ANALYSIS OF THE BOUNDARY VALUE AND CONTROL PROBLEMSFOR NONLINEAR REACTION-DIFFUSION-CONVECTION EQUATION Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-4-452-462 УДК 517.9

Analysis of the Boundary Value and Control Problems for Nonlinear Reaction—Diffusion—Convection Equation

Gennady V. Alekseev* Roman V. Brizitskii^

Institute of Applied Mathematics FEB RAS Vladivostok, Russian Federation

Received 10.03.2021, received in revised form 05.04.2021, accepted 20.05.2021 Abstract. The global solvability of the inhomogeneous mixed boundary value problem and control problems for the reaction-diffusion-convection equation are proved in the case when the reaction coefficient nonlinearly depends on the concentration. The maximum and minimum principles are established for the solution of the boundary value problem. The optimality systems are derived and the local stability estimates of optimal solutions are established for control problems with specific reaction coefficients.

Keywords: nonlinear reaction-diffusion-convection equation, mixed boundary conditions, maximum principle, control problems, optimality systems, local stability estimates.

Citation: G.V. Alekseev, R.V. Brizitskii, Analysis of the Boundary Value and Control Problems for Nonlinear Reaction-Diffusion-Convection Equation, J. Sib. Fed. Univ. Math. Phys., 2021, 14(4), 452-462. DOI: 10.17516/1997-1397-2021-14-4-452-462.

1. Introduction. Solvability of the boundary value problem

In recent years, there has been an increasing interest in the study of inverse and control problems for models of heat and mass transfer, electromagnetism and acoustics. A number of papers are devoted to the theoretical analysis of these problems, of which we note [1-16]. In these papers, the solvability of boundary value problems, inverse and extremum problems for the specified models was proved, and the questions of uniqueness and stability of their solutions were studied. Related problems for models of complex heat transfer were studied in [17,18].

This paper which continues a series of papers by the authors [10-14] is devoted to the theoretical analysis of the boundary value and control problems for the nonlinear reaction-diffusion-convection equation, considered under inhomogeneous mixed boundary conditions on the boundary of the domain.

In bounded domain Q c R3 with boundary r, consisting of two parts r^ and , the following boundary value problem for nonlinear reaction-diffusion-convection equation is considered:

-div(A(x)Vp) + u -Vp + k(p, x)p = f in Q, (1.1)

p = ^ on r^, A(x)(dp/dn + a(x)p) = x on • (1.2)

Here the function p means the concentration of the substance, u is a given vector of velocity, f is a volume density of external sources of substance, A(x) is a diffusion coefficient, function k(p, x) is a reaction coefficient, x £ Q. Below we will refer to the problem (1.1), (1.2) for the given functions A, k, f, ^, a and x as Problem 1.

* alekseev@iam.dvo.ru t mlnwizard@mail.ru © Siberian Federal University. All rights reserved

In this paper, we first prove the global solvability of Problem 1 and the nonlocal uniqueness of its solution in the case, when the reaction coefficient k(p, x) is sufficiently arbitrarily depends on both the concentration p and the spatial variable x, and the nonlinearity k(p, x)p is monotone. Under additional conditions on the functions A, f, x, a, ^ and the reaction coefficient k the minimum and maximum principles are established for the concentration p. Further, a control problem is formulated, in which the role of controls is played by the diffusion coefficient A, the volume density of external sources of substance f and the density of boundary sources x and its solvability is proved. For the mentioned problems, with specific reaction coefficients, an opti-mality system is derived and, based on its analysis a theorem on the local stability estimates of optimal solutions is formulated. This theorem can be proved according to the scheme described in detail in [11-16].

When analyzing the problems under study, we will use the Sobolev functional spaces Hs(D), s e R. Here D means either a domain Q, or some subset Q C Q, or part rD of the boundary r. By || • ||s,q, I • |s,Q and (■, -)s,Q we will denote the norm, seminorm and scalar product in Hs(Q). The norms and scalar products in L2(Q), L2(Q) or in L2(rN) will be denoted by || ■ ||q and (■, ^)q, || ■ ||q and (■, ■) or || ■ ||rN and (■, OrN, respectively. Let L+(D) = {k e Lp(D) : k > 0}, p > 3/2, Z = {v e L4(Q)3 : div v = 0 in Q, v ■ n|rN = 0}, H{o (Q) = {h e Hs(Q) : h > A0 > 0 in Q}, s > 3/2, T = {p e H1 (Q) : p|r^ = 0}. Here and below p|ro denotes the trace of a function p e H 1(Q) on the part r0 of the boundary r. For any function p e T the Friedrichs-Poincare inequality ||Vp||Q > ^o||p||2 Q holds, where positive constant S0 does not depend on p.

Let the following conditions hold:

(i) Q is a bounded domain in R3 with boundary r e C0>1, consisting of closures of two non-intersecting open parts rD and rN (r = rD U rN, rD n rN = 0), and meas r^ > 0;

(ii) A e H{0(Q), S > 3/2, f e l2(Q), x e L2(rw);

(iii) u e Z, ^ e H 1/2(rD), a e L2+(rN).

(iv) The function k : R x Q ^ R is nonnegative. In addition, for any function v e H 1(Q) the embedding k(v, ■) e L+(Q) holds for some p > 3/2, independent of v, and on any ball Br = {v e H 1(Q) : ||v|1,n ^ r} of radius r the following inequality holds:

||k(v1, ■) - k(v2, ■)|LP(n) < Lr 11 v 1 - v2 ||i4 (Q) Vv1,v2 e Br. (1.3)

Here the constant Lr depends on r but does not depend on v1, v2 e Br;

(v) (k(p1, ^)p1 - k(p2, ^)p2,p1 - p2) > 0 for all p1, p2 e H 1(Q);

(vi) Hk(p, -)||lp(q) < Q + B for all p e H 1(Q), where number p is defined in (iii), r > 0 is a fixed number, A and B are nonnegative constants.

Let us note that the condition (iv) describes an operator acting from H 1(Q) to Lp(Q), p > 3/2, allowing to take into account the rather arbitrary dependence of the reaction coefficient k on both the concentration p and the spatial variable x. Condition (v) means that the nonlinearity k(p, ^)p is monotone [19, p. 182], and condition (vi) restricts the growth in p of the reaction coefficient by a power function with exponent r.

The specified conditions will provide a proof of the solvability of Problem 1 considered under the inhomogeneous Dirichlet condition on the part rD of the boundary r. As an example of the function k(p, ■) satisfying (iv)-(vi) we give the function k : R x Q ^ R, such that k(p, x) = p2 for x e Q where Q is a subdomain of domain Q, k(p, x) = k0(x) e L+/2(Q \ Q) for x e Q \ Q.

Let us also remind that, by the Sobolev embedding theorem, the space H 1(Q) is embedded into the space Ls (Q) continuously at s < 6 and compactly at s < 6 and, with a certain constant Cs, depending on s and Q, we have the estimate

||p|U-(n) < Cs||p||1,Q Vp e H 1(Q). (1.4)

The following technical lemma holds (see details in [7]).

Lemma 1.1. Let, in addition to condition (i)-(iii), u € Z, ki(-) € L+(Q), p ^ 3/2. Then the following relations hold:

\(XVp, VV)\ < 7jAMMMnUi,n V^ € HW,

(1.5)

(XVh, Vh) > X*\\h\\2in Vh €T,X* = SqXq, \(u < Yi\\u\\lw\\p\\i,n\M\i,n V p,n € Hi(Q), (u ■Vh,h) = 0 V h €T, (1.6)

\(x,p)fn\ < yMvn\\p\\i,n Vx € L2(rN), p € H i(Q), (1.7)

\(Xap,V)rN \ < 73\Ays,nyayrN\\p\\i,n\M\i,n Vp,V € Hi(Q), (1.8)

\(kiy,n)\ < YP\\ki\\Lp(n)Mi,Mi,n Vp,v € Hi(Q). (1.9)

Here X* = ó0X0, constants Yi and 72 depend on Q, constants Ys and yS depend on Q and s, yp depends on Q and p.

Let us multiply the equation (1.1) by h € T and integrate over Q using Green's formulae. Taking into account (1.2), we obtain

(XVp, Vh) + (k(p, h) + (uVp, h) + (Xap, h)rN = (f, h) + (x, h)rN Vh € T, p\rD = (1.10)

Definition 1.1. The function p € Hi(Q), which satisfies (1.10), will be called a weak solution of Problem 1.

To prove the solvability of Problem 1, we need the following lemma [12].

Lemma 1.2. Let condition (i) holds. Then for any function ^ € H i/2(r d ) there exists a function p0 € Hi(Q), such that p0 = ^ on rD and with some constant Cr, depending on Q and rD, the estimate \\po\iin ^ Cr^W^^r^ holds.

We represent the solution to Problem 1 as the sum p = p + p0 where p0 is a given function from Lemma 1.2 and p €T is unknown function. Substituting p = p + p0 in (1.10) we will have

(XVp, Vh) + (k(p + po, ■)(p + po), h) + (u ■ Vp, h) + (Xap, h)r n =

= (f, h) + (x, h)rN - (XVpo, Vh) - (u ■ Vp0, h) - (Xapo, hK V h €T. (1.11)

Adding the term -(k(p0, ■)p0, h) to both parts of (1.11), we obtain

(XVp, Vh) + (k(p + po, ■)(p + po) - k(po, ■)po, h) + (u ■ Vp, h) + (Xap, h)rN =

(l, h) = (f, h) + (x, h)rN-(XVpo, Vh)-(uVpo, h)-(k(po, ■)p0, h)-(Xapo, hK Vh€T. (1.12)

Using the Holder inequality, Lemmas 1.1, 1.2, estimate (1.4) and condition (vi), it is easy to show that l € T* and, moreover, the following estimate holds:

\\l\\T. < M = \\f\h + Y2\\x\\tn + Cr(7s\\X\\s,n + Yi\\u\\LW)\\^\\i/2,YD +

+cr[YP(ACT\m Ti/2,TD + B) + Y3\\X\\s,n\\a\\rN]\\^\\i/2,rD . (1.13)

Let us introduce the nonlinear operator A : T ^ T* by

(A(p), h) = (XVp, Vh) + (k(p + p0, ■)(p + po) - k(po, ■)po, h) + (u ■ Vp, h)+

+(Xap,h)TN Vp, h € T. (1.14)

It is clear that the problem (1.12) is equivalent to the operator equation A(p) = l. According to [19, p. 182], to prove the existence of a solution p e T of problem (1.12) it suffices to show that: 1) the operator A is monotone on T, that is (A(u)-A(v),u-v)^0 for all u,v e T; 2) the operator A : T^T* is continuous and bounded; 3) the operator A is coercive on T.

To prove the monotonicity of the operator A we subtract the relation (1.14) for p = p2 from (1.14) for p = p1 where p1 and p2 e T are arbitrary elements. We obtain

(A(p1) - A(p2), h) = (AV(p1 - p2), Vh) + (k(p1 + p0, -)(p! + p0) - k(p2 + p0, ■)(p2 + p0), h)+

+(u ■ V(p1 - p2), h) + (Aa(p1 - p2), h)rn Vh eT. (1.15)

For h = p1 - <p2 all terms in the right-hand side of (1.15) are nonnegative due to the properties of the functions A, a, u indicated in (ii), (iii) and monotonicity of nonlinearity k(p)p. Therefore

(A(p1) - A(p2), <p1 - <f2) > 0 Vp1, p2 e T.

To prove the continuity and boundedness of the operator A we rewrite (1.15) in the form

(A(p{) - A(^), h) = (AV(p1 - 922), Vh) + (k(p! + p0, ■) - k(p2 + p0, ■), p + p0, h)+

+ (k(^2 + p0, ■)(<p1 - 22), h) + (u ■ V((p1 - 22), h) + (Aa(pp1 - 22), h)rN Vh e T. (1.16)

Using the estimates of Lemma 1.1, the estimates (1.4), (1.9), and condition (iii), from (1.16) we deduce that

|(A(p1) - A(22),h)I < (7s|A|s,n + YpLC\UpdMHp - p2M|h||1,n+

+(7p|k(p2 + p0, ■)|lp(q) + 71|u|L4(Q)3 + 7l|A|s,Q|a|rN )|| 21 - <p2|Mh||1,n Vh eT. (1.17)

The inequality (1.17) implies the continuity and boundedness of the operator A. Finally, setting h = p in (1.14) and using conditions (ii), (iv), and (1.6), we arrive at the following inequality which implies the coercivity of the operator A:

(A(p), p) = (AVp, Vp) + (k(p + p0, ■)(2 + p0) - k(p0, ■)p0, P) +

+ (Aap, p)rN > A*!P||2,q Vp e T. (1.18)

As a result we conclude that the solution p eT of the problem (1.11) exists and the estimate ||p|1,n ^ C* HIHt*, C* = A;1 takes place. In this case, the function p = p0 + p is the desired weak solution to Problem 1 and the following estimate holds:

||p||1,Q < Mv = C*M + CrM1/2,rD (C* = A;1). (1.19)

Here the constant Mi was defined in (1.13) and Cr is the constant from Lemma 1.2.

Let us show that the solution to Problem 1 is unique. Let p1 and p2 e H 1(Q) be any two solutions to Problem 1. Then their difference = 1 - 2 e T satisfies the identity

(AVp, Vh) + (k(p 1, ■) p1 - k( p2, ■) p2, h) + (u ■ V p, h) + (Aa( p1 - p2), h)rN =0 V h eT.

Setting here h = p, by virtue of conditions (iii), (v) and (1.6) we arrive at the inequality A*||p||1jQ < 0, from which it follows that p1 = p2 in Q. This proves the following theorem.

Theorem 1.1. Let conditions (i)-(vi) hold. Then there exists a unique weak solution p e H 1(Q) of Problem 1 and the estimate (1.19) holds.

Within the framework of the approach of [20] we prove the maximum and minimum principles for a weak solution p to Problem 1. To this end, we assume, in addition to (i)-(vi), that the following conditions are satisfied:

(vii) ^min < ^ < ^max a.e. on rD, fmin < f < fmax and Amin < A < AmaX a.e. in 0, amin < a < «max and Xmin < X ^ Xmax a.e. on rw.

Here ^min, ^max, fmin, fmax, Xmin, Xmax are nonnegative numbers, while «min, «max and Amin, Amax are positive numbers;

Besides, we will assume also that the reaction coefficient k satisfies the following conditions:

(viii) the reaction coefficient k has the form k = ki(p) where ki(-) : R ^ R is a continuous nonnegative function, satisfying conditions (iv)-(vi), in which one should set k(p, ■) = k1(p), and every of functional with respect to M1 and m1 equations

ki(Mi)Mi = fmax and ki(mi)mi = fmin (1.20)

has at least one solution. We set

M = maxj^max, Xmax/Aminamin,Mi}, m = mínj^min, Xmin/Amax«max, mi}. (1.21)

Theorem 1.2. Let conditions (i)-(iii), (vii), (viii) hold. Then for the solution p £ Hi (0) of Problem 1 the following maximum and minimum principle holds:

m < p < M a.e. in 0. (1.22)

Here the constants m and M are defined in (1.21) where Mi is a minimum root of the first equation in (1.20) and mi is a maximum root of the second equation in (1.20).

Proof. Firstly, we prove the validity of the maximum principle in the form of the estimate p ^ M in 0. For this purpose we introduce a nonnegative function v = maxjp — M, 0}. From the definition of v it follows that the estimate p ^ M holds if and only if v = 0 in 0. We denote by 0M C 0 a measurable subset of 0, at the points of which the inequality p > M holds, by rM we denote the measurable subset of the part , at the points of which the condition v\rM > 0 is satisfied. Set 0i = 0 \ 0M, ri = \ rM. From [21, p. 152] and [22] it follows by the definition of the constant M in (1.21) that v £ T, and the following relations hold:

v = p — M > 0 and Vv = Vp in 0M; v =0 and Vv = 0 in 0i; v\ri = 0,

(AVp, Vv) = (AVv, Vv)nM = (AVv, Vv), (u ■ Vp, v) = (u ■ Vp, v)nM = (u ■ Vv, v) = 0.

We set h = v in (1.10) at k(p) = ki(p) and add to both sides of the resulting equality the term — (ki(M)M,v)qm — (AaM,v)rM. Taking into account the properties of v we obtain

(AVv, Vv) + (ki(v + M)(v + M) — ki(M)M,v)qm + (Aav,v)rM =

= (f — ki(M)M, v)qm + (X — AaM, v]pm . (1.23)

From the definition of the constant M in (1.21), relations (1.20) and conditions (ii),(iii), (vi) and (vii) it follows that the right-hand side in (1.23) is non-positive while the second and third terms in the left-hand side are nonnegative. Taking into account this fact and the second inequality in (1.5) from (1.23) we arrive at the estimate ||v||i n < 0, from which it follows that v = 0. This means the validity of the estimate of p ^ M in 0.

To prove the minimum principle in the form of the estimate p ^ m in 0 we introduce a non-positive function w = minjp — m, 0} and note that the validity of the minimum principle is equivalent to the condition w = 0 in 0. Let us denote by 0m a measurable subset of 0, at the

points of which p < m. By rm we denote a measurable subset of the part rN, at the points of which p\rm < m. Set Q2 = Q \ Qm, r2 = rN \ rm. By definition of Qm and rm we have

w = p — m < 0 and Vw = V p in Qm; w = 0 and Vw = 0 in Q2, w = 0 on T2.

Setting h = w in (1.10) at k(p) = kl(p) we add to both sides of the resulting relation the term — (kl(m)m, w)Qm — (Xam, w)rm. Taking into account the properties of the function w we obtain

(XVw, Vw) + (ki(w + m)(w + m) — ki(m)m, w)Qm + (Xaw, w)rm =

= (f — ki(m)m,w)Qm + (x — Xam,w)rm. (1.24)

From the definition of the constant m in (1.21), (1.20) and conditions (ii), (iii) (v), (vii) it follows that the right-hand side in (1.24) is non-positive while the second and third terms in the left-hand side are nonnegative. Taking into account this fact, from (1.24) we derive that w = 0. □

Remark 1.2. For power-law reaction coefficients, the parameters M1 and ml are easily calculated. For example, for kl(p) = p2, we easily deduce that M1 = fUaL, ml = f^.

2. Formulation and solvability of control problem

To formulate the control problem we divide the set of initial data of Problem 1 into two groups: a group of fixed data, to which we assign the functions u, k(p,-),a and and the control group, to which we assign the functions X, f and x, assuming that they can change in some sets K1, K2 and K3 satisfying the condition

(j) K1 c HS0 (Q), K2 C L2(Q) and K3 c L2(rN) are nonempty convex closed sets. Define the space Y = T* x H1/2 (rD). Setting u = (X, f,x), K = K1 x K2 x K3 we introduce the operator F = (F1, F2) : H 1(Q) x K ^ Y by formulae: F2(p) = p\rD — ^ and

(Fi(p ,u),h) = (XV p, Vh) + (k( p, ■) p, h) + (u ■Vp, h) + (Xa p, h)rN — (f, h) — (x, h)rN

and rewrite (1.10) in the form F(p,u) = 0. Considering this equality as a conditional restriction on the state p € H 1(Q) and control u e K, we introduce the cost functional I and formulate the following conditional minimization problem:

J(p, u) = ^I(p) + ^ WX\\la + f \\f Ift + f WxWIn ^ inf,

F(p, u) = 0, (p, u) e H 1(Q) x K.

We denote by Zad = {(p,u) e H 1(Q) x K : F(p,u) = 0, J(p,u) < to} the set of admissible pairs for the problem (2.1) and suppose that the following condition is satisfied:

(jj) l^o > 0, Hi > 0, i = 1,2,3, and K is a bounded set or ¡n > 0, i = 0,1,2, 3 and functional I is bounded below.

We use the following cost functionals:

Ii(p) = \\p — pdWQ = i \p — pd\2dx, I2(p) = \\p — pdWl,Q. (2.2)

Q

Here pd e L2(Q) (or pd e H 1(Q)) is a given function in some subdomain Q c Q.

Theorem 2.1. Let, in addition to conditions (i), (iii)-(vi), and (j), (jj), I : H 1(Q) ^ R be a weakly semicontinuous below functional and let Zad = Then there exists at least one solution (p,u) e H 1(Q) x K of the control problem (2.1).

Proof. Let (pm, um) e Zad be a minimizing sequence for which the following is true

lim J(pm,um)= inf J(p, u) = J*.

(^,u)eZad

Condition (jj) and Theorem 1.1 yield the following estimates:

||Am||s,Q < C1, HfmHn < C2, ||Xm||rN ^ C3, || pm||1,Q < C4 (2.3)

where the constants Cj, i = 1, 2, 3,4 don't depend on m.

From the estimates (2.3) and from the condition (j) it follows that there exist weak limits A* e K1, f * e K2, x* e K3 and p* e H 1(Q) of some subsequences of sequences {Am}, {fm}, {Xm} and {pm}, respectively. Corresponding subsequences will be also denoted by {Am}, {fm}, {Xm} and {pm}. Moreover, due to the compactness of the embeddings H 1(Q) C Lp(Q) for p < 6, H 1/2(Tn) C Lq(rN) for q< 4, Hs(Q) C L~(Q) and Hs~1/2(rN) C L^^n) for s > 3/2 we can assume for m ^ to, that

pm ^ p* weakly in H 1(Q), weakly in L6(Q) and strongly in Ls(Q), s < 6, pmIrN ^ p*|rN weakly in H1/2 (r n), weakly in L4(rN) and strongly in Lq(rN), q < 4,

fm ^ f * weakly in L2(Q), Xm ^ X* weakly in L2(rN), (2.4)

Am ^ A* weakly in Hs(Q) and strongly in LTO(Q), Am|rN ^ A*IrN weakly in Hs~1/2(rN) and strongly in Lto(Tn), s > 3/2.

It is clear, that F2(p*) = 0. Let us show that F1(p*,u*) = 0, that is, that (A* Vp*, Vh) + (k(p*, ■)p*, h) + (u ■ Vp*, h) + (A*a p*, h)rn = (f*, h) + (x*, h)rN Vh eT. (2.5) To this end we note that the pair (pm,um) satisfies the identity

(AmVpm, Vh) + (k(pm, ■)pm, h) + (u ■ Vpm, h) + (Amapm, h)rN =

= (fm,h) + (xm ,h)rN V h eT. (2.6)

Let us pass to the limit in (2.6) as m ^ to. From (2.4) it follows that all linear terms in (2.6) pass into corresponding ones in (2.5).

Let us study the behaviour of nonlinear terms for m ^ to starting with (k(pm, ■)pm, h). To prove the convergence

(k( pm, ■) pm,h) ^ (k( p*, ■) p*,h) as m ^toV h eT (2.7)

it is enough to show that k(pm, ■)pm ^ k(p*, ■)p* weakly in L6/5(Q) as m ^ to. From (1.3) it follows that k(pm, ■) ^ k(p*, ■) strongly in L3/2(Q), and from (2.4) it follows that pm ^ p* weakly in L6(Q) as m ^ to. We derive from these properties that k(pm, ■)pm ^ k(p*, ■)p* weakly in L6/5(Q) and therefore (2.7) also holds.

For the term (AmVpm, Vh) the following equality holds:

(AmVpm, Vh) - (A*V p*, Vh) = ((Am - A*)V pm, Vh) + (V( pm - p*), A*Vh). (2.8)

Since A*Vh e L2(Q)3, then from (2.4) it follows that (V(pm - p*),A*Vh) ^ 0 as m ^ to for all h eT. Using Holder's inequality, (2.3) and (2.4) we easily deduce for the first term in the right-hand side of (2.8) that

I ((Am - A*)V pm, Vh)I < || Am - A* ||L»(n) ||V pm||d|Vh||Q ^ 0 as m ^to V h eT.

Then from (2.8) we obtain that (XmVpm, Vh) ^ (X*Vp*, Vh) as m ^ toV h eT. Similarly, for the nonlinear term (Xmapm, h)rN we have that

(Xmapm, h)rN — (X*ap*, h)rN = ((Xm — X*)apm, h)rN + (X*a(pm — p*), h)rN. (2.9)

Since X*ah € L4/3(r n ) then by virtue of (2.4) (pm — p*,X* ah)rN ^ 0 for all h eT as m ^ to. Using Holder inequality, (2.4) and the uniform boundedness of the quantity \\pm\\L4(rN) for any m, we deduce for the first term in the right-hand side of (2.9), that

\ ((Xm — X*)apm,h)rN \ < \\ Xm — X*WL~(rN )\\a\\rN\\p^l^^n )\\hWL4(rN) ^ 0 as m ^ TO.

To complete the proof notice that the fact J( * , u* ) = J* follows from aforesaid and from the weakly continuity below on H 1(Q) x Hs(Q) x L2(Q) x L2(rN) of the functional J. □

Remark 2.1. The functionals defined in (2.2) satisfy the conditions of Theorem 2.1.

3. Derivation of the optimality system and stability estimates

The next stage in the study of the control problem (2.1) is the derivation of the optimality system. It provides valuable information about additional properties of optimal solutions for specific reaction coefficients, for example, in the case when k(p, ■) = p2\p\. Based on its analysis, one can establish, in particular, the uniqueness and stability of the optimal solutions More details about the method for deriving estimates of local stability of optimal solutions can be found in [11-16].

Based on the theory developed in [11-16] we introduce the space Y* = T x Hl/2(rD)* dual of the space Y. It is easy to show that for the case k(p, ■) = p2\p\ the Frechet derivative of the operator F = (F1,F2) : H 1(Q) x K ^ Y with respect to p at any point (p,u) = (p,X,f,X) is a linear continuous operator F^(p,u) : H 1(Q) ^ Y that maps each element t e H 1(Q) into an element F'(0,u)(t) = (yl,y2) e Y. Here the elements yl e T* and y2 e Hl/2{rD) are defined by p, X and t with the help of the following relations:

(yi, h) = (XVt, Vh) + 4(02\0\t, h) + (XaT, h)rN + (u ■ Vt, h) V h e T, y2 = t^. (3.1)

By F^(p,u)* : Y* ^ H1 (Q)* we denote an operator adjoint of F'(p,u).

According to the general theory of smooth-convex extremum problems [23], we introduce an element y* = (9, Z) e Y*, to which we will refer as to an adjoint state and we will define the Lagrangian L : H 1(Q) x K x Y* ^ R by

L( p,u, y*) = J (p,u) + (y*,F (p,u))y,xy=J (p,u) + (Fi( p,u),9)T*xT + (Z,F2 (p,u))rD,

where (C = (C ■)R1/2(rD )*XH1/2(rD ).

Since (p2\(p\ e L2+(Q) then from [12] it follows that for any f eT* and ^ e H1/2 (rD) there exists a unique solution t e H 1(Q) of the linear problem

(XVt, Vh) + 4( pp 2\p\T,h) + (Xa.T,h)rN + (u ■VT,h) = (f,h)V h eT, t\rD = (3.2)

Therefore the operator F^(<p,u) : H 1(Q) ^ Y is an isomorphism and from [23] the following assertion follows.

Theorem 3.1. Let, under conditions (i), (iii)-(vi) and (j), (jj), k(p, ■) = p2\p\, the functional I : H 1(Q) ^ R is continuously differentiable with respect to p at the point ¡p and let an element

(<p,û) G H x K be a local minimizer for the problem (2.1). Then there exists a unique Lagrange multiplier (adjoint state) y* = (0,Z) G Y*, such that the Euler-Lagrange equation F'(cp,û)*y* = —J^(<p , û) in H 1(^)* takes place which is equivalent to the relation

(AVt, V0) + 4(p\p\r, 0) + (Xar, 0)rN + (u • Vt, 0) + (Z, t)Fd =

= —(mg/2)(/;(p),T) VT G H 1(Q), (3.3)

and the minimum principle L(p,û, y* ) ^ L(p,û, y* ) Vû G K holds which is equivalent to the inequalities

Mi(Â, A — A)s,n + ((A — A)Vip, V0) + ((A — A)ap 0K > 0 VA G Ki, (3.4)

n(p, f — Z)n — (f — f, 0) > 0 Vf G K2, (3.5)

M3(X, X — x)rn — (X — X, 0)rN > 0 VX G K3- (3.6)

The relations (3.3)-(3.6) together with the operator restriction F(p, û) = 0 comprise an optimality system for problem (2.1). It plays an important role in the study of uniqueness and stability of its solutions.

In conclusion, we formulate a theorem on the local stability of optimal solutions of problem (2.1) for I(p) = \\p — pd\\Q, which is proved according to the scheme proposed in [11].

Theorem 3.2. Assume that the conditions (i), (iii)-(vi) and (j), (jj) take place and k(p, •) = = p2\p\. Let the quadruple (pi, Ai,fi,Xi) G X x K be a solution of the problem (2.1) at I(p) = = \\p — pd\\Q, which corresponds to a specified function pd G L2(il), i = 1, 2. Let the data of the problem (2.1) or parameters and be such that the following condition hold:

nl^G < (1 — £)M, nl^G < (1 — e)M2, nl^o < (1 — e)M3, (3.7)

where £G(0,1) is an arbitrary number, the parameters nk, k=1, 2, 3, 4, monotonically depend on the norms of the initial data of the problem (2.1). Then the following local stability estimates hold:

\\Ai — A2\\s,n < vW(£Mi)(0.5 + %)\\pd — pd\\Q, (3.8)

\\fi — f2\\n < vW(£M2)(0.5 + n4)\pi — pd\\Q, (3.9)

\\Xi — X2\fn Wmg/(£M3)(0.5 + n4)\\pi — pd\Q, (3.10)

\\pi — p2\i,n < (^WMG/(£MI) + U2VMo/(£M2) + V3VMG/(£M3))(0.5 + n4)\\pdi — pd\\Q. (3.11)

Here wi = C*(Y|\\a\\rNMv + YsMv), w2 = C*, = Y2C*, where A*,Y2,Yi,Ys,C* = 1/A* are the constants from Lemma 1.1 and Mv is defined in (1.19).

A similar theorem can be formulated and proved for the functional I2(p) in (2.2). The authors plan to devote a separate paper to a more detailed study of the issues of uniqueness and stability of optimal solutions.

The work was carried out within the framework of the state assignment of the Institute of Applied Mathematics, FEB RAS (Theme no. 075-01095-20-00).

References

[1] Yu.Ya.Belov, Inverse problems for parabolic equations, J. Inv. Ill-Posed Probl., 1(1993), no. 4, 283-306.

[2] K.Ito, K.Kunish, Estimation of the convection coefficient in elliptic equations, Inverse Problems,, 14(1997), 995-1013.

[3] Yu.Ya.Belov, T.N.Shipina, The problem of determining a coefficient in the parabolic equation and some properties of its solution, J. Inv. Ill-Posed Probl., 9(2001), no. 1, 31-48.

[4] G.V.Alekseev, D.A.Tereshko, On solvability of inverse extremal problems for stationary equations of viscous heat conducting fluid, J. Inv. Ill-Posed Probl., 6(1998), no. 6, 521-562.

[5] G.V.Alekseev, E.A.Adomavichus, Theoretical analysis of inverse extremal problems of admixture diffusion in viscous fluids, J. Inv. Ill-Posed Probl., 9(2001), no. 5, 435-468.

[6] P.A.Nguyen, J.-P.Raymond, Control problems for convection-diffusion equations with control localized on manifolds, ESAIM: Control, Optimisation and Calculus of Variations, 6(2001), 467-488.

[7] G.V.Alekseev, O.V.Soboleva, D.A.Tereshko, Identification problems for a steady-sate model of mass transfer, J. Appl. Mech. Tech. Phys., 49(2008), no. 4, 537-547.

[8] G.V.Alekseev, V.G.Romanov, One class of nonscattering acoustic shells for a model of anisotropic acoustics, Journal of Applied and Industrial Mathematics, 6(2012), no. 1, 1-5. DOI: 10.1134/S1990478912010012

[9] Yu.Ya.Belov, V.G.Kopylova, On some identification problem for source function to one semievolutionary system, J. Inv. Ill-Posed Probl., 20(2012), no. 5-6, 723-743.

DOI: 10.1515/jip-2011-0027

[10] R.V.Brizitskii, Z.Y.Saritskaya, A.I.Byrganov, Multiplicative control problems for nonlinear convection-diffusion-reaction equation, Sib. El. Math. Rep., 13(2016), 352-360.

DOI: 10.17377/semi.2016.13.030

[11] G.V.Alekseev, R.V.Brizitskii, Z.Y.Saritskaya, Stability estimates of solutions to extremal problems for a nonlinear convection-diffusion-reaction equation, J. Appl. Industr. Math., 10(2016) no. 2, 155-167. DOI: 10.1134/S1990478916020010

[12] R.V.Brizitskii, Z.Y.Saritskaya, Stability of solutions to extremum problems for the nonlinear convection-diffusion-reaction equation with the Dirichlet condition, Comp. Math. Math. Phys., 56(2016), no. 12, 2011-2022. DOI: 10.1134/S096554251612006X

[13] R.V.Brizitskii, Z.Y.Saritskaya, Stability of solutions of control problems for the convection-diffusion-reaction equation with a strong nonlinearity, Dif. Eq., 53(2017), no. 4, 485-496. DOI: 10.1134/S0012266117040073

[14] R.V.Brizitskii, Zh.Yu.Saritskaya, Optimization analysis of the inverse coefficient problem for the nonlinear convection-diffusion-reaction equation, J. Inv. Ill-Posed Probl., 26(2018), no. 6, 821-833. DOI: 10.1515/jiip-2017-0011

[15] G.V.Alekseev, Stability estimates in the problem of cloaking material bodies for Maxwell's equations, Comp. Math. Math. Phys., 54(2014), no. 12, 1788-1803.

DOI: 10.1134/S0965542514120069

[16] G.V.Alekseev, Analysis and optimization in problems of cloaking of material bodies for the Maxwell equations, Dif. Eq., 52(2016), no. 3, 361-372. DOI: 10.1134/S0012266116030101

[17] A.E.Kovtanyuk, A.Y.Chebotarev, N.D.Botkin, K.-H.Hoffmann, Solvability of P1 approximation of a conductive-radiative heat transfer problem, Appl. Math. Comp., 249(2014), 247-252.

[18] A.E.Kovtanyuk, A.Yu.Chebotarev, N.D.Botkin, K.-H.Hoffmann, Optimal boundary control of a steady-state heat transfer model accounting for radiative effects, J. Math. Anal. Appl., 439(2016), 678-689.

[19] J.-L.Lions, Some Methods of Solving Non-Linear Boundary Value Problems, Dunod-Gauthier-Villars, Paris, 1969; Russian translation: Mir, Moscow, 1972.

[20] O.A.Ladyzhenskaya, N.N.Uraltseva, Linear and quasilinear elliptic equations, New York-London, Academic Press, 1968.

[21] D.Gilbarg, M.Trudinger, Elliptic partial differential equations of second order, Springer, 1984.

[22] H.Berninger, Non-overlapping domain decomposition for the Richards equation via superposition operators, In: Domain Decomposition Methods in Science and Engineering XVIII, Springer, 2009, 169-176. DOI: 10.1007/978-3-642-02677-5_17

[23] A.V.Fursikov, Optimal control of distributed systems. Theory and applications, American Mathematical Society, 2000.

Анализ краевых задач и задач управления для нелинейного уравнения реакции-диффузии-конвекции

Геннадий В. Алексеев Роман В. Бризицкий

Институт прикладной математики ДВО РАН Владивосток, Российская Федерация

Аннотация. Доказывается глобальная разрешимость неоднородной смешанной краевой задачи и задач управления для уравнения реакции-диффузии-конвекции в случае, когда коэффициент реакции нелинейно зависит от концентрации. Для решения краевой задачи устанавливаются принципы максимума и минимума. Для задач управления с конкретными коэффициентами реакции выводятся системы оптимальности и устанавливаются оценки локальной устойчивости оптимальных решений.

Ключевые слова: нелинейное уравнение реакции-диффузии-конвекции, смешанные граничные условия, принцип максимума, задачи управления, системы оптимальности, оценки локальной устойчивости.