Optimization in Nonlinear Models of Mass Transfer

Abstract

Optimal control problem for convection{di usion{reaction equation, in which reaction coe cient depends nonlinearly on substance's concentration, is considered. Numerical algorithms for solving nonlinear boundary value and optimal control problems are proposed for the equation under study. Separately, the results of the numerical experiments about nonlinear boundary value problems' solvability are presented. For this purpose the FreeFem++ solver is used. These studies allow to understand better the process of pollution's spread in the atmosphere and ght against its consequences. Particularly, they give an opportunity to reveal and eliminate the sources of pollution using the measured impurity's concentration in some available domain. Also the correctness of mathematical models of mass transfer and optimal control problems, which are considered in the paper, is justi ed.

Keywords

• FreeFem++
• nonlinear convection-di usion-reaction equation
• optimal control problem
• multiplicative control problems
• optimality system
• numerical algorithm

In a bounded domain is considered

R3 with boundary

the following boundary value problem ' + u r' + k' = f in ; ' = 0 on : ( 1 ) Here function ' means polluting substance's concentration, u is a given vector of velocity, f is a volume density of external sources of substance, { constant di usion coe cient, function k = k(') is a reaction coe cient. This problem ( 1 ) will be called problem 1 below.

This study of optimal control problems for a model ( 1 ) is intended to develop e cient mechanisms to control chemical reactions' behavior. The decision to choose a velocity vector u as a control can signify the regularization of combustion process at Copyright c by the paper's authors. Copying permitted for private and academic purposes. In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org the expense of fuel feed's intensity changing (see [1]). The e ciency criterion for such kind of control is the measured concentration of unburned fuel in a subdomain.

It should be mentioned that some inverse problems can be reduced to the optimal control ones as from the mathematical point of view optimal control problems are the problems of cost functionals' minimization on weak boundary problem's solutions. At the same time one or several functions can be changed in some certain convex closed sets. The mentioned functions are usually called controls, cause their changing is exactly the thing that in uences on the minimum of the cost functional.

From the other side, one can thought that such functions are searched on the assumption of the minimum of corresponding cost functionals, which attaches these extremum problems the meaning of indenti cation problems of the functions and inverse problems. See [2{10] about similar methods and approaches.

Particularly, the optimal control problem, which is considered in this paper, can represent the indenti cation problem for a velocity and a direction of wind or a fuel, depending on the case and the situation.

With the help of this approach it's possible to reveal the hidden sources of pollution, which are located in the places, inaccessible for observation (under water, on the territory of the adjacent country). Then the data about the concentration of polluting substance in the domain, which is accessible for measurement, about the direction and the velocity of the wind or of the ow in the basin are used.

The results of numerical experiments, executed in FreeFEM++, are given for the solving of nonlinear boundary value problem. We should note a quick convergence of the simple iteration method while the initial approximation of the boundary value problem's solution was chosen not very successful. The computations are conducted for a number of reaction coe cients, which depend nonlinearly on the substance's concentration at di erent boundary conditions. The chosen geometry of the domain and the given velocity eld simplify the understanding of numerical experiments' results.

The numerical algorithm for solving the optimal control problem is presented. This algorithm is based on using of optimality system, obtained for the extremum problem. Su cient conditions of such algorithms' convergence were obtained in [20]. But these conditions have the meaning of either the smallness conditions for the initial data of a boundary value problem or demands greater values of the regularizer in an optimal control problem, which spoils the quality of the last one. That's why it's interesting to analyse the convergence of such algorithm depending on the initial approximation of the optimal control problem's solution. As in the case of the boundary value problem.

The reasoning of the correctess of the considered mathematical model implies the following. The global solvability of problem 1, when reaction coe cients belong to rather wide class of functions, is proved in [10{12]. In this paper it is shown that power coe cients from [13{15] are particular cases of the reaction coe cients considered in [10{12], with which nonlocal uniqueness of boundary value problem's solution takes place. The solvability of multiplicative control problem with common reaction coe cients is proved further. For a quadratic reaction coe cient optimality system is obtained, on the analysis of which su cient conditions for local uniqueness of multiplicative control problems' solutions for particular cost functionals are received.

While studing problem 1 and optimal control problems Sobolev spaces will be used: Hs(D), Hs(D) Hs(D)3; s 2 R and Lr(D), 1 r 1, where D is either a domain or its boudary . Scalar products in L2( ), H1( ) and H1( ) are denoted by ( ; ) and ( ; )1, scalar products in L2( ) { by ( ; ) , norm in L2( ) { by k k, norm or semi-norm in H1( ) { by k k1 or j j1.

It will be assumed that the domain and its boundary satisfy the following: (i) is a bounded domain in the space R3 with boundary 2 C0;1.

Let D( ) be the space of in nitely di erentiable functions with nite support in , Lp+( ) = fk 2 Lp( ) : k 0g, p 3=2. Also let Z = fv 2 L4( ) : div v = 0 in g, V Z \ H1( ).

From Poincare-Friedrichs inequality and from the continuity of the embedding operator H1( ) L4( ) this lemma follows:

Lemma 1. If conditions (i) hold, then there are such positive constants C0, , C4 and , depending on , that for any functions '; S 2 H1( ), k 2 Lp+( ), where p 3=2, u 2 Z these relations are correct: ( 2 ) ( 3 ) ( 4 ) ( 5 ) j(r'; rS)j k'k1kSk1; k'kL4( )

C4k'k1; j(k'; S)j

C0kkkLp( )k'k1kSk1; j(u r'; S)j kukL4( )k'k1kSk1

C4kuk1k'k1kSk1; (u

r'; ') = 0 8' 2 H01( ); and for any function S 2 H01( ) the inequality takes place (rS; rS) kSk12: From lemma 1 follows that while conditions (i) are satis ed with the constant = when k 2 L3+=2( ), then the coercitive inequality is met (rS; rS) + (kS; S) kSk12 8S 2 H01( ): Let in addition to (i) the conditions hold: (ii) f 2 L2( ), u 2 Z.

(iii) k 2 Lp+( ), p 3=2, wherein function k = k(') is Lipschitz continuous of ', i.e. if k'1k1 c and k'2k1 c, then kk('1) k('2)kLp( )

Lk'1

'2kL4( ) 8'1; '2 2 H01( ):

Let's multiply the equation in ( 1 ) by S 2 H01( ) and integrate over . The following will be got (r'; rS) + (k(')'; S) + (u r'; S) = (f; S) 8S 2 H01( ): ( 6 ) As a result, the weak formulation of problem 1 is obtained. It consists in nding function ' 2 H01( ) from ( 6 ).

De nition 1. A function ' 2 H01( ) which satis es ( 6 ) will be called a weak solution of problem 1.

The following theorem takes place [12].

Theorem 1. If conditions (i){(iii) hold, then a weak solution ' 2 H01( ) of problem 1 exists and the estimate takes place: ( 7 ) ( 8 ) If, besides, this condition is met k'k1

M' = (1= )kf k: C0Lkf k 2; then the problem 1's weak solution is unique.

From [12{14] it ensues that the power dependence is interesting as an example of particular cases of function k = k('), k = '2 and k(') = '2j'j, for instance. As the case of quadratic reaction coe cient was analysed in detail in [12]. In particular, it was shown that a function k = '2 satis es conditions (iii) and for this function there is nonlocal uniqueness of problem 1's weak solution, so let's consider the function k = '2j'j.

For k = '2j'j the equality is true: k('1) k('2) = '12(j'1j j'2j) + ('1 '2)('1 + '2)j'2j a.e. in and also an estimate takes place:

Z ('1 '2)3=2'31 d 2=3 k'1

2 '2kL3( )k'1kL6( ): In such case function k = '2j'j satis es the condition (iii).

When k = '2j'j nonlocal uniqueness of problem 1's solution takes place. Actually, let k = '2j'j and '1; '2 2 H1( ) be two solutions of problem 1. Then their di erence ' = '1 '2 2 H01( ) satis es the ratio (r'; rh) + ('13j'1j '32j'2j; h) + (u r'; h) = 0 8h 2 H01( ): ( 9 ) It's clear that ('13j'1j '32j'2j)('1 '2) = '41j'1j '32j'2j'1 '31j'1j'2 + '42j'2j a.e. in and on the strength of Young's inequality

4 '2'1 (4=5)'25 + (1=5)'15; '1'2 4 (4=5)'15 + (1=5)'25 a.e. in :

In such case ('31j'1j '32j'2j; ') 0 a.e. in . Assuming h = ' in ( 9 ), on the strength of lemma 1 it can be concluded that ' = 0 or '1 = '2 in .

From aforesaid and [12] follows

Theorem 2. Let conditions (i), (ii) hold. Then when k = '2 and k = '2j'j, there is a unique weak solution ' 2 H01( ) of problem 1 and the estimate ( 7 ) is met.

Let's separately consider the reaction coe cient k('), which generalizes the forth power, but is not a function of ' in a common sense. Let k(') be an operator acting form T to Lp+( ), where p 3=2 and satisfying the following conditions: ( 1 ) for all w1; w2 2 Br = fw 2 T : kwk1; rg the following estimate holds: kk(w1) k(w2)kLp( )

Lkw1 w2kL6( ); where L is a constant, depending on r, but not depending on w1; w2; ( 2 ) k(')' satis es monotony condition (k('1)'1 k('2)'2; '1 '2) 0 8'1; '2 2 H01( ):

Let's consider a simple example of the operator k('), satisfying the conditions ( 1 ), ( 2 ) and generalizing numerical functions: k(') = '4 in subdomain Q and k(') = k0 in n Q, where k0 2 L3+=2( ).

This example takes into account the in uence on the chemical reaction's velocity not only of substance's concentration, but also of inhomogeneity of chemical reaction's behavior in the considered domain. Conditions ( 1 ), ( 2 ) are also met for the reaction coe cient k(') = (x)'4, where (x) 2 L+1( ).

It should be noted that the reaction coe cient k(') = '4 gives the convectiondi usion-equation the maximum possible nonlinearity of 5th power for the solution ' 2 H1( ). For such strong nonlinearity the theory of problem 1's solvability proving, which was stated above , is unacceptable in view of the fact that the boundary value problem's operator is not compact. The solvability of problem 1 at k('), satisfying the conditions ( 1 ), ( 2 ) follows from the results of [16].

The following theorem holds

Theorem 3. Let conditions (i), ( 1 ), ( 2 ) hold. Then there is a unique weak solution ' 2 H01( ) of problem 1 and the estimate ( 7 ) is met. 2

Statement of optimal control problem and its solvability Let's formulate an optimal control problem for problem 1. For this purpose the whole set of initial data will be devided into two groups: the group of xed functions, in which function f is included, and the group of controlling functions, in which u will be included, assuming that it can be changed in some subset K.

Let's introduce an operator F : H01( ) K ! H 1( ) by formula hF ('; u); Si = (r'; rS) + (u r'; S) + (k(')'; S) (f; S): Then ( 6 ) can be rewritten in the following form:

F ('; u) = 0: ( 10 ) Let's suppose that these conditions hold (j) K V is a nonempty convex closed set; (jj) i 0; i = 1; 2 and K is a bounded set l > 0; l = 0; 1 and the functional I is bounded below.

Treating ( 10 ) as a conditional restriction on the state ' 2 H01( ) and on the control u 2 K, the problem of conditional minimization can be formulated as follows: J ('; k) 0 I(') + 2 21 kuk12 ! inf; F ('; u) = 0; ('; u) 2 H01( ) The following cost functionals can be used in the capacity of the possible ones:

Z I1(') = k' 'dk2Q = j' 'dj2dx; I2(') = k' 'dk12;Q: Here 'd 2 L2(Q) is a given function in some subdomain - Q . The set of possible pairs for the problem ( 11 ) is denoted by Zad = f('; u) 2 H01( ) K : F ('; u)=0; J ('; u)<1g.

Theorem 3. Let conditions (i){(iii) and (j), (jj) hold. Then there is at least one solution of the optimal control problem ( 11 ).

Proof. Let ('m; um) be a minimizing sequence, for which the following is true: lim J ('m; um) = m!1

inf ('m;um)2Zad

J ('m; um)

J : That and the conditions of theorem for the functional J from ( 11 ) imply the estimate kumk1 c1. From theorem 1 follows directly that k'mk1 c2, where constant c2 doesn't depend on m.

Then the weak limits ' 2 H01( ) and u 2 V of some subsequences of sequences f'mg and fumg exist. Corresponding sequences will be also denoted by f'mg and fumg. With this in mind it can be considered that ( 12 ) ( 13 ) ( 14 ) ( 15 ) 'm ! ' 2 H1( ) weakly in H1( ) ang strongly in L4( ); um ! u 2 H1( ) weakly in H1( ) and strongly in L4( ): Let's show that F (' ; u ) = 0, i.e.

(r' ; rS) + (k(' )' ; S) + (u r' ; S) = (f; S) 8S 2 H01( ): And it should be taken into account that 'm and um satisfy the relations (r'm; rS) + (k('m)'m; S) + (um r'm; S) = (f; S) 8S 2 H01( ): Let's pass to the limit in ( 15 ) at m ! 1. All linear summands in ( 15 ) turn into corresponding ones in ( 14 ). For the nonlinear summand (k('m)'m; S) the inequality takes place j(k('m)'m; S) (k(' )' ; S)j j(k('m)('m ' ); S)j + j(k('m) k(' ); ' S)j: On the strength of lemma 1 and condition (iii) for the function k = k(') it is obtained that j(k('m) k(' ); ' S)j

Lk'm

' kL4( )k' kL4( )kSkL4( ) ! 0 at m ! 1: To apply the property ( 12 ) for the summand j(k('m)('m ' ); S)j, embedding density by norm k k1 will be used. Let fSng 2 D( ) be such a sequence of functions that kSn Sk1 ! 0 at n ! 1.

This inequality holds: j(k('m)('m ' ); Sn)j kk('m)kL3=2( )kSnkL12( )k'm ' kL4( ) ! 0 at m ! 1: As far as

Then

jj(k('m)('m ' ); Sn)j j(k('m)('m ' ); S)jj j(k('m)('m ' ); Sn

S)j kk('m)kL3=2( )k'm ' kL6( )kSn

SkL6( ) ! 0 at n ! 1; m = 1; 2; ::: lim (k('m)'m; S) = (k (' )' ; S): m!1 For the nonlinear summand (um r'm; S) this relation is satis ed (um r'm; S) (u

r' ; S) = = (u r('m ' ); S) + ((um u ) r'm; S) 8S 2 H01( ): ( 16 )

On the strength of ( 12 ) a weak convergence takes place: r'm ! r' in L2( ), according to which (u r('m ' ); S) = (r('m

' ); u S) ! 0 at m ! 1 8S 2 H01( ); and from ( 13 ) follows that j((um u ) r'm; S)j

kr'mkL2( )kum ukL4( )kSkL4( ) ! 0 as m ! 1 8S 2 H01( ): Then, taking ( 16 ) into account, it is obtained lim (um r'm; S) = (u m!1 r' ; S):

As the functional J is weakly semicontinuous below on H01( ) aforesaid follows that

V, then from

J = lim J ('m; um) = limm!1J ('m; um) m!1

J (' ; u ) Further the case of k(') = '2j'j will be considered and the principle of Lagrange multipliers for the problem ( 11 ) will be justi ed.

Let's introduce a Lagrange multiplier ( 0; ) 2 R H01( ) and a Lagrangian L : H01( ) V R H01( ) ! R by formula

L('; u; 0; ) = 0J ('; u) + h ; F ('; u)i 0J ('; u) + hF ('; u); i: ( 17 ) A common analysis shows that Frechet derivative of the operator F with respect to ' in ( 10 ) for k = '2j'j in the point ('^; u^) 2 H01( ) V is a linear continuous operator F '0('^; u^) : H01( ) ! H 1( ), which assigns to each element 2 H01( ) an element ^l 2 H 1( ), where h^l; Si = (r ; rS) + 4('^2j'j ; S) + (u r ; S): From lemma 1 follows that the operator F '0('^; u^) is an isomorphism. Then according to [18, 19] the theorem takes place:

Theorem 5. While conditions (i), (ii) and (j), (jj) hold, let ('^; u^) 2 H01( ) V be an element, on which the local minimum is achieved in the problem ( 11 ) if k = '2j'j. Then there is a unique nonzero Lagrange multiplier (1; ), where 2 H01( ), such as Euler{Lagrange equation is satis ed

hJ '0('^; u^); i + hF '0('^; u^) ; i = 0 8 2 H01( ); and is equivalent to (r ; r ) + 4('^2j'j ; ) + (u r ; ) = 0('^ 'd; )Q 8 2 H01( ); and also the minimum principle is true: which is equivalent to the inequality hL0u('^; u^; 1; ); u u^i 0 8 u 2 V; ( 18 ) ( 19 ) ( 20 ) 1(u^; u u^)1 + ((u u^) r'^; ) 0 8u 2 V:

The relation ( 19 ) together with the variational inequality ( 20 ) and the operational restriction ( 10 ), which is equivalent to the ratio ( 6 ), are the optimality system for the problem ( 11 ) when k = '2j'j. 4

Computations Two di erent nonlinear boundary value problems were solved in cases of reaction coeffecients k(') = '2 and k(') = j'j. For both cases the exact solution 'e = 0:2(x2 + y2) was taken. Also, the same u = (0; 0), '0 = 0:1(x + y); = 10. The function f was simply calculated in each case by substituting the function ' in the equation by the exact solution. The domain is [ 1; 1] [0; 1]. The relative error is obtained by formula k'k'e'ke2k2 .

FreeFem++ listing border c1(t=-1,1){x = t; y = 0; label=1;}; border c2(t=0,1){x = 1; y = t;label=2; }; border c3(t=-1,1){x=t;y=1;label=3;}; border c4(t=0,1){x=-1;y=t;label=4;}; int n=3; mesh Th = buildmesh(c1(20*n) + c2(10*n) + c3(-20*n) + c4(-10*n)); fespace Vh(Th,P2); Vh u1, u2, phi, phi1, s, err0, phi0; u1 = 0; u2 = 0; int lambda = 10; Vh f=-lambda*0.8+(0.2*(x^2+y^2))^3; Vh phiex = 0.2*(x^2+y^2); phi0 = 0.1*(x+y); plot(phi0, cmm="phi0", wait = true, value = 1, fill = 1); problem equation1(phi1,s)= int2d(Th)(lambda*(dx(phi1)*dx(s) + dy(phi1)*dy(s))) + int2d(Th)(phi0^2*phi1*s) + int2d(Th)((u1*dx(phi1) + u2*dy(phi1))*s) - int2d(Th)(f*s) + on(1,phi1=0.2*x^2) + on(2,phi1=0.2+0.2*y^2) + on(3,phi1=0.2+0.2*x^2) + on(4,phi1=0.2+0.2*y^2); Let's rst consider the case of k(') = j'j. Then the initial equation's weak formulation will obtain the form (r'; rS) + (j'j'; S) + (u r'; S) = (f; S) 8S 2 H01( ): (21) Then, taking into account that the exact solution is ' = 0:2(x2 + y2) and u = (0; 0), the following f = 0:8 + (0:2(x2 + y2))2. 4.2

Case 2 Let's now consider the case of k(') = '2. Then the initial equation's weak formulation will obtain the form (r'; rS) + ('3; S) + (u r'; S) = (f; S) 8S 2 H01( ): (22) Then, simillary, f = 0:8 + (0:2(x2 + y2))3.

Here we are presenting some computations for both cases. It should be mentioned that the relative error for the rst case at the last step of the loop is 1:1914e 10 and for the second one is 1:0646e 10. For the numerical research of extremum problems the algorithm from [20] will be used in future. Here are the recurrence relations, which stand for the algorithm for the case of the considered optimal control problem: (r ; r k) + 4('2kj'kj ; k) + (uk r ; k 1) = 0('k 'd; ) 8 2 H01; 1(uk; u uk)1 + ((u uk) r'k 1; k 1) 0 8u 2 K; (r'k; rS) + (uk r'k; S) + ('3k 1j'kj; S) = (f; S) 8S 2 H01: (23) (24)

Under the condition 1 > 0 and the fact that the set K V is convex and closed, we can introduce the projection operator P : V ! K. Then the variational inequality (24) is equivalent to the equation uk = P ('k 1 k 1= 1), k 0.

The result about this algorithm's convergence was obtained similarly to the [20]. Acknowledgments. This work was supported by the Russian Foundation for Basic Research (project no. 16-01-00365) and Ministry of Education and Science of Russian Federation (contract no. 14.Y26.31.0003).

References

1. Becker , V. , Braack , M. , Vexler , B. : Numerical parameter estimation for chemical models in multidimensional reactive ows . Combust. Theory Modelling . 8 , 661 { 682 ( 2004 )

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

3. Alekseev , G.V. , Shepelov , M.A. : On stability of solutions of the coe cient inverse extremal problems for the stationary convection{di usion{reaction equation . Journal of Applied and Industrial Mathematics . 1 , 1 { 14 ( 2013 )

4. Alekseev , G.V. , Adomavichus , E.A. : Theoretical analysis of inverse extremal problems of admixture di usion in viscous uid . J. Inverse Ill-Posed Probl. 9 , 435 { 468 ( 2001 )

5. Alekseev , G.V. : Inverse extremal problems for stationary equations in mass transfer theory . Comp. Math. Math. Phys. 42 , 363 { 376 ( 2002 )

6. Alekseev , G.V. : Coee cient inverse extremum for stationary heat and mass transfer equations . Comp. Math. Math. Phys. 47 , 1007 { 1028 ( 2007 )

7. Alekseev , G.V. , Soboleva , O.V. , Tereshko , D.A. : Identi cation problems for a steady-state model of mass transfer . J. Appl. Mech. Tech. Phys. 49 , 537 { 547 ( 2008 )

8. Alekseev , G.V. , Tereshko , D.A. : Two parameter extremum problems of boundary control for stationary thermal convection equations . Comp. Math. Math. Phys. 51 , 1539 { 1557 ( 2011 )

9. Alekseev , G.V. , Vakhitov , I.S. , Soboleva , O.V. : Stability estimates in identi cation problems for the convection{di usion{reaction equation . Comp. Math. Math. Phys. 52 , 1635 { 1649 ( 2012 )

10. Alekseev , G.V. , Brizitski , R.V. , Saritskaya , Zh.Yu.: Extremum problem's solutions' stability estimates for nonlinear convection-di usion-reaction equation . Journal of Applied and Industrial Mathematics . 19 , 3 { 16 ( 2016 )

11. Brizitski , R.V. , Saritskaya , Zh.Yu.: Boundary value and optimal control problems for nonlinear convection-di usion-reaction equation . Key Engineering Materials . 685 , 13 { 17 ( 2016 )

12. Brizitskii , R.V. , Saritskaya , Zh.Yu.: Boundary value and optimal control problems for nonlinear convection-di usion-reaction equation . Siberian Electronic Mathematical Reports . 12 , 447 { 456 ( 2015 )

13. Kovtanyuk , A.E. , Chebotarev , A.Yu. , Botkin , N.D. , Ho mann, K.-H.: The unique solvability of a complex 3D heat transfer problem . J. Math. Anal. Appl . 409 , 808 - 815 ( 2014 )

14. Kovtanyuk , A.E. , Chebotarev , A. Yu .: Steady-state problem of complex heat transfer . Comp. Math. Math. Phys. 54 , 719 { 726 ( 2014 )

15. Kovtanyuk , A.E. , Chebotarev , A. Yu. , Botkin , N.D. , Ho man , K.-H.: Unique solvability of a steady-state heat tranfer model . Commun Nonlinear Sci. Number Simulat . 20 , 776 { 784 ( 2015 )

16. Lions , J.-L.: Some methods of solutions of nonlinear boundary value problem , Moscow ( 1972 )

17. Grisvard , P. : Elliptic problems in nonsmooth domains. Monograph and studies in mathematics . Pitman, London ( 1985 )

18. Io e, A.D., Tikhomirov , V.M. : Theory of extremal problems . Elsevier, Amsterdam ( 1978 )

19. Cea , J. Lectures on Optimization. Theory and Algorithms . Springer{Verlag, BerlinHeidelberg-New York ( 1978 ).

20. Brizitskii , R.V. , Savenkova , A.S.: Asymptotic behavior of solutions to multiplicative control problems for elliptic equations . Comp. Math. Math. Phys. 48 , 1570 { 1580 ( 2008 )