IJDE International Journal of Differential Equations 1687-9651 1687-9643 Hindawi Publishing Corporation 10.1155/2016/9428128 9428128 Research Article Existence of Optimal Control for a Nonlinear-Viscous Fluid Model http://orcid.org/0000-0002-1514-4475 Baranovskii Evgenii S. http://orcid.org/0000-0001-8356-5418 Artemov Mikhail A. Kaikina Elena Department of Applied Mathematics Informatics and Mechanics Voronezh State University Universitetskaya Ploshchad 1 Voronezh 394006 Russia vsu.ru 2016 2762016 2016 05 04 2016 05 06 2016 2016 Copyright © 2016 Evgenii S. Baranovskii and Mikhail A. Artemov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the optimal control problem for a mathematical model describing steady flows of a nonlinear-viscous incompressible fluid in a bounded three-dimensional (or a two-dimensional) domain with impermeable solid walls. The control parameter is the surface force at a given part of the flow domain boundary. For a given bounded set of admissible controls, we construct generalized (weak) solutions that minimize a given cost functional.

1. Introduction

The control and optimization problems in hydrodynamics have been the focus of attention of the control theory specialists for a long time. Flow boundary control problems have attracted increasing interest in recent years (see, e.g., ). Such problems are of interest from a theoretical perspective and are beneficial to applications as boundary control is easy to implement in practice.

In this paper, we study the optimal boundary control problem for a mathematical model describing steady flows of a nonlinear-viscous incompressible fluid in a bounded domain of space R n , n = 2,3 , with impermeable solid walls. A distinguishing feature of the problem under consideration is that the surface force at the flow domain boundary is used as a control parameter instead of the nonhomogeneous Dirichlet boundary condition for the velocity field. Such an approach makes it possible to consider the case of flow control in a domain with impermeable solid walls without using external body forces as control parameters.

It should be mentioned at this point that a lot of studies have been conducted towards mathematical models of nonlinear-viscous fluids (see monograph  and ). Nevertheless, there are very few results on the existence and properties of solutions of control problems for nonlinear-viscous fluid flows. To the best of our knowledge, some results have only been obtained for the two-dimensional case (see [13, 14]).

Also, we would mention that there are many mathematical results concerning optimal control problems for the classical Navier-Stokes equations (see  and the references therein).

The aim of this paper is to prove the solvability of the optimal control problem, which is discussed above. More precisely, for a given bounded set of admissible boundary controls, we will construct generalized (weak) solutions that minimize a given lower weakly semicontinuous cost functional.

2. Problem Formulation and Main Result

Let Ω be a bounded domain in R n ( n = 2 or 3) with boundary Γ C 2 . Consider the following optimal boundary control problem: (1) v · v - div S + p = f in Ω , (2) · v = 0 in Ω , (3) S = ψ I 2 v D v in Ω , (4) v · n = 0 on Γ , (5) v = 0 on Γ Γ c , (6) S n τ = u on Γ c , (7) u U , (8) J v , S , u i n f , where v is the velocity field, p is the pressure function, S is the extra-stress tensor, f is the body force, the symbol denotes the gradient with respect to the spatial variables x 1 , , x n , the divergence div S is the vector with coordinates (9) div S i = j = 1 n S j i x j , D ( v ) is the rate of deformation tensor, (10) D i j v = 1 2 v i x j + v j x i , I 2 ( v ) is the second invariant of D ( v ) , (11) I 2 v = i , j = 1 n D i j v 2 , ψ is a given function, n is the unit vector of the outer normal to Γ , u is the control, v · n is the scalar product of the vectors v and n in space R n , the symbol [ · ] τ denotes the tangential component of a vector, that is, (12) w τ = w - w · n n , Γ c is a part of Γ from which the control is realized, U is the set of admissible controls, and J is a given cost functional.

From here on, the following notations will be used. M s n × n denotes the space of symmetric n × n -matrices with the norm (13) A M s n × n = i , j = 1 n A i j 2 1 / 2 .

We use the standard notations L q ( Ω , E ) and W m , q ( Ω , E ) for the Lebesgue and Sobolev spaces of vector functions defined on Ω with values in a finite-dimensional space E (for details, see ). The scalar product in the space L 2 ( Ω , E ) is denoted by ( · , · ) .

By definition, put (14) L τ 2 Γ c , R n = w L 2 Γ c , R n : w · n = 0 . Moreover, we introduce the space (15) X Ω , R n = v W 1,2 Ω , R n : · v = 0 , v Γ · n = 0 , v Γ Γ c = 0 with the following norm: (16) v X Ω , R n = D v L 2 Ω , M s n × n . In the right-hand side of (15), the restriction of a vector function v : Ω R n to Γ is defined by the formula (17) v Γ = γ 0 v , where γ 0 : W 1,2 ( Ω , R n ) L 2 ( Γ , R n ) is the trace operator.

It follows from Korn’s inequality (see ) that the norm · X ( Ω , R n ) is equivalent to the norm induced from W 1,2 ( Ω , R n ) . Furthermore, we have the following estimates: (18) v L 2 Γ , R n C 1 v X Ω , R n , v L 2 Ω , R n C 2 v X Ω , R n , where C 1 and C 2 are positive constants.

Suppose the following:

the function ψ is measurable and there exist constants a 1 and a 2 such that (19) 0 < a 1 ψ t a 2 , t 0 , + ,

for any A , B M s n × n , we have (20) i , j = 1 n ψ A M s n × n 2 A i j - ψ B M s n × n 2 B i j A i j - B i j 0 ,

the set U is bounded and sequentially weakly closed in L τ 2 ( Γ c , R n ) ,

the functional J : X ( Ω , R n ) × L 2 ( Ω , M s n × n ) × L τ 2 ( Γ c , R n ) R is lower weakly semicontinuous; that is, for any sequence { ( v k , S k , u k ) } k = 1 such that v k v weakly in X ( Ω , R n ) , S k S weakly in L 2 ( Ω , M s n × n ) , and u k u weakly in L τ 2 ( Γ c , R n ) , we have (21) J v , S , u lim inf k J v k , S k , u k .

Example 1.

Let us consider the following cost functionals: (22) J 1 v , S , u = λ 1 v - v ~ L 2 Ω , R n 2 + λ 2 S - S ~ L 2 Ω , M s n × n 2 + λ 3 u - u ~ L τ 2 Γ c , R n 2 , J 2 v , S , u = - λ 1 v - w ~ L 2 Ω , R n 2 + λ 2 S - S ~ L 2 Ω , M s n × n 2 + λ 3 u - u ~ L τ 2 Γ c , R n 2 , where v ~ is a favorable velocity field; w ~ is an unfavorable velocity field, that is, a velocity field whose appearance is undesirable; S ~ is a favorable extra-stress tensor; u ~ is a favorable surface force at Γ c ; and λ 1 , λ 2 , and λ 3 are positive cost parameters. It is obvious that condition (iv) holds for the functionals J = J i , i = 1,2 .

Remark 2.

We do not assume that the set of admissible controls is convex. As is known, the convexity condition is widely used in studying of optimal control problems (see, e.g., ). However, this condition does not always hold in applications. Obviously, condition (iii) is weaker than the convexity condition. For example, (iii) is satisfied if the set U can be represented as the union of finite number of convex closed sets in the space L τ 2 ( Γ c , R n ) .

Now we introduce the concept of admissible triplets of (1)–(8) by analogy with the definition of generalized (weak) solutions to hydrodynamic models with slip boundary conditions (see, e.g., [8, 19, 20]).

Let f L 2 ( Ω , R n ) .

Definition 3.

One says that a triplet ( v , S , u ) X ( Ω , R n ) × L 2 ( Ω , M s n × n ) × L τ 2 ( Γ c , R n ) is an admissible triplet of control system (1)–(8) if the equality (23) - i = 1 n v i v , φ x i + ψ I 2 v D v , D φ = f , φ + Γ c u · φ d Γ c holds for any φ X ( Ω , R n ) and if conditions (3) and (7) hold.

Remark 4.

Equation (23) appears for the following reasons. Let us assume that ( v , S , p , u ) is a classical solution of (1)–(7). We take the L 2 -scalar product of (1) with φ X ( Ω , R n ) . By integrating by parts, we obtain (24) - i = 1 n v i v , φ x i + S , D φ - Γ c S n · φ d Γ c = f , φ . Combining this with (3) and (6), we get (23).

On the other hand, it is not difficult to prove that if an admissible triplet ( v , S , u ) is sufficiently smooth, then there exists a function p such that ( v , S , p , u ) is a classical solution to (1)–(7).

Let M be the set of admissible triplets to problem (1)–(8).

Definition 5.

A triplet ( v , S , u ) M is called a solution of optimization problem (1)–(8) if the equality (25) J v , S , u = inf v , S , u M J v , S , u holds.

Our main result provides existence of solutions to (1)–(8).

Theorem 6.

If conditions (i), (ii), (iii), and (iv) hold, then optimization problem (1)–(8) has at least one solution.

3. Proof of Theorem <xref ref-type="statement" rid="thm1">6</xref>

The proof of Theorem 6 is based on the Galerkin method and monotonicity methods , as well as the following lemma.

Lemma 7.

Let B R = { x R m : x R m R } be a closed ball. Suppose the continuous mapping F : B R × [ 0,1 ] R m satisfies the following conditions:

F ( x , λ ) 0 for any ( x , λ ) B R × [ 0,1 ] ,

F ( x , 0 ) = A x for any x B R ,

where A : R m R m is an isomorphism;

then for any λ [ 0,1 ] the equation F ( x , λ ) = 0 has at least one solution x λ B R .

Lemma 7 can be proved by methods of topological degree theory (see, e.g., ).

Proof of Theorem <xref ref-type="statement" rid="thm1">6</xref>.

First we show that the set of admissible triplets is nonempty. Let us fix an element u 0 = ( u 1 0 , , u n 0 ) U . Suppose { φ j } j = 1 is an orthonormal basis of the space X ( Ω , R n ) .

For an arbitrary fixed number m N , we consider the following auxiliary problem.

Find a vector ( α m 1 , , α m m ) R m such that (26) - λ i = 1 n v i m v m , φ j x i + ψ λ I 2 v m D v m , D φ j = λ f , φ j + λ Γ c u 0 · φ j d Γ c , j = 1 , m , (27) v m = j = 1 m α m j φ j , where λ is a parameter, λ [ 0,1 ] .

First we prove some a priori estimates of solutions to problem (26) and (27). Let ( α m 1 , , α m m ) be a solution of system (26) and (27) with a fixed parameter λ [ 0,1 ] . We multiply (26) by α m j and add the corresponding equalities for j = 1 , , m . Taking into account the equality (28) i = 1 n v i m v m , v m x i = 0 , we obtain (29) ψ λ I 2 v m D v m , D v m = λ f , v m + λ Γ c u 0 · v m d Γ c . Using (18) and (19), from (29) we obtain the estimate (30) a 1 v m X Ω , R n 2 f L 2 Ω , R n v m L 2 Ω , R n + u 0 L τ 2 Γ c , R n v m L τ 2 Γ c , R n C 2 f L 2 Ω , R n + C 1 u 0 L τ 2 Γ c , R n v m X Ω , R n . This yields that (31) v m X Ω , R n a 1 - 1 C 2 f L 2 Ω , R n + C 1 u 0 L τ 2 Γ c , R n .

Applying Lemma 7 to system (26) and (27), we see that problem (26) and (27) is solvable for any λ [ 0,1 ] and m N .

Let { v m } m = 1 be a sequence of vector functions that satisfy (26) and (27) with λ = 1 . It is clear that (32) - i = 1 n v i m v m , φ j x i + ψ I 2 v m D v m , D φ j = f , φ j + Γ c u 0 · φ j d Γ c , j = 1 , , m .

Note that estimate (31) is independent of m . This shows the existence of a vector function v 0 X ( Ω , R n ) and a subsequence m such that v m v 0 weakly in X ( Ω , R n ) . For the sake of simplicity, we assume that (33) v m v 0 weakly  in X Ω , R n   as m . Moreover, by the Sobolev embedding theorems, we have (34) v m v 0 strongly  in L 4 Ω , R n   as m .

Using (34), we get (35) i = 1 n v i m v m , φ j x i i = 1 n v i 0 v 0 , φ j x i as m . Therefore we can pass to the limit m in equality (32) and obtain (36) lim m ψ I 2 v m D v m , D φ j = i = 1 n v i 0 v 0 , φ j x i + f , φ j + Γ c u 0 · φ j d Γ c for any j N . Since { φ j } j = 1 is a basis of the space X ( Ω , R n ) , it follows that equality (36) remains valid if we replace φ j by an arbitrary vector function φ X ( Ω , R n ) : (37) lim m ψ I 2 v m D v m , D φ = i = 1 n v i 0 v 0 , φ x i + f , φ + Γ c u 0 · φ d Γ c .

Now we multiply (32) by α m j and add the corresponding equalities for j = 1 , , m . The result is (38) ψ I 2 v m D v m , D v m = f , v m + Γ c u 0 · v m d Γ c . Hence we find in the limit (39) lim m ψ I 2 v m D v m , D v m = f , v 0 + Γ c u 0 · v 0 d Γ c . Taking into account (20), (33), (37), and (39), we obtain the estimate (40) - μ i = 1 n v i 0 v 0 , φ j x i + μ ψ I 2 v 0 - μ φ j D v 0 - μ φ j , D φ j - μ f , φ j - μ Γ c u 0 · φ j d Γ c = - f , v 0 - Γ c u 0 · v 0 d Γ c + i = 1 n v i 0 v 0 , v 0 - μ φ j x i + f , v 0 - μ φ j + Γ c u 0 · v 0 - μ φ j d Γ c + μ ψ I 2 v 0 - μ φ j D v 0 - μ φ j , D φ j = - lim m ψ I 2 v m D v m , D v m + lim m ψ I 2 v m D v m , D v 0 - μ φ j + lim m ψ I 2 v 0 - μ φ j D v 0 - μ φ j , D v m - D v 0 - μ φ j = - lim m ψ I 2 v m D v m - ψ I 2 v 0 - μ φ j D v 0 - μ φ j , D v m - D v 0 - μ φ j 0 for any number μ > 0 . Multiplying the obtained inequality by μ - 1 , we get (41) - i = 1 n v i 0 v 0 , φ j x i + ψ I 2 v 0 - μ φ j D v 0 - μ φ j , D φ j - f , φ j - Γ c u 0 · φ j d Γ c 0 for any j N and μ > 0 .

Using Krasnoselskii’s theorem  on continuity of Nemytskii operators, we can pass to the limit μ 0 in (41): (42) - i = 1 n v i 0 v 0 , φ j x i + ψ I 2 v 0 D v 0 , D φ j - f , φ j - Γ c u 0 · φ j d Γ c 0 . Since { φ j } j = 1 is a basis of the space X ( Ω , R n ) , it follows that inequality (42) remains valid if we replace φ j by an arbitrary vector function φ X ( Ω , R n ) . Furthermore, since φ is an arbitrary vector function from the space X ( Ω , R n ) , we have (43) - i = 1 n v i 0 v 0 , φ x i + ψ I 2 v 0 D v 0 , D φ - f , φ - Γ c u 0 · φ d Γ c = 0 . This implies that the triplet ( v 0 , ψ ( I 2 ( v 0 ) ) D ( v 0 ) , u 0 ) is an admissible triplet of problem (1)–(7) and thus M .

We will show that M is bounded in the space X ( Ω , R n ) × L 2 ( Ω , M s n × n ) × L τ 2 ( Γ c , R n ) . Suppose ( v , S , u ) is an arbitrary triplet from M and φ = v . It follows from (23) that (44) ψ I 2 v D v , D v = f , v + Γ c u · v d Γ c . This yields that (45) v X Ω , R n a 1 - 1 C 2 f L 2 Ω , R n + C 1 sup w U w L τ 2 Γ c , R n . Moreover, taking into account (19), we obtain (46) S L 2 Ω , M s n × n = ψ I 2 v D v L 2 Ω , M s n × n a 2 D v L 2 Ω , M s n × n = a 2 v X Ω , R n a 2 a 1 - 1 C 2 f L 2 Ω , R n + C 1 s u p w U w L τ 2 Γ c , R n . Recall that the set U is bounded in L τ 2 ( Γ c , R n ) . Therefore from estimates (45) and (46) it follows that the set M is bounded in the space X ( Ω , R n ) × L 2 ( Ω , M s n × n ) × L τ 2 ( Γ c , R n ) .

Now we will show that the set M is sequentially weakly closed. Take a sequence { ( v k , S k , u k ) } k = 1 M such that v k v ^ weakly in X ( Ω , R n ) , S k S ^ weakly in L 2 ( Ω , M s n × n ) , and u k u ^ weakly in L τ 2 ( Γ c , R n ) as k . Let us check that ( v ^ , S ^ , u ^ ) M .

By definition, we have (47) - i = 1 n v i k v k , φ x i + ψ I 2 v k D v k , D φ = f , φ + Γ c u k · φ d Γ c for any φ X ( Ω , R n ) . Arguing as above, we conclude that (48) - i = 1 n v ^ i v ^ , φ x i + ψ I 2 v ^ D v ^ , D φ = f , φ + Γ c u ^ · φ d Γ c . From condition (iii), we get u ^ U . Thus, it remains to show that (49) S ^ = ψ I 2 v ^ D v ^ .

Since v k v ^ weakly in X ( Ω , R n ) , we see that (50) D v k D v ^ weakly  in L 2 Ω , R s n × n   as k . Note also that (51) ψ I 2 v k D v k , Φ = S k , Φ S ^ , Φ as k , for any Φ L 2 ( Ω , R s n × n ) .

Using the equality S k = ψ ( I 2 ( v k ) ) D ( v k ) , we rewrite (47) as follows: (52) - i = 1 n v i k v k , φ x i + S k , D φ = f , φ + Γ c u k · φ d Γ c . Passing to the limit k in this equality, we obtain (53) - i = 1 n v ^ i v ^ , φ x i + S ^ , D φ = f , φ + Γ c u ^ · φ d Γ c . Substituting v ^ for φ in (53), we get (54) S ^ , D v ^ = f , v ^ + Γ c u ^ · v ^ d Γ c . Further, substituting v k for φ in (47), we find (55) lim k ψ I 2 v k D v k , D v k = f , v ^ + Γ c u ^ · v ^ d Γ c . Combining this with (54), we obtain (56) lim k ψ I 2 v k D v k , D v k = S ^ , D v ^ . By [21, Chapter III, Lemma 1.3 ] and (50), (51), and (56), we get (49).

Applying the generalized Weierstrass theorem (see ), we conclude that there exists an element ( v , S , u ) M such that (57) J v , S , u = inf v , S , u M J v , S , u . This proves Theorem 6.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

The work of the first author was partially supported by Grant 16-31-00182 of the Russian Foundation of Basic Research.

Hinze M. Kunisch K. Second order methods for boundary control of the instationary Navier-Stokes system Zeitschrift für Angewandte Mathematik und Mechanik 2004 84 3 171 187 10.1002/zamm.200310094 2-s2.0-1542615357 Fursikov A. V. Flow of a viscous incompressible fluid around a body: boundary-value problems and minimization of the work of a fluid Journal of Mathematical Sciences 2012 180 6 763 816 10.1007/s10958-012-0670-1 2-s2.0-84856533258 Doubova A. Fernández-Cara E. On the control of viscoelastic Jeffreys fluids Systems and Control Letters 2012 61 4 573 579 10.1016/j.sysconle.2012.02.003 2-s2.0-84858715255 Baranovskii E. S. Solvability of the stationary optimal control problem for motion equations of second grade fluids Siberian Electronic Mathematical Reports 2012 9 1 554 560 10.13140/RG.2.1.3228.9364 2-s2.0-84875636699 Liu H. Boundary optimal control of time-periodic Stokes-Oseen flows Journal of Optimization Theory and Applications 2012 154 3 1015 1035 10.1007/s10957-012-0026-5 2-s2.0-84865443837 Baranovskiǐ E. S. An optimal boundary control problem for the motion equations of polymer solutions Siberian Advances in Mathematics 2014 24 3 159 168 10.3103/S105513441403002X 2-s2.0-84907376824 Artemov M. A. Optimal boundary control for the incompressible viscoelastic fluid system ARPN Journal of Engineering and Applied Sciences 2016 11 5 2923 2927 Litvinov V. G. Motion of a Nonlinear-Viscous Fluid 1982 Moscow, Russia Nauka Sobolevskii P. E. The existence of solutions of a mathematical model of a nonlinear viscous fluid Doklady Akademii Nauk SSSR 1985 285 1 44 48 Kuz'min M. Yu. A mathematical model of the motion of a nonlinear viscous fluid with the condition of slip on the boundary Russian Mathematics 2007 51 5 51 60 10.3103/S1066369X07050064 Zhikov V. V. New approach to the solvability of generalized Navier-Stokes equations Functional Analysis and Its Applications 2009 43 3 190 207 10.1007/s10688-009-0027-9 2-s2.0-71449092205 Litvinov W. G. Model for laminar and turbulent flows of viscous and nonlinear viscous non-Newtonian fluids Journal of Mathematical Physics 2011 52 5, article 053102 10.1063/1.3578752 2-s2.0-79957896608 Slawig T. Distributed control for a class of non-Newtonian fluids Journal of Differential Equations 2005 219 1 116 143 10.1016/j.jde.2005.03.009 2-s2.0-27944472740 Wachsmuth D. Roubíček T. Optimal control of planar flow of incompressible non-Newtonian fluids Zeitschrift für Analysis und ihre Anwendung 2010 29 3 351 376 10.4171/zaa/1412 2-s2.0-77954240012 Lions J. L. Control of Distributed Singular Systems 1985 Paris, France Gauthier-Villars Abergel F. Temam R. On some control problems in fluid mechanics Theoretical and Computational Fluid Dynamics 1990 1 6 303 325 10.1007/BF00271794 ZBL0708.76106 2-s2.0-34249958144 Fursikov A. V. Optimal Control of Distributed Systems 2000 Providence, RI, USA AMS Adams R. A. Fournier J. J. F. Sobolev Spaces 2003 40 Amsterdam, The Netherlands Academic Press Pure and Applied Mathematics Baranovskii E. S. On steady motion of viscoelastic fluid of Oldroyd type Sbornik: Mathematics 2014 205 6 763 776 10.1070/SM2014v205n06ABEH004397 Artemov M. A. Baranovskii E. S. Mixed boundary-value problems for motion equations of a viscoelastic medium Electronic Journal of Differential Equations 2015 2015 252 1 9 2-s2.0-84943240610 Gaevskii H. Greger K. Zaharias K. Nonlinear Operator Equations and Differential Operator Equations 1978 Moscow, Russia Mir Skrypnik I. V. Methods for Analysis of Nonlinear Elliptic Boundary Value Problems 1994 139 American Mathematical Society Translations of Mathematical Monographs Zeidler E. Nonlinear Functional Analysis and Its Applications, III: Variational Methods and Optimization 1985 New York, NY, USA Springer