Optimal control over solutions of a multicomponent model of reaction-diffusion in a tubular reactor

Take a finite connected oriented graph G = G( V; E) with vertex set V = {Vi} M 1 and edge set E={Ej} K= 1, 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 v1 jt = «1 v1 jss + /1 j (v1 j,v 2 j ,-, vmj ) + u1 j, v2 jt = a2 v2 jss + f2 j (v1 j, v2 j, -, vmj ) + u 2 j, vkjt = «kvkjss + fjj (v1 j,v 2 j,-, vmj ) + ukj, 0 = ak+1 v(k+1) jss + f(k+1) j ( v1 j, v2 j,—, vmj ) + u (k+1) j ’

0  amvmjss + fmj (v1 j,v2 j,—, vmj ) + umj for all s e (0, lj), t e R, j = 1, K, with positive parameters ai, i = 1 ,m and some functions/je C”(Rm,R) for i = 1 ,m and j = 1 ,K . Here the functions vi = vi(s, t), i = 1 ,k and vi = vi(s, t), i = k +1 ,m characterize the concentrations of reagents (activator and inhibitor); αi, i = 1,m are the diffusion coefficients; the functions fi correspond to the interaction between the reagents; the prescribed functions ui = ui(s, t) characterize exterior actions. For (1) at each vertex Vi for i = 1,m impose flow balance and continuity conditions

^    djvjs (0, t) -     ^    drvrs (lr, t ) = 0,(2)

• j : Ej e E « (V i)             r : Er e E ® ( V )

vr(0,t)=vj(0,t)=vh(lh,t)=vn(ln,t)(3)

for all Er, Eje E( Vi) and Eh, Ene E“(Vi). Here E

Initially, a nondegenerate system of equations of the reaction-diffusion type εv_t = α∆v + f₁(v,w) + u₁, w_t = β∆w + f₂ (v, w) + u₂.

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 f1(v,w) =γ- (δ+1)v +v2w, f2(v,w) = δv-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)=β1w-κ1v, f2(v, w) =β2w -κ2v - 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

0 = α∆v+ f₁(v, w) +u₁, w_t = β∆w + f₂ (v, w) + u₂

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

Lx+M(x) = u, 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,u) → 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 u∈U_ad, where U_ad 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 β2 ≤ 0 and β1 = κ2.

Математика

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 p-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⊂H ⊂A^*                                  (13)

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

H⊃kerL ≡ coker L⊂H^*, A = kerL⊕coim L, A^* = coker L⊕imL.           (14)

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

Take an s-monotone and p-coercive operator M∈ Cr(A; A*) with r ≥ 1 (thapt- 1 is, [M'yx,x]>0 ∀x,y∈ A\{0} and ∃ CM, CM ∈ ℝ+ such that [M(x), x] ≥ CM x p and M(x) *≤CM x , where p ≥ 2) possessing symmetric Fréchet 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 co- ercive.

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

(I -Q)u is independent of t∈(0,T).                              (15)

Denote by P the projection along ker L onto coim L. Given a point x₀∈M, put x ¹₀ = Px₀∈coim L.

Definition 1. [8] Call a set M a Banach C^r-manifold at x0 ∈M whenever there exist neighborhoods O⊂M and O1 ⊂coim L of the points x0 and x¹₀ = Px0 respectively and a C^r-diffeomorphism D: O1→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 M is called a Banach C^r-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 C^r-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 t∈[0,T] 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 {φ_i} which is complete in A. Construct Galerkin approximations to the solution to (9), (10) as

n xn (s, t ) = Ea^ t ^ s),                               (17)

i=1

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

[ Lxt ,Ф^ + [ M (xn U>J = [ u ,^t ],                             (18)

[ L (xn (0) - x о),ф-] = 0, i = 1,..., n,                                    (19)

Lxⁿ (0) → Lx₀ for n → +∞ strongly in the subspace im L.

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 Tn∈ ℝ+, Tn = Tn(x0), and An = span{φ1,φ2,…,φn}.

Lemma 1. [17] Let M be s-monotone and p-coercive operator. For every x0∈A there exists a unique local solution xⁿ∈C^r(0, T_n; 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 | x∈L_∞(0,T;coimL) ∩ L_p(0,T;A), xɺ¹∈L₂(0,T; coim L)}.

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

T

T

t d                        t

Jф(t) —Lx + M(x),w dt = Jф(t)[u,w]dt,Vwe A,Vфe L2(0,T).

0 L^dt               J 0

Call a solution to (9) a solution to the Showalter–Sidorov problem whenever it satisfies (10).

Theorem 2. [11] Let M be s-monotone and p-coercive operator. For every x₀∈A, T∈ℝ₊, u∈L₂(0, T;

A^*) there exists a unique solution x∈X 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 ⊂ H is com- pact. Construct the space U = Lq(0, T; A ), + = 1 and define in it a nonempty closed convex set Uad. pq

Consider the optimal control problem

J(x, u) → inf, u∈Uad                                    (22)

defining the objective functional as TT

J(x,u) = eJi|x(t) - z_d (t)|A dt + (1 - в)J|U(t)|A_z dt, ве (0,1),                   (23)

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

Definition 3. [11] Refer to a pair (xɶ,uɶ)∈X×Uad as a solution to the optimal control problem (9), (10), (22) if

J(xɶ,uɶ) = inf J(x, u),

(x,u)

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

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

J(x,u) < +∞.

If Uad ≠ ∅then for every u∈Uad⊂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 x0∈A and T∈ℝ₊, 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

Математика

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,h〉= ∑ d_j∫g_j(s)h_j(s)ds.

Ej e E °

Construct the Banach space

H = {g = (g 1, g2, -, gj, -, gK): gj e W2 (°, lj) and conditions (3) holds} with the norm lj

II g|H = E dj j⁽g 2s⁽s)⁺gj²⁽s)) ds.

Eje E °

Put

Lp(G) = {g = (g 1,g2,-,gj,-,gK): gj e Lp(°,lj)}.

The set Lp(G) is a Banach space with the norm lj g Lp (G)= ∑ dj∫ |gj(s)|p ds. p     ej∈E  °

By the Sobolev embedding theorem, the space W12 (0,lj) 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) = E dj j⁽gjs⁽s) hjs⁽s)⁺agj⁽s)hj⁽s))^ds, g, h ^{e H}.

EjeE °

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

Consider the Hilbert space

H = L^(G) = {v = (vi,v2,-,Vm):vie L2(G)} equipped with the inner product m [ v^,z ]=E< vi^,z i=1

and identified with its adjoint. By analogy, construct the space A = H^m and denote by A^* the adjoint to A with respect to the inner product in H. Writing x = (v1, v2,…,vm), ζ = (ζ1, ζ2,…, ζm), and u = (u1, u2,…,um), define the operators

[Lx,Z] = <v^) + - + <v_k,n_k>, x,ze A

[ M (x ),Z] = ai<V1 s ,Z1 s > + < /1( x ),Z1> + «2 < V 2 s ,Z2 s > + < f2( X), Z2 > + -

^+am <^Vms, ^Zms>⁺< fm^(x^Cm >, ^{x,Z e}A.

Lemma 2. (i) The operator Le L(A; A ) is selfadjoint and nonnegative definite.

• (ii)    Suppose thatfje C”(R^m, K) for i = 1 ,m andj = 1 ,K . Then Me C(A; A*).

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

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).

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) if f_i take as (6). Proceed to a mathematical model of neural signal transmission based on the FitzHugh–Nagumo system

• ^v1 jt ^{- a}1 ^v1 jss⁺Д1^v1 j⁺P12v2j⁺...⁺P1 mvmj^{+ K}1 v³j = ^u1 j,

v2 jt - a2v2 jss + P2\ v1 j + P22v2 j + ...+ P2mvmj + K2v2 j = u2 j ’ vkjt -«kvkjss + Pk 1 v1 j + Pk2v2j +... + Pkmvmj + Kkvkj = ukj,

-ak+1 v (k+1) jss + P( k+1)1 v1 j + P( k+1)2 v 2 j + ". + P( k+1) mvmj = u (k+1) j, amvmjss + Pm1 v1 j + Pm 2 v 2 j + ". + Pmmvmj umj 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 = {вч} mj=1 has the property

3CB,C_B > 0: C_B [x,x] < [Bx,x] < CB [x,x].                           (25)

By analogy with Section 2, consider the Hilbert space H=(L₂^m (G), [·,·]) and the Banach space A = H^m. By the Sobolev embedding theorem, there are dense continuous embeddings (13); furthermore, the embedding A c H is compact. Write x = (v1, v₂,...,vm), Z = (Zi, Z2,—, Zm), and u = (u 1, u2,^, um)• Then the operator M = M1 + M2 becomes

^[M 1⁽x), ^Z]= ^a1(^v1 s, ^Z1 s) + <Рц ^v1 + 312v2 ⁺— ⁺P1 mvm, ft) ^{+ a}2(^v2s, ^Z2s^{} +}

⁺<^в21 ^v1 ⁺P22^v 2 ⁺— ⁺P2 mvm , ^Z2 ^{) +}— ^{+ a}m< vms ,Cms > ^{+ +}^Pm1 ^v1 ⁺Pm2^v2 ^{+ —}■ ⁺Pmmvm, ^Zm⁾, ^{x,Z e}A,

^[M 2 ^(x), ^{Z ]}= ^K1 < vX^y}^{+ K}2 < v 2, ^Z2 > ⁺— ^{+ K}k < vk, ^Z1 >, x^{,Z e}A, where v_m³ = (v_m³₁,v_m³₂,…,v_m³_k) .

Lemma 3. (i) Suppose that aie R ₊for i = 1 ,m and condition (25) is satisfied. Then the operator M1e C”(A; A*) is s-monotone and 2-coercive.

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

coercive.

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

[M\^, Z] = «1<^1 s, Z1 s) +

11^1 + P12^2 + — + P1 m^m, Z1) +

+«2 <^2 s ,Z2 s ) +

21Z1 + P22Z2 + — + P2 mZm, Z ) + — +

^+am ms, ^Zms^{} +}

m 1Z1⁺Pm2^2⁺ • ^{— +}PmmZm, ^Zm ⁾, ^{x,Z e}A,

[m2xZ,Z] = 3K1<V2Z1,Z1) + 3K2<v 1^2,Z2) + — + 3Kk<v2Zk,Zk),x,Ze A.

Then the continuous embedding W1 (G) c L₂(G) yields

1 [ M 1xz,Z]k cizAiziia .

1 [ M2 xZ,Z]k 3CI zAw 1A1 zA,

I [м2,tiZ₂),z]<6C|Z1IIAIZ2IlAlxlAIIZIA -I[М2,(ZvZ2,Z3).Z]I<⁶C|Z,IIAIZ21IAIZ31IAIZIA ,C = maxK,

Математика

where M'1x and M'2x stand for the Fréchet derivatives of M1 and M2 at x. Since M₁⁽²_x⁾ ≡O and M⁽₂⁴_x⁾ ≡O, the operators M1 and M2 are C^∞-smooth. Since

[ M ^] = a1<£s ,£ s ) + <^1 + в <2 + • + A m^m ,£> + ^+a2 <^2 s ,£ s > ⁺<^21^1⁺^22^2 ⁺• •^{+ в}m^m ,£ > ⁺• • ^{+ +a}m <^ms ^ms > ⁺<РпЛ⁺Pm 2^2⁺• • •⁺Pmm^m ,^m > > 0, ^x,^^eA, [M^,£1 = 3^<v²^) + 3K2<v&,£) + + 3^<v^,&) >0, x,^e A, it follows that the operators M1 and M2 are s -monotone. Since

C1||x| A<[M 1(x), x]< C21|x| A,

^[M2 ^(x)^x]= ^K1 Iv11144 ^{(G) +K}2 Iv21144 ^{(G) +}• ^{+ K}k IIvk 1144 ^(G), 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} xHxHx. ..xH, km coimL = HxHx„,xH x{0}x{0}x„.x{0},

k cokerL = {0}x{0}x...x{0} xH*xH*x^xH*

km imL = H*xH*x^xH* x{0}x{0}x^x{0}.

k

Hence, condition (14) is satisfied.

Construct the set

M = {^{x e}A^:<«k+1 v(k+1)s, ^(k+1)s > ⁺<^в(k+1)1 ^v1 ^{+ в}(k+1)2v2 ⁺• • • ⁺

^+в(k+1)m^vm , ^k+1>⁺• ⁺<«mvms, ^ms ) ⁺

m1 ^v1 ⁺Pm 2^v 2 ⁺• ■ ⁺

⁺Pmmvm, ^m ^ <u(k+1), ^k+1 ^ ⁺• ⁺<um, ^m ^^}.

Condition (15) becomes

(0, — ,0,u_k+i,—,u_m) is independent of te (0,T)                         (26)

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

Theorem 4. Suppose that aie R ₊for i = 1 ,m , Kie R ₊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,...,v_k,v_k+1,...,v_m): vi e L_(0,T;H) nL₄(0,T;H)), dvi- e L2(0, T; H), i = 1,..., k; v e L_TC(0, T; H) n L2(0, T; H)), dt

U = {u = (U1,u2,...,um): u e L4(0,T;L4(G)),i = 1,-,k;

ui e L₂(0,T;H*),i = k +1,...,m}.

By analogy with Section 2, construct an orthonormal system {φ1, φ2,…, φi}, where {φi} 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 ) = j^«' (t )>qii (s), i = 1,^, m, l=1

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

<Vn - aivnss + в11 vn + в12v2 + ... + в1 mV"m + *i(Vn )3,Ф,->= <«1,ф-), < V nt — a₂V 2s + в21 Vn + в22 V 2 + ... + в2 mVm + K2( V 2 )3,Ф,->= <« 2,Ф>,

...

< < Vn^{- a}kVⁿ_kss + ^в 1 Vn + ^в 2 Vn + ... + ^e_kmVⁿm + ^Kk⁽Vn ^)ф = < u_k ф\

<^-ak+1^V(k+1) ss^{+ в}( k+1)1 ^V1 ^{+ в}( k+1)2 ^V 2 ⁺ "^{. + в}( k+1) m^Vm ,^‘) = < u (k+1),Ф1),

...

<^-amVⁿmss^{+ в}т 1 Vn^{+ в}т 2 V n⁺...^{+ в}тт^"т ,Фг > = < «m ,Фг >,

_                                     = 1,П, and the Showalter–Sidorov conditions

< V1 (0) — V 01, Ф > = 0,..., < Vk (0) - v0 k Ф = 0, i = 1, n.

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

Theorem 5. Suppose that α∈ℝ₊for = 1,m , κ∈ℝ₊for = 1,k and conditions (25) are satisfied.

Given x0∈A and u∈U, there exists a unique solution x∈X to problem (2)–(4), (24).

Choose a nonempty closed convex set Uad ⊂ U. Consider the optimal control problem J(x,u) →inf(27)

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

J(X, u)-e'V) V„-:f^ (G) dt + e ^ J |V,-zd^

=1 0           2            =k +10

kTmT

• ^{+(1 - e)}EjlluilL₂(G) dt^{+ (1}"^в)Z JIlumllL(G) dt,^ве(0,¹), =1 0                         =k +10

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

Theorem 6. Suppose that α , κ∈ℝ₊for = 1,m and conditions (26) are satisfied. Then for every x0 ∈A problem (2)–(4), (24), (27) admits optimal control.

The author s grateful to Professors N.A. Manakova and G.A. Sv r dyuk for sett ng the problem and product ve d scuss ons.