Optimal control over solutions of a multicomponent model of reaction-diffusion in a tubular reactor Текст научной статьи по специальности «Математика»

DOI: 10.14529/mmph200102

OPTIMAL CONTROL OVER SOLUTIONS OF A MULTICOMPONENT MODEL OF REACTION-DIFFUSION IN A TUBULAR REACTOR

O.V. Gavrilova

South Ural State University, Chelyabinsk, Russian Federation E-mail: [email protected]

This article studies a mathematical model of reaction-diffusion in a tubular reactor based on degenerate equations of reaction-diffusion type defined on a geometric graph. It is precisely the degenerate case that is studied, since when building the mathematical model it is taken into account that the speed of one sought function is significantly higher than the speed of the other. This model belongs to a wide class of semilinear Sobolev-type equations. We give sufficient conditions for the simplicity of the phase manifold of the abstract Sobolev-type equation in the case of s-monotone and ^-coercive operator; we prove the existence and uniqueness of a solution to the Showalter-Sidorov problem in the weak generalized sense, and the existence of optimal control over weak generalized solutions to this problem. On the basis of the abstract theory, we find sufficient conditions for the existence of optimal control for a mathematical model of neural signal transmission.

Keywords: Sobolev-type equations; phase manifold; Showalter-Sidorov problem; reaction-diffusion equations; optimal control problem.

Introduction

Take a finite connected oriented graph G = G(V; E) with vertex set V = {Vi} f= i and edge set

E={Ej} K = i, where each edge is of length lj > 0 and transverse cross-section area dj > 0. Consider on G the multicomponent system of reaction-diffusion equations

Vijt = aivijss + fi j (vij , V2 j Vmj ) + uij ,

V2 jt = a2V2jss + f2 j (vij , V2 j Vmj ) + u2j ,

Vj = akVkjss + fkj (vij , V2 j Vmj ) + ukj , < (i) 0 = ak+i v(k+i) jss + f(k+i) j (vi j, v2 j, • • •, vmj ) + u(k+i) j,

0 = amVmjss + fmj (vij , V2j , •, Vmj ) + umj

for all se (0,lj), te R, j = i,K,

with positive parameters a, i = i,m and some functions f^e for i = i,m and j = i,K . Here the

functions vi = vi(s, t), i = i,k and vi = vi(s, t), i = k + i,m characterize the concentrations of reagents

(activator and inhibitor); a, i = i,m are the diffusion coefficients; the functions f correspond to the interaction between the reagents; the prescribed functions ui = ui(s, t) characterize exterior actions. For (i) at each vertex Vi for i = i,m impose flow balance and continuity conditions

I djVjs (0, t) - X drvrs (lr, t ) = 0, (2)

j-.Ej.eEa(VI) r-.EreEw(Vi)

Vr (0, t) = Vj (0, t) = vh (lh, t) = Vn (ln, t) (3)

for all Er, Ej-e Ea(V) and Eh, Ene E°(V). Here Ea<m)(V1) stands for the set of edges starting (ending) at V. Conditions (2), (3) and the system (i) constitute our mathematical model of reaction-diffusion in a tubular reactor. Complement (2), (3) with the Showalter-Sidorov initial conditions

Vj (s,0) = V0j (s) for all s e (0,L), i = U, j = i,K. (4)

Gavrilova O.V. Optimal Control over Solutions of a Multicomponent Model

of Reaction-Diffusion in a Tubular Reactor

Initially, a nondegenerate system of equations of the reaction-diffusion type

\evt = cDv + fi(v, w) + Mj, [wt = fiAw + f2(v, w) + m2.

was obtained in [1-3], depending on the two desired functions v = v(s, t) and w = w(s, t). These systems model a large class of processes. In the case

fi(v, w) = g-(d + 1)v + v2w, f2(v, w) = dv - v2w (6)

the system (5) describes the Lefever-Prigogine Brusselator [1], proposed as a model of an autocatalytic reaction with diffusion. The FitzHugh-Nagumo model [2, 3] is of this type with

f1(v, w) = b1w -kjv, f2(v, w) = p2w -k2v - w3. (7)

The first qualitative analysis of the system (5) appeared in [4] under the assumption that the rate of change of one concentration is much greater than that of the other. This assumption leads to the degenerate system

f 0 = cDv + f1(v, w) + u1,

I , 1 (8)

[wt = pDw + f2 (v, w) + m2.

The analysis of the morphology [4] of the phase spaces of the degenerate FitzHugh-Nagumo model (7), (8) and the Lefever-Prigogine Brusselator (6), (8) on open bounded regions showed that these phase spaces contain fold and cusp singularities [4]. Multicomponent reaction-diffusion models are studied in [5, 6]. They usually involve many inhibitors. Models with three and four components, one activator and two or three inhibitors, and their stability were studied in [5]. Goal of this article is reseach of multicomponent reaction-diffusuin with different numbers of inhibitors and activators not only inhibitors.

Consider two Banach spaces X and U. The preimage of the degenerate system (1) with conditions (2), (3) is the abstract semilinear Sobolev-type equation

dLx+M(x) = m, kerL * {0}. (9)

dt

Here L is a continuous linear operator and M is a smooth nonlinear operator to be specified. The analytic and qualitative aspects of initial (multipoint initial-final) value problems for linear and semilinear Sobolev-type models are studied in [7-12]. Complement (9) with the Showalter-Sidorov initial condition

L( x(0) - x0) = 0. (10)

Considering this initial condition instead of the classical Cauchy condition

x(0) = x0 (11)

in the case of degenerate equation (9), we can avoid the lack of existence of a solution for arbitrary initial data [8]. Condition (10) directly generalizes condition (11) since Cauchy and Showalter-Sidorov problems are equivalent in the case that L_1 exists and is continuous. However, condition (10) fails to guarantee the uniqueness of solution to problem (9), (10), for instance in the cases that the phase manifold of (9) lies in a Banach manifold with singularities [4, 8]. Thus, to find conditions under which the solution is unique, we must study the structure of the phase manifold.

Our goal is to study the optimal control problem

J(x,m) ® min (12)

by the solutions to (9), (10) in the weak generalized sense [13, 14]. Here J(x,u) is a certain purpose-built quality functional with control ue Uad, where Uad is a closed convex set in the control space U. The optimal control problem for linear Sobolev-type equations with the Cauchy initial condition was originally posed and studied in [9]. That article initiated a series of studies of optimal control problems for linear Sobolev-type equations with various initial conditions [10-12]. Sufficient conditions for the existence of a solution to problem (9), (10), (12) when L is a Fredholm operator were obtained in [11]. We give sufficient conditions for the simplicity of the phase manifold of problem (1)-(3) in case L is not Fredholm operator. Optimal control problems in various reaction-diffusion models are studied in [12]. The Showalter-Sidorov problem and the optimal control problem for degenerate two-component FitzHugh-Nagumo model (7), (8) is considered in [12] in the case that < 0 and /31 = k2.

This article is organized as follows. In the first section we talk about abstract semilinear Sobolev-type equations and discuss sufficient conditions for the simplicity of the phase manifold of the abstract equations (9) in the case of s-monotone and /»-coercive operator M and prove the existence and uniqueness of a solution to (9), (10) in the weak generalized sense using the Galerkin method. In the second section we construct a mathematical model of reaction-diffusion in a tubular reactor basing on the initial-boundary value problem for degenerate reaction-diffusion equations defined on a geometric graph. In the third section we study the optimal control problem for a mathematical model of neural signal transmission and give sufficient conditions for the existence of a solution to it.

1. Abstract semilinear Sobolev-type equation in the case of s-monotone and p-coercive operator

Consider abstract semilinear Sobolev-type equation (9) with the Showalter-Sidorov initial condition (10). All our arguments in this section will be based on the general theory of abstract Sobolev-type equations, which is described in sufficient detail in [8, 11]. Take a separable real Hilbert space H = (H;[-,-]) identified with its adjoint, as well as an adjoint pair (A; A*) with respect to [•,•] of reflexive separable Banach spaces such that the embeddings

A c H c A* (13)

are dense and continuous. Take a selfadjoint nonnegative definite operator Le L(A;A ) with

H 3 ker L ° coker L c H*, A = ker L © coim L, A*=coker L © im L. (14)

Remark 1. Condition (14) is satisfied, for instance, in the case that LeL(A; A*) is a selfadjoint nonnegative definite Fredholm operator [11].

Take an s-monotone and /-coercive operator MeCr(A; A) with r> 1 (that is, [M'yx,x]>0

II IIP"

"x,ye A\{0} and $ CM, CM e R + such that [M(x), x] > Cm\\x\\p and ||M(x)||* ^FH , where p > 2)

possessing symmetric Frechet derivative. Note that every strongly monotone operator is s-monotone, while every s-monotone operator is strictly monotone. In turn, every p-coercive operator is strongly coercive.

By condition (14), there exists a projection Q along coker L onto im L. Make the assumption that

(I- Q)u is independent of te (0,7). (15)

Consider the set

{xe A: (I -Q)Mx = (I - Q)u}, if kerL * {0};

(16)

A, if ker L = {0}. Introduce

coimL = {xe A: [x, j] = 0"je kerL \ {0}}.

Denote by P the projection along ker L onto coim L. Given a point x0e M, put x0 = Px0e coim L.

Definition 1. [8] Call a set Ma Banach Cr-manifold at x0eM whenever there exist neighborhoods Oc M and Oi c coim L of the points x0 and x0 = Px0 respectively and a Cr-diffeomorphism D: O^O such that D-1 is the restriction of the projection P to O. Refer to the pair (D, O1) as a chart. The set Mis called a Banach Cr-manifold modeled on the space coim L whenever each of its points admits a chart.

Theorem 1. [8] Suppose that condition (15) is met and M is s-monotone and p-coercive operator. Then the set M is a Banach Cr-manifold projecting diffeomorphically along ker L onto coim L everywhere with the possible exception of the origin.

The proof of the Theorem 1 is analogous to the proof of Theorem 1 in [8].

Remark 2. Observe that if x = x(t) for te [0,7] is a solution to (9) then it must lie in M. Refer to M as the phase manifold of equation (9).

Since the space A is separable, there is an orthonormal system (in the sense of H) of functions which is complete in A. Construct Galerkin approximations to the solution to (9), (10) as

x" (s, t) = fa, (t)j (s), (17)

i=1

where the coefficients ai = ai(t) for i = 1,...,« are determined from the following problem:

Lx? j + M (xn ),j = [u,j ], (18)

L( xn (0) - x0 ),j ] = 0, i = 1,..., n, (19)

Lxn (0) ® Lx0 for n ® +¥ strongly in the subspace im L. (20)

In the case dimker L < it is necessary to have n > dimker L. Equation (18) constitute a degenerate

system of ordinary differential equations. Suppose that Tne H.+, Tn = Tn(x0), and An = spanf^,^,...,^}.

Lemma 1. [17] Let M be s-monotone and /»-coercive operator. For every x0 e A there exists a unique local solution xne Cr(0, Tn; An) to problem (18), (19).

The proof rests on the existence Theorem for solutions to a system of algebraic-differential equations with the Showalter-Sidorov condition [17]. Construct the space

X = {x | xe L¥(0,T;coimL) nLp (0,T; A), x1 e L2(0,T;coim L)}.

Definition 2. [11] Call a weak generalized solution to (9) the vector function xeX satisfying the condition

T r d 1 T

j(t) —Lx+M(x),w dt = j(t)[u,w]dt,"we A,"je L2(0,T). (21)

0 Ldt J 0 Call a solution to (9) a solution to the Showalter-Sidorov problem whenever it satisfies (10).

Theorem 2. [11] LetMbe s-monotone andp-coercive operator. For every x0eA, Te H.+, ueL2(0, T; A ) there exists a unique solution xeX to problem (9), (10).

This goes in several stages and relies on the monotonicity method of [13, 14]. Assume that all requirements of the previous section are satisfied and the embedding A c H is com* 1 1

pact. Construct the space U = Lq(0, T; A), —+ — = 1 and define in it a nonempty closed convex set Uad.

p q

Consider the optimal control problem

J(x, u) ® inf, u eUad (22)

defining the objective functional as

J(x,u) = bf||x(t) - (t)||A dt + (1 -b)J||«(0||q* dt, (0,1), (23)

0 0

Here zd = zd(t) is the required state.

Definition 3. [11] Refer to a pair (x, U) eX*Uad as a solution to the optimal control problem (9), (10), (22) if

J ( x, U) = inf J ( x, u ),

( x,u)

where the pairs (x, u) e X*X*Uad satisfy (9), (10) in the sense of Definition 2; call the vector function U the optimal control.

Remark 3. Refer as an admissible element of problem (9), (10), (23) to a pair (x,u)e X*Uad, satisfying problem (9), (10) with

J(x, u) < +¥.

If Uad 0 then for every u eUad с U by Theorem 2 there exists a unique solution x = x(u) to problem (9), (10). Hence, the set of admissible elements of the problem is nonempty. Using the results obtained in the paper [11] we can show that

Theorem 3. [11] Let M be s-monotone and p-coercive operator. Given x0eA and Te M +, there exists a solution to problem (9), (10), (22).

2. A mathematical model of reaction-diffusion in a tubular reactor

In this section we construct a mathematical model of reaction-diffusion in a tubular reactor basing on the initial-boundary value problem for degenerate reaction-diffusion equations defined on a geometric graph and reduce it to the abstract Showalter-Sidorov problem (9), (10), we construct

Вестник ЮУрГУ. Серия «Математика. Механика. Физика» 17

2020, том 12, № 1, С. 14-23

functional spaces and establish the main properties of operators. Take on finite connected oriented graph G the multicomponent system of reaction-diffusion equations (1) with flow balance and continuity conditions (2), (3) and the Showalter-Sidorov initial conditions (4).

Consider the Hilbert space

L2(G) = {g = (gi,g2,...,gj,...,gK): gj e ¿2(0,lj)} equipped with the inner product

i.

< g, h) = I dj J g j (s)hj (s)ds.

eÎE о

Construct the Banach space

H = {g = (gi,g2,—,gj, — ,gK): gj î W(0,lj) and conditions (3) holds}

with the norm

lj

l|g|H = I dj J(g2 (s) + g2 (s))ds. ejîE о

Put

Lp(G) = {g = (gi,g2,—,gj,—,gK): gj î Lp(0,lj)}. The set Lp(G) is a Banach space with the norm

lj

Ml CG)= I dJ J I gj (s)|p

Lp EjeE 0

By the Sobolev embedding theorem, the space W\ (0,l;) consists of absolutely continuous functions, and so H is well-defined, dense, and compactly embedded into L2 (G). Fix a>0 and construct the operator

lj

< A g, h) = X dj j (gj, (s)hjS (s) + agj (s)hj (s))ds, g, h e H.

Ej.eE 0

The operator AeL(H;H*) is bijective, its spectrum is real, discrete, of finite multiplicity, and accumulates only at +w, while its eigenfunctions constitute a basis for the space H [15]. Denote by a sequence of eigenfunctions of the homogeneous Dirichlet problem for the operator A on the graph G, and by {^i} the associated sequence of eigenvalues in decreasing order with multiplicities taken into account. Consider the Hilbert space

H = Lm(G) = {v = (vi,V2,...,v„):Vi e ¿2(G)} equipped with the inner product

m

[v,C] = X<v ,C)

i=1

and identified with its adjoint. By analogy, construct the space A = H and denote by A the adjoint to A with respect to the inner product in H. Writing x = (v1, v2,...,vm), C = (Zi, Z2,. • •, Cm), and u = (u1, u2,...,um), define the operators

[Lx,C] = <vi,hi) + • •• + <vkhk), x,Ce A

[M (x), C] = a <vis ,Cis ) + < fl (x), Cl) + «2 <v2s , ^2s ) + < /2 (x), C2 ) + - •

+am <vms , Cms ) + < fm (x), Cm X X, C e A.

Lemma 2. (i) The operator Le L(A; A ) is selfadjoint and nonnegative definite. (ii) Suppose that j C"(Im, I) for i = and j = 1K . Then Me C%4; A*). Proof. Claim (i) follows from the construction of L. The containment Me C^(A; A*) is a classical result [16].

Thus, problem (1)-(4) reduces to the Showalter-Sidorov problem (9), (10).

3. Optimal control problem for a mathematical model of neural signal transmission

In the this section we apply the abstract results of the second section to study the optimal control problem for a mathematical model of neural signal transmission, which can be obtained from a multicomponent reaction-diffusion model (1) iff take as (6). Proceed to a mathematical model of neural signal transmission based on the FitzHugh-Nagumo system

V1 jt - aiv1 jss + Alv1 j + b12v2 j + ... + p1mVmj + k1V1j = U1 j , V2jt - a2V2jss + b21V1 j + p22V2j + - + b2mVmj + k2V\j = U2j ,

Vkjt - akVkjss + bk1V1 j + Pk2V2 j + - + pkmVmj + kkVl = Ukj , ~ak+1V(k+1) jss +b(k+1)1v1 j +b(k+1)2 v2 j + ••• + fi(kV= U'

(24)

J(k+1)m mj "(k+1)j-

~amVmjss +pm1V1j + Pm2V2j + ••• + PmmVmj =!

mj

for all s e (0, lj), t e R, j = 1, K,

defined on a finite connected oriented graph G and complemented with conditions (2), (3), where the matrix B = [fly] mj=1 has the property

$CB, CB > 0: CB [x, x] £ [Bx, x] £ CB [x, x] •

(25)

By analogy with Section 2, consider the Hilbert space H=(L m (G), [•,•]) and the Banach space A = Hm. By the Sobolev embedding theorem, there are dense continuous embeddings (13); furthermore, the embedding Ac H is compact. Write x = (v1, v2,...,vm), Z = (Zu Z2,., Zm), and u = (u1, u2,..., um). Then the operator M=M1 + M2 becomes

[Ml (x), Z] = a {Vis ,Zis > + <bl 1V1 + bl2V2 + - • + blmVm , Z > + «2 <V2s , ^2s > +

+<b21V1 +P22V2 + - + b2mVm ,Z2 > + - + «m {Vms ,Zms > + +<bm1V1 + Pm2V2 + - + PmmVm ,Zm X X,Z e A

[M2(x),Z] = k<Vi3,Zi> + k<v3,Z2> + - + k<vI,Q, x,Ze A,

where v3m = (vh

3 3 )

Vm 2, * *', Vmk ) •

Lemma 3. (i) Suppose that aie R + for i = 1,m and condition (25) is satisfied. Then the operator M3e C°(A; A*) is s-monotone and 2-coercive.

(ii) Suppose that Kie R + for i = 1,k . Then the operator M2e C°(l4(G), Lkk (G)) is s-monotone and 43

coercive.

Proof. The Frechet derivatives of M1 and M2 at xeA are defined as

[M^X, Z] = «1 <£s ,! > + {bxX + P12X2 + - + p1mXm , Z > +

+«2<X2s ,Z2s > + <b21X1 + b22X2 + - + b2mXm , Z2 > + - +

+«m <Xms , Cms > + <bm1X + Pm2X2 + - + PmmXm , Cm

>, x, Z e A,

[M2xX,C] = 3kl<Vl2Xl,Cl> + 3k2<V22X2,Z2> + - + 3kk<V2kXk,Zk>, x,Ze A. Then the continuous embedding W\ (G) c L2(G) yields

| [M1xX,Z^|£ C| x A| Id A,

112IIZII

| LM2 xX,Z \ |£ 3C| x jxil^iu:

i[m;;xxxxxz] |£ 6C| 1x11 A IX2IIMJZlla ,

| Гм;х (Х1,Х2,Хз),!^ |£ 6C| 1X11AIX21| AI Хз || A| III A, C = maxk,

where M'1x and M'2x stand for the Frechet derivatives of M1 and M2 at x. Since M=O and M fx =O, the operators M1 and M2 are C-smooth. Since

[M1x£ £] = «1 <x1s , x1s ) + +&2x + . + frmtm ,X1) +

+«2 <X*2s , X ) + <p21^1 + P22Z2 + - + b2mXm , X ) + - +

+«m <Xms ,Xms ) + <Pm1^1 + Pm2^2 + - + Pmmtm ,Xm ) > 0 ^ X e A, [M'2xX,X] = 3k1 <v?x,x)+3k2)+-+3kk<v2Xk,Xk)>0, x,Xe A, it follows that the operators M1 and M2 are s -monotone. Since

q|| x|| A £[M1( x), x]£ C2I |x|| A, [M2(x),x] = k|Iv^4 (G) +k*21|v214 (G) +- + kk||vk|I4 (G),

11 IIl4 11 IIl4 11 IIl4

it follows that M1 is 2-coercive and M2 is 4-coercive.

Remark 4. By the construction of L, the sets ker L, coim L, coker L, and im L are defined as kerL = {0}x{0}x...x{0} xHTxHx...xH ,

k m-k coimL = HxHx...xH x{0{x{0}x...x{0} ,

k m-k

cokerL = {0}x{0}x...>{{0} xH*xH* x...xXH ,

k m-k imL = H* xXX x...xXH x{0{x{0}x...x{0} .

k m-k

Hence, condition (14) is satisfied. Construct the set

K2 + ... +

M = {x e A : <ak+1v(k+1)s , x(k+1)s ) + <b(k+1)1v1 + P(k+1)2v2 +b(k+1)mvm , X+1) + . + <«mv ms , xms ) + <bm1v1 + Pm 2 v2 + ••• + + PmmVm Xm ) = <u(k+1), Xk+1) + " • + <um Xm )} .

Condition (15) becomes

(0,.,0,uk+1,...,um) is independent of te (0,T) (26)

Then Theorem 1 and Lemmas 2 and 3 imply the following theorem.

Theorem 4. Suppose that aie M + for i = 1,m, Kie M + for i = 1,k and conditions (25) and (26) are satisfied. Then the set M is a simple Banach C-manifold modeled on the subspace coim L. Construct the spaces

X ={x = (v1,v2,...,vk,vk+1,...,vm): v, e L^(0,T;H) n¿4(0,7;H)),

dvLe ¿2(0,7;H),i = 1,...,k; v, e L^(0,T;H) n^(0,T;H)), dt

U ={u = (u1,u2,...,um): ui e L4 (0,T;L4(G)),i = 1,.,k;

3

ut e L2(0,T;H*),i = k +1,...,m}. By analogy with Section 2, construct an orthonormal system {^1, ^2,..., where are eigenvectors of A, which in view of the embedding (13) constitutes a basis for the space H. Construct Galerkin approximations to the solution to problem (2), (3), (24) as

vn (s, t ) = Ial (t )j (s), i = 1,..., m,

i=1

where the coefficients a i for i = 1,m and l = 1,« are determined by the system

v -ml +№2 +...+Amv"m + k(vn )3, j > = {uxj >, (v"2t - WL + bV + fe" + ... + p2mV"m +k V )3 , j > = U, j >,

<v" - akv"kss + bv" + fik2V2n +... + pkmvnm +кк (v")3 , j > = (uk, j >,

(-ak+lV"k+1W + Ь(к+1)1"V" + Р(к+1)2 V" + ... + Ь(к+l)mVm j > = <U(k+1), j X

<-«mV

"

m ' mss

+ Pm1Vn +^2 + ... + Pmmv"m j > = j ),

i = 1, n,

and the Showalter-Sidorov conditions

<vi (0) - voi, j > = 0,..., {vk (0) - v0k, j > = 0, i = 1,n. Then Theorem 2 and Lemmas 2 and 3 imply the following theorem.

Theorem 5. Suppose that aie R + for i = 1,m , Kie R + for i = 1,k and conditions (25) are satisfied. Given x0 e A and ueU, there exists a unique solution xe X to problem (2)-(4), (24).

Choose a nonempty closed convex set Uad c U. Consider the optimal control problem

J (x, u) ® inf

by solutions to problem (2)-(4), (25), where the objective functional is defined as

(27)

L2(G)

к T

J(x,u) = |v, -zd i=10

к T m T

+(1 -b)Zj| Ui| IL2(G)dt+(1 -b) zill

и

+b zi| V - zd

i=к+1o

L4(G)

f +

i=10

bî (0,1),

i=к+1o

m "L4 (G) 3

Then Theorem 3 and Lemmas 2 and 3 imply the following Theorem.

Theorem 6. Suppose that akî R + for i = 1,m and conditions (26) are satisfied. Then for every x0e A problem (2)-(4), (24), (27) admits optimal control.

The author is grateful to Professors N.A. Manakova and G.A. Sviridyuk for setting the problem and productive discussions.

References

1. Prigogine I., Lefever R. Symmetry breaking instabilities in dissipative systems II. The Journal of Chemical Physics, 1968, Vol. 48, Iss. 4, pp. 1695-1700. DOI: 10.1063/1.1668896

2. FitzHugh R. Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophysical Journal, 1961, Vol. 1, no. 6, pp. 445-466. DOI: 10.1016/s0006-3495(61)86902-6

3. Nagumo J., Arimoto S., Yoshizawa S. An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 1962, Vol. 50, Iss. 10, pp. 2061-2070. DOI: 10.1109/JRPR0C. 1962.288235

4. Bokareva T.A, Sviridiuk G.A. Whitney assemblies of phase spaces of certain semilinear equations of Sobolev type. Mathematical Notes, 1994, Vol. 55, no. 3, pp. 237-242. DOI: 10.1007/BF02110776

5. Gubernov V.V., Kolobov A.V., Polezhaev A.A., Sidhu H.S., McIntosh A.C., Brindley J. Stabilization of combustion wave through the competitive endothermic reaction. Proceeding of the Royal Society A, 2015, Vol. 471, no. 2180, pp. 20150293. DOI: 10.1098/rspa.2015.0293

6. Savchik J., Chang Br., Rabitz H. Application of moments to the general linear multicomponent reaction-diffusion equation. Journal of Physical Chemistry, 1983, Vol. 87, no. 11, pp. 1990-1997. DOI: 10.1021/j 100234a031

7. Zagrebina S.A. A Multipoint Initial-Final Value Problem for a Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, Vol. 7, no. 3, pp. 5-22. DOI: 10.14529/mmp140301

4

8. Sviridyuk G.A., Manakova N.A. The Phase Space of the Cauchy-Dirichlet Problem for the Oskolkov Equation of Nonlinear Filtration. Russian Mathematics, 2003, Vol. 47, no. 9, pp. 33-38.

9. Sviridyuk G.A., Efremov A.A. Optimal Control of Sobolev Type Linear Equations with Relativity p-Sectorial Operators. Differential Equations, 1995, Vol. 31, no. 11, pp. 1882-1890.

10. Zamyshlyaeva A.A., Tsyplenkova O.N., Bychkov E.V. Optimal Control of Solutions to the Initial-Final for the Sobolev Type Equation of Higher Order. Journal of Computational and Engineering Mathematics, 2016, Vol. 3, no. 2, pp. 57-67. DOI: 10.14529/jcem1602007

11. Sviridyuk G.A., Manakova N.A. An optimal control problem for the Hoff equation. Journal of Applied and Industrial Mathematics, 2007, Vol. 1, no. 2, pp. 247-253.

12. Manakova N.A., Gavrilova O.V. Optimal Control for a Mathematical Model of Nerve Impulse Spreading. Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming & Computer Software, 2015, Vol. 8, no. 4, pp. 120-126. DOI: 10.14529/mmp150411

13. Al'shin A.B., Korpusov M.O., Sveshnikov A.G. Blow-up in nonlinear sobolev-type equations. Berlin, N.Y., Walter de Gruyter GmbH and Co. KG, 2011, 648 p. DOI: 10.1515/9783110255294

14. Lions J.-L. Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris, Dunod, 1969, 554 p. (in French).

15. Bayazitova A.A. The Sturm-Liouville problem on geometric graph. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2010, no. 16 (192), Issue 5, pp. 4-10. (in Russ.).

16. Hassard D.D., Kazarinoff N.D., Wan Y.-H. Theory and Applications of Hopf Bifurcation. Cambridge, New York, Cambridge University Press, 1981, 311 p.

17. Sviridyuk G.A. O razreshimosti singulyarnoy sistemy obyknovennykh differentsial'nykh uravneniy (On the Solvability of Singular Systems of Ordinary Differential Equations). Differentsial'nye Uravneniya, 1987, Vol. 23, no. 9, pp. 1637-1639. (in Russ.).

Received December 26, 2019

Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2020, vol. 12, no. 1, pp. 14-23

УДК 517.9 DOI: 10.14529/mmph200102

ОПТИМАЛЬНОЕ УПРАВЛЕНИЕ РЕШЕНИЯМИ МНОГОКОМПОНЕНТНОЙ МОДЕЛИ РЕАКЦИИ-ДИФФУЗИИ В ТРУБЧАТОМ РЕАКТОРЕ

О.В. Гаврилова

Южно-Уральский государственный университет, г. Челябинск, Российская Федерация E-mail: [email protected]

Статья посвящена изучению математической модели реакции-диффузии в трубчатом реакторе на основе вырожденных уравнений типа реакции-диффузии, заданных на геометрическом графе. Исследуется именно вырожденный случай, так как при построении математической модели учитывается, что скорость одной искомой функции значительно превышает скорость другой. Изучаемая модель относится к широкому классу полулинейных моделей соболевского типа. Приводятся достаточные условия простоты фазового многообразия абстрактного уравнения соболевского типа в случае s-монотонного и р-коэрцитивного оператора; доказываются существование и единственность решения задачи Шоуолтера-Сидорова в слабом обобщенном смысле и существование оптимального управления слабыми обобщенными решениями рассматриваемой задачи. На основе абстрактной теории найдены достаточные условия существования оптимального управления для математической модели передачи импульса по нейронам.

Ключевые слова: уравнения соболевского типа; фазовое многообразие; задача Шоуолтера-Сидорова; система уравнений реакция-диффузия; задача оптимального управления.

Литература

1. Prigogine, I. Symmetry breaking instabilities in dissipative systems II / I. Prigogine, R. Lefever // The Journal of Chemical Physics. - 1968. - Vol. 48, Iss. 4. - P. 1695-1700.

2. FitzHugh, R. Impulses and physiological states in theoretical models of nerve membrane / R. FitzHugh // Biophysical Journal. - 1961. - Vol. 1, no. 6. - P. 445-466.

3. Nagumo, J. An active pulse transmission line simulating nerve axon / J. Nagumo, S. Arimoto, S. Yoshizawa // Proceedings of the IRE. - 1962. - Vol. 50, Iss. 10. - P. 2061-2070.

4. Бокарева, Т.А. Сборки Уитни фазовых пространств некоторых полулинейных уравнений типа Соболева / Т.А. Бокарева, Г.А. Свиридюк // Математические заметки. - 1994. - Т. 55, № 3. -С.3-10.

5. Stabilization of combustion wave through the competitive endothermic reaction / V.V. Gubernov, A.V. Kolobov, A.A. Polezhaev et al. // Proceeding of the Royal Society A. - 2015. - Vol. 471, no. 2180. - P.20150293.

6. Savchik, J. Application of moments to the general linear multicomponent reaction-diffusion equation / J. Savchik, Br. Chang, H. Rabitz // Journal of Physical Chemistry. - 1983. - Vol. 87, no. 11. -P.1990-1997.

7. Zagrebina, S.A. A Multipoint Initial-Final Value Problem for a Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid / S.A. Zagrebina // Вестник ЮУрГУ. Серия «Математическое моделирование и программирование». - 2014. - Т. 7, № 3. - P. 5-22.

8. Свиридюк, Г.А. Фазовое пространство задачи Коши - Дирихле для уравнения Осколкова нелинейной фильтрации / Г.А. Свиридюк, Н.А. Манакова // Известия вузов. Серия: Математика. -2003. - № 9. - С. 36-41.

9. Свиридюк, Г.А. Оптимальное управление линейными уравнениями типа Соболева с относительно р-секториальными операторами / Г.А. Свиридюк, А.А. Ефремов // Дифференциальные уравнения. - 1995. - Т. 31, № 11. - С. 1912-1919.

10. Zamyshlyaeva, A.A. Optimal control of solutions to the initial-final for the Sobolev type equation of higher order / A.A. Zamyshlyaeva, O.N. Tsyplenkova, E.V. Bychkov // Journal of Computational and Engineering Mathematics. - 2016. - Vol. 3, no. 2. - P. 57-67.

11. Свиридюк, Г.А. Задача оптимального управления для уравнения Хоффа / Г.А. Свиридюк, Н.А. Манакова, // Сибирский журнал индустриальной математики. - 2005. - Т. 8, № 2. - С. 144151.

12. Манакова, Н.А. Оптимальное управление для одной математической модели распростренения нервного импульса / Н.А. Манакова, О.В. Гаврилова // Вестник ЮУрГУ. Серия «Математическое моделирование и программирование». - 2015. - Т. 8, № 4. - C. 120-126.

13. Al'shin, A.B. Blow-up in Nonlinear Sobolev-type Equations / A.B. Al'shin, M.O. Korpusov, A G. Sveshnikov. - Berlin; N.Y.: Walter de Gruyter GmbH and Co. KG, 2011. - 648 p.

14. Лионс, Ж.-Л. Некоторые методы решения нелинейных краевых задач / Ж.-Л. Лионс. -М.: Мир, 1972. - 588 p.

15. Баязитова, А.А. Задача Штурма-Лиувилля на геометрическом графе / А.А. Баязитова // Вестник ЮУрГУ. Серия «Математическое моделирование и программирование». - 2010. -Вып. 5. - № 16 (192). - С. 4-10.

16. Хэссард, Б. Теория и приложения бифуркации рождения цикла / Б. Хэссард, Н. Казаринов, И. Вэн. - М.: Мир, 1985. - 280 с.

17. Свиридюк, Г.А. О разрешимости сингулярной системы обыкновенных дифференциальных уравнений / Г.А. Свиридюк // Дифференциальные уравнения. - 1987. - Т. 23, № 9. - С. 1637-1639.

Поступила в редакцию 26 декабря 2019 г.