dxdt + v J p(Y,n)dt + J g(Y,Y,n)dt = 9t 0 0 (30)

= J u(x)n(x, 0)dx + f f (x,t)n(x,t)dxdt

9 9t

for any n(x,t) € ^¿(ST), n(x,T) = 0, and p(x,t) is an arbitrary element of C(ST). Minimizing functional J(u) has the form (25), u(x) € U = H = L2(S)n.

Theorem 4. Let u is the optimal control of the system (1)-(4), then the state Y(u)(x,t) of this system satisfies the identity (30), where u(x) replaced by u(x), and the inequality

(DYt(u) - (Yt(u) - Yt(u)))H + (Yt(u) - Yt(u) - Yt(u))H > 0

for any u(x) G Ug = U. The function p(u)(x,t), like above, belongs to space C(9T).

The proof of the statement of the theorem repeats verbatim the proof of Theorem 3, since the representation of the minimizing functional J(u) does not change.

Remark 4. Statements of Theorems 3 and 4 are necessary conditions for the existence of optimal distributed and starting controls. For a linearized Navier — Stokes system

dY - vAY + grad p = f

the necessary and sufficient conditions for optimal control can be established using a conjugate system to this one.

5. Conclusion. The study of the problem of optimal distributed and starting control of the Navier — Stokes evolutionary differential system is considered in the Sobolev spaces of functions with carriers in the network-like region of the n-dimensional Euclidean space (n > 2 ). The paper presents the results of two main areas of research: obtaining conditions of weak solvability of the initial boundary problem for the Navier — Stokes system; the formation and solution of optimal control problems of the Navier — Stokes system. When analyzing the weak solvability of the initial boundary-boundary problem, it is reduced to the differential-difference system and the construction of a priori estimates for weak solutions of this system is carried out. Based on the Galerkin method with a special basis, an algorithm is formed for the actual construction of a weak solution to the initial boundary problem for the Navier — Stokes system. The obtained results are effectively used in the analysis of optimal control problems not only for network hydrodynamic processes, but also for the analysis of inverse problems of mathematical physics, the problems of determining the minimax of controlled systems, stability and stabilization of mechanical systems [13-17].

References

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Received: May 12, 2023. Accepted: October 12, 2023.

A u t h o r s' i n fo r m a t i o n:

Aleksei P. Zhabko — Dr. Sci. in Physics and Mathematics, Professor; a.zhabko@spbu.ru Vyacheslav V. Provotorov — Dr. Sci. in Physics and Mathematics, Professor; wwprov@mail.ru Sergey M. Sergeev — PhD in Engineering, Associate Professor; sergeev2@yandex.ru

Оптимальное управление системой Навье — Стокса с пространственной переменной в сетеподобной области

А. П. Жабко1, В. В. Провоторов2, С. М. Сергеев3

1 Санкт-Петербургский государственный университет,

Российская Федерация, 199034, Санкт-Петербург, Университетская наб., 7-9

2 Воронежский государственный университет,

Российская Федерация, 394006, Воронеж, Университетская пл., 1

3 Санкт-Петербургский политехнический университет Петра Великого, Российская Федерация, 195251, Санкт-Петербург, ул. Политехническая, 29

Для цитирования: Zhabko A. P., Provotorov V. V., Sergeev S. M. Optimal control of the Navier — Stokes system with a space variable in a network-like domain // Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления. 2023. Т. 19. Вып. 4. С. 549-562. https://doi.org/10.21638/l1701/spbu10.2023.411

Проведено исследование задачи оптимального управления эволюционной дифференциальной системой Навье — Стокса, рассматриваемой в пространствах Соболева, элементы которых — это функции с носителями в те-мерной сетеподобной области. Такая область состоит из конечного числа подобластей, взаимно примыкающих определенными частями поверхностей своих границ по типу графа. Для функций, являющихся элементами указанных пространств, представлены условия существования следов на поверхностях примыкания и рассмотрены условия примыкания подобластей, которым эти функции удовлетворяют. В прикладных вопросах анализа процессов переноса сплошных сред условия примыкания описывают закономерности протекания потоков жидкостей через границы примыкающих подобластей. Приведены результаты двух основных вопросов исследования: слабая разрешимость начально-краевой задачи для системы Навье — Стокса и получение условий существования слабого решения этой задачи; формирование и решение задач оптимального управления разного типа системой Навье — Стокса. Основополагающим подходом анализа слабой разрешимости начально-краевой задачи является редукция ее к дифференциально-разностной (полудискретизация исходной системы по временной переменной) и последующее использование априорных оценок для слабых решений полученных краевых задач. Такие оценки используются для доказательства теоремы существования слабого решения исходной дифференциальной системы и указывают путь фактического построения этого решения. Представлен универсальный подход к решению задач оптимального распределенного и стартового управления эволюционной системой Навье — Стокса. Последнее существенно расширяет возможности анализа нестационарных сетеподобных процессов прикладной гидродинамики (например, процессов транспортировки разного типа жидкостей по сетевым или магистральным трубопроводам) и оптимального управления этими процессами. Ключевые слова: дифференциально-разностная система, эволюционная система На-вье — Стокса, сетеподобная область, разрешимость, оптимальное управление.

Контактная информация:

Жабко Алексей Петрович — д-р физ.-мат. наук, проф.; a.zhabko@spbu.ru Провоторов Вячеслав Васильевич — д-р физ.-мат. наук, проф.; wwprov@mail.ru Сергеев Сергей Михайлович — канд. техн. наук, доц.; sergeev2@yandex.ru