Stability of Optimal Controls for the Stationary Boussinesq Equations

The stationary Boussinesq equations describing the heat transfer in the viscous heat-conducting fluid under inhomogeneous Dirichlet boundary conditions for velocity and mixed boundary conditions for temperature are considered. The optimal control problems for these equations with tracking-type functionals are formulated. A local stability of the concrete control problem solutions with respect to some disturbances of both cost functionals and state equation is proved.


Introduction
Much attention has been recently given to the optimal control problems for thermal and hydrodynamic processes.In fluid dynamics and thermal convection, such problems are motivated by the search for the most effective mechanisms of the thermal and hydrodynamic fields control 1-4 .A number of papers are devoted to theoretical study of control problems for stationary models of heat and mass transfer see e.g., [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] .A solvability of extremum problems is proved, and optimality systems which describe the necessary conditions of extremum were constructed and studied.Sufficient conditions to the data are established in 16,18,19 which provide the uniqueness and stability of solutions of control problems in particular cases.
Along with the optimal control problems, an important role in applications is played by the identification problems for heat and mass transfer models.In these problems, unknown densities of boundary or distributed sources, coefficients of model differential equations, or boundary conditions are recovered from additional information of the original boundary value problem solution.It is significant that the identification problems can be reduced to appropriate extremum problems by choosing a suitable tracking-type cost functional.As a result, both control and identification problems can be studied using International Journal of Differential Equations an unified approach based on the constrained optimization theory in the Hilbert or Banach spaces see 1-4 .The main goal of this paper is to perform an uniqueness and stability analysis of solutions to control problems with tracking-type functionals for the steady-state Boussinesq equations.We shall consider the situation when the boundary or distributed heat sources play roles of controls and the cost functional depends on the velocity.Using some results of 2 we deduce firstly the optimality system for the general control problem which describes the first-order necessary optimality conditions.Then, based on the optimality system analysis, we deduce a special inequality for the difference of solutions to the original and perturbed control problems.The latter is obtained by perturbing both cost functional and one of the functions entering into the state equation.Using this inequality, we shall establish the sufficient conditions for data which provide a local stability and uniqueness of solutions to control problems under consideration in the case of concrete tracking-type cost functionals.
The structure of the paper is as follows.In Section 2, the boundary value problem for the stationary Boussinesq equations is formulated, and some properties of the solution are described.In Section 3, an optimal control problem is stated, and some theorems concerning the problem solvability, validity of the Lagrange principle for it, and regularity of the Lagrange multiplier are given.In addition, some additional properties of solutions to the control problem under consideration will be established.In Section 4, we shall prove the local stability and uniqueness of solutions to control problems with the velocity-tracking cost functionals.Finally, in Section 5, the local uniqueness and stability of optimal controls for the vorticity-tracking cost functional is proved.

Statement of Boundary Problem
In this paper we consider the model of heat transfer in a viscous incompressible heatconducting fluid.The model consists of the Navier-Stokes equation and the convectiondiffusion equation for temperature that are nonlinearly related via buoyancy in the Boussinesq approximation and via convective heat transfer.It is described by equations Here Ω is a bounded domain in the space R d , d 2, 3 with a boundary Γ consisting of two parts Γ D and Γ N ; u, p, and T denote the velocity and temperature fields, respectively; p P/ρ, where P is the pressure and ρ const > 0 is the density of the medium; ν is the kinematic viscosity coefficient, G is the gravitational acceleration vector, β is the volumetric thermal expansion coefficient, λ is the thermal conductivity coefficient, g is a given vector-function on Γ, ψ is a given function on a part Γ D of Γ, χ is a function given on another part Γ N Γ \ Γ D of Γ, n is the unit outer normal.We shall refer to problem 2.1 -2.3 as Problem 1.We note that all quantities in 2.1 -2.3 are dimensional and their dimensions are defined in terms of SI units.
International Journal of Differential Equations 3 We assume that the following conditions are satisfied: Below we shall use the Sobolev spaces H s D and L 2 D , where s ∈ R, or H s D and L 2 D for the vector functions where D denotes Ω, its subset Q, Γ or a part Γ 0 of the boundary Γ.In particularly we need the function spaces

2.4
The inner products and norms in Let in addition to condition i the following conditions hold: The following technical lemma holds see 2, 20 .

International Journal of Differential Equations
Besides the following identities take place: in addition to i , ii .We multiply the equations in 2.1 , 2.2 by test functions v ∈ H 1 0 Ω and S ∈ T and integrate the results over Ω with use of Green's formulas to obtain the weak formulation for the model

2.15
Following theorem see 2 establishes the solvability of Problem 1 and gives a priori estimates for its solution.

2.16
Here M u , M p and M T are nondecreasing continuous functions of the norms If, additionally, f, g, χ, f, ψ, α are small in the sense that where δ 0 , δ 1 , γ 0 , γ 1 and β 1 are constants entering into 2.5 -2.7 , then the weak solution to Problem 1 is unique.

Statement of Control Problems
Our goal is the study of control problems for the model 2.1 -2.3 with tracking-type functionals.The problems consist in minimization of certain functionals depending on the state and controls.As the cost functionals we choose some of the following ones: Here Q is a subdomain of Ω.The functionals I 1 , I 2 , and I 3 where functions are interpreted as measured velocity or vorticity fields are used to solve the inverse problems for the models in questions 2 .
In order to formulate a control problem for the model 2.1 -2.3 we split the set of all data of Problem 1 into two groups: the group of controls containing the functions χ ∈ L 2 Γ N , ψ ∈ H 1/2 Γ D , and f ∈ L 2 Ω , which play the role of controls and the group of fixed data comprising the invariable functions f, b, and α.As to the function g entering into the boundary condition for the velocity in 2.3 , it will play peculiar role since the stability of solutions to control problems under consideration see below will be studied with respect to small perturbations, both the cost functional and the function g in the norm of Denote by I : H 1 Ω → R a weakly lower semicontinuous functional.We assume that the controls χ, ψ, and f vary in some sets Here μ 0 , μ 1 , μ 2 , μ 3 are nonnegative parameters which serve to regulate the relative importance of each of terms in 3.2 and besides to match their dimensions.Another goal of introducing parameters μ i is to ensure the uniqueness and stability of the solutions to control problems under study see below .
We assume that following conditions take place: iii Considering the functional J at weak solutions to Problem 1 we write the corresponding constraint which has the form of the weak formulation 2.13 -2.15 of Problem 1 as follows:

3.4
The mathematical statement of the optimal control problem is as follows: to seek a pair x, u , where x u, p, T ∈ X and u χ, ψ, f

3.7
Here and below ζ, Γ D and κ is an auxiliary dimensional parameter.Its dimension κ is chosen so that dimensions of ξ, σ, θ at the adjoint state coincide with those at the basic state, that is, Here L 0 , T 0 , M 0 , K 0 denote the SI dimensions of the length, time, mass, and temperature units expressed in meters, seconds, kilograms, and degrees Kelvin, respectively.As a result ξ, σ, and θ can be referred to below as the adjoint velocity, pressure, and temperature.Simple analysis shows see details in 16 that the necessity for the fulfillment of 3.8 is that κ is given by κ L 2 0 T −2 0 K −2 0 .The following theorems see, e.g., 2 give sufficient conditions for the solvability of control problem 3.5 , the validity of the Lagrange principle for it, and a regularity condition for a Lagrange multiplier.
for the adjoint state y * is satisfied and the minimum principle holds which is equivalent to the inequality L x, u, λ 0 , y * , g ≤ L x, u, λ 0 , y * , g ∀u ∈ K.

3.10
Theorem 3.3.Let the assumptions of Theorem 3.2 be satisfied and condition 2.17 holds for all u ≡ χ, ψ, f ∈ K. Then any nontrivial Lagrange multiplier satisfying 3.9 is regular, that is, has the form 1, y * and is uniquely determined.
We note that the functional J and Lagrangian L given by 3.7 are continuously differentiable functions of controls χ, ψ, f and its derivatives with respect to χ, ψ, and f are given by

3.11
Here for example L χ x, u, λ 0 , y * , g is the Gateaux derivative with respect to χ at the point •, λ 0 , y * , g the following conditions are satisfied see 22 :

3.12
We also note that the Euler-Lagrange equation 3.9 is equivalent to identities

3.13
Relations 3.13 , the minimum principle which is equivalent to the inequalities 3.10 or 3.12 , and the operator constraint 3.3 which is equivalent to 2.13 -2.15 constitute the optimality system for control problem 3.5 .Theorems 3.1 and 3.2 above are valid without any smallness conditions in relation to the data of Problem 1.The natural smallness condition 2.17 arises only when proving the uniqueness of solution to boundary problem 2.1 -2.3 and Lagrange multiplier regularity.However, condition 2.17 does not provide the uniqueness of problem 3.5 solution.

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Therefore, an investigation of problem 3.5 solution uniqueness is an interesting and complicated problem.Studying of its solution stability with respect to small perturbations of both cost functional I entering into 3.2 and state equation 3.3 is also of interest.In order to investigate these questions we should establish some additional properties of the solution for the optimality system 2.13 -2.15 , 3.12 , 3.13 .Based on these properties, we shall impose in the next section the sufficient conditions providing the uniqueness and stability of solutions to control problem 3.5 for particular cost functionals introduced in 3.1 .
Let us consider problem 3.5 .We assume below that the function g entering into 2.3 can vary in a certain set

3.14
It is obtained by replacing the functional I in 3.5 by a close functional I depending on u and by replacing a function g ∈ G by a close function g ∈ G.
By Theorem 3.1 the following estimates hold for triples u i , p i , T i :

3.15
Here where M u , M p , and M T are introduced in Theorem 3.1.We introduce "model" Reynolds number Re, Raley number Ra, and Prandtl number P by They are analogues of the following dimensionless parameters widely used in fluid dynamics: the Reynolds number Re, the Rayleigh number Ra, and the Prandtl number Pr.We can show that the parameters introduced in 3.17 are also dimensionless if u , |u| 1 , and u 1 where u is an arbitrary scalar are defined as Here l is a dimensional factor of dimension l L 0 whose value is equal to 1. Assume that the following condition takes place:

3.19
Let us denote by 1, y * i , where , Lagrange multipliers corresponding to solutions x i , u i .By Theorems 3.2 and 3.3 and 3.12 they satisfy relations We renamed

International Journal of Differential Equations
Set further v ξ in 3.25 , S κθ in 3.26 , and subtract obtained relations from 3.29 .Using inequality 3.28 and arguing as in 18 , we obtain

3.30
Thus we have proved the following result.
Below we shall need the estimates of differences Here C 0 is a constant depending on Ω.The existence of u 0 follows from 20, page 24 .We present the difference u ≡ u 1 − u 2 as u u 0 u, where u ∈ V is a new unknown function.Set u u 0 u, v u in 3.25 .Taking into account 2.9 we obtain Using estimates 2.5 , 2.6 , 2.7 , and 3.15 , we deduce from 3.31 that

3.32
It follows from 3.19 that

3.33
Rewriting the inequality 3.32 by 3.33 as

3.35
Taking into account the relation u u 0 u, we come to the following estimate u 1 via g 1/2,Γ and T 1 :

3.36
Denote by

3.37
Using estimates 2.5 -2.8 and 3.15 we deduce that

3.39
Taking into account the relation T T 0 T , we obtain from this estimate that

3.40
Using further the estimate 3.36 for u, we deduce from 3.40 that

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From this inequality and 3.17 , 3.19 we come to the following estimate: Using 3.42 , we deduce from 3.36 that

3.43
Taking into account 3.17 we come to the following estimate for u 1 : An analogous estimate holds and for the pressure difference p p 1 − p 2 .In order to establish this estimate we make use of inf-sup condition 2.10 .By 2.10 for the function p p 1 − p 2 and any small number δ > 0 there exists a function v 0 ∈ H 1 0 Ω , v 0 / 0, such that − div v 0 , p ≥ β 0 v 0 1 p where β 0 β − δ > 0. Set v v 0 in the identity for u in 3.25 and make of this estimate and estimates 2.6 , 2.7 , 3.15 .We shall have

3.46
Using 3.42 and 3.44 , we come to the following final estimate for p :

3.47
International Journal of Differential Equations 13 Remark 3.5.Along with three-parametric control problem 3.5 we shall consider and oneparametric control problem which corresponds to situation when a function u χ is a unique control.This problem can be considered as particular case of the general control problem 3.5 , for which the set K 2 consists of one element ψ 0 ∈ H 1/2 Γ D and the set K 3 consists of one element f 0 ∈ L 2 Ω .For this case the conditions f ≡ f 1 − f 2 0, ψ ≡ ψ 1 − ψ 2 0 take place, and the estimates 3.42 -3.47 and inequality 3.30 take the form 48 3.51

Control Problems for Velocity Tracking-Type Cost Functionals
Based on Theorem 3.4 and estimates 3.42 -3.47 or 3.48 -3.50 , we study below uniqueness and stability of the solution to problem 3.5 for concrete tracking-type cost functionals.We consider firstly the case mentioned in Remark 3.5 where I I 1 and the heat flux χ on the part Γ N of Γ is a unique control; that is, we consider one-parametric control problem

4.1
In accordance to Remark 3.5 we can consider problem 4.1 as a particular case of the general control problem 3.5 , which corresponds to the situation when every of sets K 2 and K 3 consists of one element. Let in addition to 3.24 we note that under conditions of problem 4.1 we have Using identities 4.3 , 4.4 , 3.22 we estimate parameters ξ i , θ i , σ i and ζ i .Firstly we deduce estimates for norms ξ i 1 and θ i 1 .To this end we set w ξ i , τ θ i in 4.3 , 3.22 .Taking into account 2.11 , 2.12 , and condition ξ i ∈ V, which follows from 4.4 , we obtain Using estimates 2.5 -2.8 and 3.15 we have where By virtue of 4.8 -4.10 and 4.12 , we deduce from 4.7 and 4.6 that

4.15
Taking into account 4.14 , we obtain from 4.15 that Using 3.33 we deduce successfully from 4.16 , 4.14 that Let us estimate further the norms σ i and ζ i −1/2,Γ from 4.3 .In order to estimate σ i we make use of inf-sup condition 2.10 .By 2.10 for a function σ i ∈ L 2 0 Ω and any small number δ > 0 there exists a function v i ∈ H 1 0 Ω , v i / 0, such that the inequality holds.Setting in 4.3 w v i and using this estimate together with estimates 2.6 , 3.15 , 4.11 , we have

4.19
From this inequality we deduce by 4.17 that

4.20
Taking into account 4.17 , we come from 4.20 to the estimate

International Journal of Differential Equations
As ζ ζ 1 − ζ 2 we obtain from this inequality that Re Re 0 2R 2Ra 1 .

4.24
It follows from 4.24 that

4.26
Let the data for problem 4.1 and parameters μ 0 , μ 1 be such that with a certain constant ε > 0 the following condition takes place: Under condition 4.27 we deduce from 4.25 that

4.28
Taking into account 4.28 and the estimate ,Γ which follows from 4.23 , we come from 4.5 to the inequality

4.29
It follows from this inequality that Excluding nonpositive term −εμ 1 χ 2 Γ N from the right-hand side of 4.30 , we deduce from 4.30 that 4.31 Equation 4.31 is a quadratic inequality for u Q .Solving it we come to the following estimate for u Q :

4.32
As the estimate 4.32 is equivalent to the following estimate for the velocity difference u 1 − u 2 :

4.33
This estimate under Q Ω has the sense of the stability estimate in L 2 Ω of the component u of the solution u, p, T, χ to problem 4.1 relative to small perturbations of functions v d ∈ L 2 Ω and g ∈ G in the norms of L 2 Ω and H 1/2 Γ , respectively.In particular case where g 1 g 2 the estimate 4.33 transforms to "exact" a priori estimate It was obtained when studying control problems for Navier-Stokes and in 18 when studying control problems for heat convection equations.If besides u This yields together with 4.30 , 3.48 , 3.50 that χ 1 χ 2 , T 1 T 2 , p 1 p 2 .The latter means the uniqueness of the solution to problem 4.1 when Q Ω and condition 4.27 holds.
It is important to note that the uniqueness and stability of the solution to problem 4.1 under condition 4.27 take place and in the case where Q ⊂ Ω; that is, Q is only a part of domain Ω.In order to prove this fact let us consider the inequality 4.30 .Using 4.32 we deduce from 4.30 that

4.34
International Journal of Differential Equations From 4.34 and 3.48 -3.50 we come to the following stability estimates: 35 36 37 where

4.39
Thus we have proved the theorem.
Theorem 4.1.Let, under conditions (i), (ii), (iii) for K 1 and 3.19 , the quadruple u i , p i , T i , χ i be a solution to problem 4.1 corresponding to given functions

4.42
Here parameters γ and Re 0 are given by 4.13 .From 4.42 we obtain that

4.43
Here constants b, c 1 , c 2 , and c 3 are given by relations

International Journal of Differential Equations
Let the data for problem 4.40 and parameters μ 0 , μ 1 , μ 2 , and μ 3 be such that Under condition 4.45 we deduce from 4.43 that

4.46
Taking into account 4.46 and 4.23 , we come from 4.41 to the inequality It follows from this inequality that We note again that the uniqueness and stability of the solution to problem 4.40 under condition 4.45 take place and in the case Q ⊂ Ω where Q is only a part of the domain Ω.In order to establish this fact we consider inequality 4.48 which we rewrite taking into account 4.32 as

4.49
From this inequality and from 3.42 -3.47 we come to the following stability estimates: 51 52 Here a constant d depending on μ 1 , μ 2 , and μ 3 is given by and a quantity Δ is defined in 4.39 .Thus the following theorem is proved.In the same manner one can study control problem in addition to 3.24 we note that under conditions of problem 4.55 we have

4.59
Using estimates 3.15 we deduce in addition to 4.8 -4.10 that

4.60
where It follows from this inequality that

4.63
Excluding nonpositive term −εμ 1 χ 2 Γ N , we deduce from 4.63 that Equation 4.31 is a quadratic inequality relative to u 1,Q .By solving it we come to the estimate 4.65 which is equivalent to the following estimate for u 1 − u 2 : We note again that using 4.63 , 4.65 we can deduce rougher stability estimates of the solution to problem 4.55 which take place even in the case where Q / Ω.In fact we deduce from 4.63 4.65 that

4.67
From 4.67 and 3.48 -3.50 we come to the estimates 4.35 -4.38 where one should set

International Journal of Differential Equations
In the similar way one can study three-parametric control problem

4.69
It

Control Problem for Vorticity Tracking-Type Cost Functional
Consider now one-parametric control problem Using identities 5.3 , 3.22 , 4.4 we estimate parameters ξ i , θ i , σ i , and ζ i .Firstly we deduce estimates of norms ξ i 1 and θ i 1 .To this end we set w ξ i , τ θ i in 5.3 , 3.22 .Taking into account 2.11 , 2.12 and condition ξ i ∈ V, which follows from 4.4 , we obtain 4.7 and relation Using 2.9 , 3.15 we deduce in addition to 4.8 -4.10 that where Arguing as above in analysis of problem 4.1 we come to the same estimates 4.17

5.8
It follows from this inequality that Excluding nonpositive term −εμ 1 χ 2 Γ N , we deduce from 5.9 that Equation 5.10 is a quadratic inequality relative to rot u Q .Solving it we come to the estimate which is equivalent to the following estimate for the difference rot u 1 − rot u 2 :

5.12
International Journal of Differential Equations The estimate 5.12 under Q Ω has the sense of the stability estimate in the norm L 2 Ω of the vorticity curl u of the component u of the solution u, p, T, χ to problem 5.1 relative to small perturbations of functions ζ d ∈ L 2 Ω and g ∈ H 1/2 Γ in the norms of L 2 Ω and H 1/2 Γ , respectively.In particular case where ζ If Q / Ω we can deduce from 5.11 and 5.9 rougher stability estimates of the solution to problem 5.1 , which are analogous to estimates 4.35 -4.38 .In fact using 5.11 we deduce from 5.9 that .

5.13
From 5.13 and 3.48 -3.50 we come to the estimates 4.35 -4.38 where .

5.14
Thus the following theorem is proved.
Theorem 5.1.Let, under conditions (i), (ii), (iii) for K 1 and 3.19 , the quadruple u i , p i , T i ,χ i be a solution to problem 5.1 corresponding to given functions ζ i d ∈ L 2 Q and g i ∈ G, i 1, 2, where Q ⊂ Ω is an arbitrary open subset, and let parameters a and b, c be defined in relations 4.23 and 4.26 , in which γ and Re 0 are given by 5.7 .Suppose that condition 4.27 is satisfied.Then the stability estimates 5.12 and 4.35 -4.38 hold true where Δ is defined in 5.14 .
In the similar way one can study three-parametric control problem

5.15
It is obtained

Conclusion
In this paper we studied control problems for the steady-state Boussinesq equations describing the heat transfer in viscous heat-conducting fluid under inhomogeneous Dirichlet boundary conditions for velocity and mixed boundary conditions for temperature.These problems were formulated as constrained minimization problems with tracking-type cost functionals.We studied the optimality system which describes the first-order necessary optimality conditions for the general control problem and established some properties of its solution.In particular we deduced a special inequality for the difference of solutions to the original and perturbed control problem.The latter is obtained by perturbing both the cost functional and the boundary function entering into the Dirichlet boundary condition for the velocity.Using this inequality we found the group of sufficient conditions for the data which provide a local stability and uniqueness of concrete control problems with velocity-tracking or vorticity-tracking cost functionals.This group consists of two conditions: the first is the same for all control problems and has the form of the standard condition 3.19 which ensures the uniqueness of the solution to the original boundary value problem for the Boussinesq equations.The second one depends on the form of control problem under study.In particular for the one-parametric problem 4.1 corresponding to velocity-tracking functional I 1 v it has the form of estimates 4.27 of the parameters μ 0 and μ 1 included in 4.1 , while for the three-parametric problem 4.40 it has the form of estimates 4.45 of the parameters μ 0 , μ 1 , μ 2 , and μ 3 included in 4.40 .Similar conditions take place for another tracking-type functionals.
On the one hand, conditions 4.27 and 4.45 are similar to the uniqueness and stability conditions for the solution to the coefficient identification problems for the linear convection-diffusion-reaction equation.On the other hand, these conditions contain compressed information on the Boussinesq heat transfer model 2.1 , 2.
where Q ⊂ Ω is an arbitrary open subset, and let the parameters a and b, c are defined in 4.23 and 4.26 in which parameters γ and Re 0 are given by 4.13 .Suppose that condition 4.27 is satisfied.Then stability estimates 4.33 and 4.35 -4.38 hold true where Δ is defined in 4.39 .Now we consider three-parametric control problem

2 d
in addition to 3.24 , we note that under conditions of problem 4.40 identities 3.20 and 3.21 transform to identities 4.3 , 4.4 , identity 3.22 does not change, while inequality 3.30 takes by 4.2 a form nonpositive terms from the right-hand side of 4.48 , we come to the inequality 4.31 where constants a and b are defined in 4.23 and 4.44 .From 4.31 we deduce the estimate 4.32 for u Q with mentioned constants a and b given by 4.23 and 4.44 .As in the case of problem 4.1 , stability in the norm L 2 Ω of the component u of the solution to problem 4.40 relative to small perturbations of functions v d ∈ L 2 Ω and g ∈ G in the norms of L 2 Ω and H 1/2 Γ , respectively, and uniqueness of the solution to problem 4.40 follow from 4.32 in the case when Q Ω and 4.45 holds.

Theorem 4 . 2 .
Let, under conditions (i), (ii), (iii), and 3.19 , an element u i , p i , T i , χ i , ψ i , f i be a solution to problem 4.40 corresponding to given functions v d u i d ∈ L 2 Q and g i ∈ G, i 1, 2, where Q is an arbitrary open subset, and let parameters a and b, c 1 , c 2 , c 3 be defined in 4.23 and 4.44 , where γ and Re 0 are given by 4.13 .Suppose that conditions 4.45 are satisfied.Then stability estimates 4.33 and 4.50 -4.53 hold where Δ and d are defined in 4.39 and 4.54 .

1 d ∈ L 2 Q 2 d ∈ L 2 Q 1 d − u 2 d
χ 1 be a solution to problem 5.1 corresponding to given functions ζ d ≡ ζ and g g 1 ∈ G, and let x 2 , u 2 ≡ u 2 , p 2 , T 2 , χ 2 be a solution to problem 4.1 corresponding to perturbed functions ζ d ≡ ζ and g g 2 ∈ G. Setting u d u in addition to 3.24 , we have under conditions of problem 4.1 ,

1 d ζ 2 d
and g 1 g 2 it follows from 5.11 that rot u 1 rot u 2 in Ω, if Q Ω.From this relation and from 4.30 , 3.48 , 3.50 it follows that χ 1 χ 2 , T 1 T 2 , p 1 p 2 .The latter means the uniqueness of the solution to problem 4.1 when Q Ω and condition 4.27 holds.

from 4 .Theorem 5 . 2 .
40 by replacing the cost functional I 1 v by I 3 v .The following theorem holds.Let, under conditions (i), (ii), (iii), and 3.19 , an element u i , p i , T i ,χ i , ψ i , f i be a solution to problem 5.15 corresponding to given functionsζ d ζ i d ∈ L 2 Q and g i ∈ G, i 1, 2, where Q ⊂ Ω isan arbitrary open subset, and let parameters a and b, c 1 , c 2 , c 3 be given by relations 4.23 and 4.44 , in which γ and Re 0 be defined in 5.7 .Suppose that conditions 4.45 are satisfied.Then the stability estimates 5.12 and 4.50 -4.53 hold where Δ and d are defined in 5.14 and 4.54 .
2 in the form of the constant c defined in 4.26 for problem 4.1 or in the form of three constants c 1 , c 2 , c 3 defined in 4.44 for problem 4.40 .An analysis of the expressions for c or c 1 , c 2 , c 3 shows that for fixed values of the parameters μ l inequality 4.27 or inequalities 4.45 represent additional constraints on the Reynolds number Re, Rayleigh number Ra, and Prandtl number P which together with 3.19 ensure the uniqueness and stability of the solution to problem 4.1 or 4.40 .We also note that for fixed values of Re, Ra, and P inequalities 4.27 and 4.45 imply that to ensure the uniqueness and stability of the solution to problem 4.1 or 4.40 the values of the parameters μ 1 , μ 2 , and μ 3 should be positive and exceed the constants on the right-hand sides of inequalities 4.27 and 4.45 .This means that the term μ 1 /2 χ 2 Γ N in the expression for minimized functional in 4.1 or the terms μ 1 /2 χ 2 Γ N , μ 2 /2 ψ 2 1/2,Γ D and μ 3 /2 f 2 in the expression for minimized functional in 4.40 have a regularizing effect on the control problem under consideration.The same conclusions hold true and for another control problems studied in this paper.
Y denotes the Fréchet derivative International Journal of Differential Equations of F with respect to x at the point x, u, g .By F x x, u, g * : Y * → X * we denote the adjoint operator of F x x, u, g which is determined by the relation Γ D * be the duals of the spaces X and Y .Let F x x, u, g : X →

Theorem 3.1. Let conditions (i)-(iv) hold and g ∈ H 1/2 Γ . Then there exists at least one solution x, u u,
Theorem 3.4.Let under conditions of Theorem 3.2 for functionals I and I and condition 3.19 quadruples u 1 , p 1 , T 1 , u 1 and u 2 , p 2 , T 2 , u 2 be solutions to problem 3.5 under g g 1 and problem 3.14 under g g 2 , respectively, y *

2
International Journal of Differential Equations Identity 3.22 for problem 4.1 does not change, while identities 3.20 , 3.21 , and inequality 3.51 take due to 4.2 a form ν ∇w, ∇ξ i 56 International Journal of Differential Equations Identity 3.22 for problem 4.1 does not change while identities 3.20 , 3.21 and inequality 3.51 transform by 4.56 to 4.4 and relations ν ∇w, ∇ξ i 4.58 Using identities 4.57 , 4.4 , and 3.22 we estimate parameters ξ i , θ i , σ i and ζ i .To this end we set w ξ i , τ θ i in 4.57 , 3.22 .Taking into account 2.11 , 2.12 and condition ξ i ∈ V which follows from 4.4 we obtain 4.7 and relation Proceeding further as above in study of problem 4.1 we come to the estimates for ξ i , θ i , σ i and ζ ζ 1 − ζ 2 .They have a form 4.17 , 4.21 , and 4.23 , where parameters γ and Re 0 are given by 4.61 .Let us assume that the condition 4.27 takes place where parameter c is defined in 4.26 , 4.61 .Using 4.27 and estimates 4.17 , 4.21 , 4.23 we deduce inequality 4.28 where parameter b is given by relations 4.26 , 4.61 .Taking into account 4.28 and 4.23 , we come from 4.58 to the inequality The estimate 4.66 under Q Ω has the sense of the stability estimate in the norm H 1 Ω of the component u of the solution u, p, T, χ to problem 4.55 relative to small perturbations of functions v d ∈ H 1 Ω and g ∈ G in the norms of H 1 Ω and H 1/2 Γ respectively.In the case where u This yields together with 4.63 , 3.48 , 3.50 that χ 1 χ 2 , T 1 T 2 , p 1 p 2 .The latter means the uniqueness of the solution to problem 4.55 when Q Ω and 4.27 holds.
Q and g i ∈ G, i 1, 2, where Q ⊂ Ω is an arbitrary open subset, and let parameters a, b, c be defined in 4.23 and 4.26 , in which γ and Re 0 are given by 4.61 .Suppose that condition 4.27 is satisfied.Then the stability estimates 4.66 and 4.35 -4.38 hold where Δ is defined in 4.68 .
is obtained from 4.40 by replacing of the cost functional I 1 v by I 2 v .Analogous analysis shows that the following theorem holds.Q and g i ∈ G, i 1, 2, where Q ⊂ Ω is an arbitrary open subset and let parameters a and b, c 1 , c 2 , c 3 are defined in 4.23 and 4.26 , in which γ and Re 0 are given by 4.61 .Suppose that conditions 4.45 are satisfied.Then the stability estimates 4.66 and 4.50 -4.53 hold where Δ is defined in 4.68 .
4.21 , and 4.23 for ξ i 1 , θ i 1 , σ i , and ζ −1/2,Γ in which parameters γ and Re 0 are given by 5.7 .Let us assume that the condition 4.27 takes place where parameter c is defined in 4.26 , 5.7 .Using 4.27 and 4.17 , 4.21 , 4.23 we deduce inequality 4.28 where parameter b is given by 4.26 , 5.7 .Taking into account 4.28 and 4.23 with parameter a defined in 4.23 , 5.7 we come from 5.4 to the inequality