Boundary Control Problem for Heat Convection Equations with Slip Boundary Condition

We analyze an optimal boundary control problem for heat convection equations in a three-dimensional domain, with mixed boundary conditions. We prove the existence of optimal solutions, by considering boundary controls for the velocity vector and the temperature. The analyzed optimal control problem includes the minimization of a Lebesgue norm between the velocity and some desired field, as well as the temperature and some desired temperature. By using the Lagrange multipliers theorem we derive an optimality system. We also give a second-order sufficient condition.


Introduction
Let Ω ⊂ R 3 be a simply connected bounded domain with Lipschitz boundary Γ.We deal with the existence of weak solutions of a boundary value problem describing the motion of a viscous heat conduction fluid in Ω, with a Navier slip condition on the boundary for the velocity vector.This model is given by the following system of partial differential equations: div u = 0 in Ω.
Here, u() is the velocity field, () represents the hydrostatic pressure, and () is the temperature of the fluid at point  ∈ Ω; the constant ] > 0 denotes the kinematic viscosity;  is the volume extension coefficient; G() is the free fall acceleration vector; the field f() represents external source of linear momentum; the constant  > 0 is the temperature conduction and () is the volume density of heat sources.Without loss of generality, we assume that density of the fluid is equal to one; thus if the variables in (1)-( 2) are nondimensionalized, then the coefficient of viscosity ] is simply the inverse of the Reynolds number Re (see, for example, [1] p. 166 and [2]).We prove the existence of weak solutions for system (1)- (3).Moreover, we study an optimal boundary control problem.To do this, we consider the following boundary conditions: Here the boundary Γ = Γ 1 ∪ Γ 2 ∪ Γ 3 = Γ 4 ∪ Γ 5 , where Γ 1 ∩ Γ 2 = Γ 1 ∩ Γ 3 = Γ 2 ∩ Γ 3 = Γ 4 ∩ Γ 5 = 0, and n denotes the outward normal vector on Γ.The function u 0 is defined on Γ 1 ;  ≥ 0 and  are functions givens on Γ 5 , and the functions g 1 ,  2 describe the Dirichlet boundary control for velocity u on Γ 2 and for temperature  on Γ 4 , respectively.The controls g 1 ,  2 lie in the closed convex sets U 1 ⊂ H 1/2 (Γ 2 ) and U 2 ⊂  1/2 (Γ 4 ), respectively.The term [(u)n + u] tang fl (u)n + u − [((u)n + u) ⋅ n]n represents the tangential component of the vector (u)n+u, where (u) fl (1/2)(∇u+∇  u) is the deformation tensor, and the real number  ≥ 0 is the friction coefficient which measures the tendency of the fluid to slip on Γ 3 .When  = 0 and the boundary is flat, the fluid slips along the Γ 3 without friction and there is no boundary layers.When  → +∞, the friction is so intense that the fluid is almost at rest near the boundary; the condition [(u)n + u] tang = 0, u ⋅ n = 0 on Γ 3 converges to the Dirichlet condition.
The case when  is a scalar function has been studied in [3], for the 2D nonstationary Navier-Stokes equations.The condition [(u)n + u] tang = 0, u ⋅ n = 0 on Γ 3 is a Navier friction boundary condition.The Navier boundary condition was proposed by Navier [4], who claimed that the tangential component of the viscous stress at the boundary should be proportional to the tangential velocity.Navier boundary condition was also derived by Maxwell [5] from the kinetic theory of gases and rigorously justified as a homogenization of the no-slip condition on a rough boundary [6].Before we describe in detail our results, let us briefly comment on previous related work.The heat convection equations, or also known as Boussinesq model, have been extensively analyzed, and, among the many articles on these equations, we just mention [7][8][9][10][11][12].In these papers are studied the existence of very weak, weak, and strong solutions and uniqueness with Dirichlet boundary conditions on the velocity and temperature or Dirichlet boundary condition on the velocity and mixed boundary conditions on the temperature.The tools used are Galerkin method, linearization, and fixed point results.The results obtained are similar to those known for the classical Navier-Stokes equations.
When Navier friction boundary condition is considered, the authors do not know results of existence for the classical Boussinesq model, except work due to Rionero and Mulone [13], where it is considered a slightly more general model on thermodiffusive mixture.
In the context of the classic Navier-Stokes equations there is a good amount of work with Navier friction boundary condition, among which we can mention [3,[14][15][16][17][18][19], and references therein, where the existence, regularity, and uniqueness of weak and strong solutions are studied.In the context of steady-state microstructure fluids, also called micropolar fluids, we can mention works [20,21].In [20] a threedimensional domain is considered where the density of the fluid is constant, while in [21] the study is performed on a two-dimensional domain with variable density.
For optimal control problems for the classical Boussinesq stationary model, see [22][23][24][25][26][27], and the references therein.In such works, standard results have been obtained, such as the existence of an optimal solution and stability as well as first-order necessary optimality conditions, from which the authors derive an optimality system.However, as far as we know, for this model there are no known studies of sufficient second-order conditions for the existence of a local extreme, except for the recent work [28], in which the authors study a problem of control of limits associated with the stationary system of Rayleigh-Bénard-Marangoni (similar to system (1)-( 3)), for which they obtain results of existence, uniqueness, and regularity of the model; in addition they provide the existence of optimal solutions, derive optimality conditions, obtain a second-order sufficient condition, and establish a result of uniqueness for the optimal solution.
The outline of this paper is as follows.In Section 2, we introduce the spaces of functions appropriate for the development of the article.In Section 3, we prove the existence and uniqueness of weak solutions to model ( 1)-( 4), by using the Galerkin method.In Section 4, we establish the optimal control problem and also we prove the existence of optimal solutions, and, using the Lagrange multipliers method, we obtain the first-order optimality conditions, from which we derive an optimality system.In Section 5, we obtain a second-order sufficient optimality condition.
In order to establish the weak formulation of problem (1)-( 4) we consider the following result.
Lemma 1 (see [19]).Let u ∈ H 2 (Ω) and k ∈ H 1 (Ω) be divergence-free vector fields tangent to the boundary Γ.Then, Then, motivated by the formula of integration by parts and Lemma 1, we establish the following definition of weak solution of system (1)-(4).

Existence of a Weak Solution to Problem (1)-(4)
In order to prove the existence of a solution to system (7)-( 9), we reduce the problem to an auxiliary problem with homogeneous conditions for the fields velocity on Γ \ Γ 3 and the temperature on Γ 4 .For this, we introduce the following results.
where  is a positive constant depending on Ω such that ‖∇u‖ ≤ ‖u‖ H1  and  is given in (11), then, there exists (û, T) ∈ X, which is a solution of system (13).
Moreover, the following estimate is satisfied: where Proof.The existence of a solution will be proved by using the Galerkin method.For this purpose, we consider an approximate solution of ( 13) and then we pass to the limit.Since H1  and  0 are separable spaces, there exist Hilbert spaces bases {k  } ∞ =1 and {  } ∞ =1 of H1  and  0 , respectively.For each  ∈ N, we consider H the space spanned by {k 1 , . . ., k  } and   the space spanned by { 1 , . . .,   }; let X  = H ×   ⊂ X.The scalar product on X  is the scalar product (⋅, ⋅) X  induced by X.
Remark 7. Since Re = 1/], then condition (15) shows us that the existence of solution for the system (7)-( 9) is guaranteed when the Reynolds number is small (the flow is laminar).
Theorem 8 (uniqueness of solutions).Under conditions of Theorem 6, if ],  are large enough and ‖‖  ∞ (Γ 5 ) is small enough such that where  is a positive constant that depends on the domain Ω and C1 , Θ are given in (30), then the solution (u, ) ∈ H 1  ×  1 (Ω) of system ( 7)-( 9) provided by Theorem 6 is unique.
Remark 9.The condition (35) assures us of the uniqueness of solution of system ( 7)-( 9) when the viscosity ] and the temperature conduction coefficient  are large enough, which implies that the Rayleigh number is small.That is, the uniqueness of weak solution is obtained when the heat transfer is produced primarily by conduction.

The Optimal Control Problem
In this section the statement of the boundary control problem to study is established.We suppose that U 1 ⊂ H1/2 00 (Γ 2 ) and U 2 ⊂  1/2 00 (Γ 4 ) are nonempty sets.We consider data ), and the functions g 1 ∈ U 1 ,  2 ∈ U 2 , describing the Dirichlet boundary controls for u on Γ 2 and  on Γ 4 , respectively.For simplicity, here and hereinbelow, we will use the product space H = H 1  ×  1 (Ω), and we consider the following objective functional  : where Ĵ : H 1  × 1 (Ω) → R is a weakly lower semicontinuous functional, and the nonnegative constants  1 ,  2 measure the cost of the controls.
The following are examples of the weakly lower semicontinuous functional in (43) most interesting from a physical viewpoint: where u  ∈ L 2 (Ω) and   ∈  2 (Ω) are the desired states.The functionals  1 and  2 describe, respectively, the deviation of the velocity of flow from a given velocity and the deviation of the temperature from a given temperature.The functional  3 describes the resistance in a fluid due to viscous friction (see [22]).
For simplicity we will do the study with Ĵ =  1 +  2 ; also the study can be done considering Ĵ as the sum of two or more functionals given in (44).Thus, we define the following constrained minimization problem related to system (7)-( 9): reaches its minimum over the weak solutions of system ( 1)-( 4). ( The of admissible solutions of optimal control problem (45) is defined by where U ad fl U 1 × U 2 .A similar control problem has been studied in [20,21] for micropolar fluids.In [20] a threedimensional domain was considered with density constant; while in [21] the problem was studied in a two-dimensional domain, where the fluid density was considered as an extra variable in the system.

Existence of Optimal Solution.
In this subsection we will prove the existence of an optimal solution for problem (45).For this purpose we introduce the following result.The proof follows a technique that is standard for optimal control problems.We will repeat it for convenience of the reader.
Theorem 10.Under the hypotheses of Theorem 6, and also assuming that the conditions are satisfied: Proof.From Theorem 6 we have that the set of admissible solutions  ad ̸ = 0, and since the functional  is bounded below, there exists a minimizing sequence {s  = (u  ,   , g  1 ,   2 )} ≥1 ∈ S ad ,  ∈ N, such that lim →∞ (s  ) = inf s∈S ad (s).Also, by definition of set S ad , we have the fact that s  satisfies system (7)- (9); that is, (50) by standard Sobolev embeddings, we obtain that → T strongly in  2 (Ω) , Moreover, since u  = u 0 on Γ 1 , u  = g  1 on Γ 2 , and   =   2 on Γ 4 , then from (51) it follows that ũ = u g1 on Γ \ Γ 3 and T = g2 on Γ 4 ; thus,s the boundary conditions given in (9).A standard procedure permits passing the limit in (51), as  goes to ∞, and proving that s is solution of system ( 7)-( 9).Consequently we have s ∈ S ad and lim →∞  (s  ) = inf s∈ ad
Recall that problem (45) is a problem with constraint; one method to obtain necessary conditions of optimality is Lagrange method, which consists of transforming the problem (45) into an unrestricted one, using an auxiliary function (Lagrange functional or Lagrangian) and additional variables, called Lagrange multipliers.The Lagrangian associated with optimal control problem (45) is given by where  = ((, ), , ) ∈ X × H−1/2 00 (Γ \ Γ 3 ) ×  −1/2 00 (Γ 4 ).Concerning differentiation of the objective functional  and the constraint operator F, we have the following results.
Definition 13.Let s = ((ũ, T), (g 1 , g2 )) ∈ H × U ad be an admissible point for the control problem (55).We say that s is a regular point of the operator F, defined by (54), if where In the following lemma, we give a condition to assure that s ∈ S ad is a regular point for optimal control problem (45).Thereafter the existence of Lagrange multipliers is shown.Lemma 14.Let s = (ũ, T, g1 , g2 ) ∈ S  be a feasible solution for the optimal problem (45).If ],  are sufficiently large and where the constant  > 0 depends on the domain Ω, then s is a regular point.

Mathematical Problems in Engineering
In order to prove the existence of a solution for system (64) we define the bilinear form  : X × X → R by  ((ω, ψ) , (k, )) fl 2] ( (ω) ,  (k)) and the linear and continuous functional F : X → R by Thus, from (66)-(67), system (64) is equivalent to the following.Find (ω, ψ) ∈ X such that We notice that the form (⋅, ⋅) is continuous.Now, by taking (k, ) = (ω, ψ) in (66), we have By applying the Hölder and Young inequalities to the terms in the right-hand side of (69), we have Then, by using the last inequalities in (69), we have where, by hypotheses, Thus, we obtain that the form (⋅, ⋅) is X-coercive; therefore, from (68), (71), and the Lax-Milgram theorem we conclude the existence of a unique (ω, ψ) ∈ X solution of system (64), and, consequently, we obtain that (, ) ∈ H is solution of (63).
In the following result we prove the existence of Lagrange multipliers provided that a local optimal solution s = (ũ, T, g1 , g2 ) ∈ S ad verifies the regular point condition provided by Lemma 14.
Remark 17.Since the set of controls U  is convex, then from (80) we deduce (87)

Second-Order Sufficient Condition
In this section, we will discuss sufficient conditions for s = (ũ, T, g1 , g2 ) ∈ S ad being a local optimal solution for control problem (45).We will establish a H × U ad −coercivity condition on the second derivative of the Lagrange functional L given in (56) associated with problem (45) in order to assure that an admissible point s is a local optimal solution.Due to Lemmas 11 and 12, the functional L is Fréchet differentiable with respect to the point s = ((u, ), (g 1 ,  2 )) ∈ H × U ad .

Conclusions
In this paper, the existence and uniqueness of weak solutions of the stationary heat convection equations with mixed boundary conditions, including the Navier slip condition, were proved.Also, an optimal boundary control problem was analyzed.Boundary controls for the velocity vector and temperature and the existence of optimal solutions were proved.The optimal control problem analyzed includes the minimization of the  2 -distance between the velocities and some desired fields.By using the theorem of Lagrange multipliers, an optimality system was derived.A secondorder sufficient condition was also given.