# Stability of Optimal Controls for the Stationary Boussinesq Equations

Authors: Gennady Alekseev; Dmitry Tereshko
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

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.


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## Body

## 1. 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., [5–19]). 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 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 Section2, 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.


## 2. Statement of Boundary Problem

In this paper we consider the model of heat transfer in a viscous incompressible heat-conducting fluid. The model consists of the Navier-Stokes equation and the convection-diffusion equation for temperature that are nonlinearly related via buoyancy in the Boussinesq approximation and via convective heat transfer. It is described by equations(2.1)-νΔu+(u⋅∇)u+∇p=f-β̃GT,div⁡u=0inΩ,(2.2)-λΔT+u⋅∇T=finΩ,(2.3)u=gon⁡Γ,T=ψon⁡ΓD,λ(∂T∂n+αT)=χon⁡ΓN.
Here Ω is a bounded domain in the space ℝd, d=2,3 with a boundary Γ consisting of two parts ΓD and ΓN; 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, β̃ 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 ΓD of Γ, χ is a function given on another part ΓN=Γ∖ΓD 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.We assume that the following conditions are satisfied:(i)
Ω is a bounded domain in ℝd, d=2,3, with Lipschitz boundary Γ∈C0,1, consisting of coupled components Γ(i), i=1,2,…,N; Γ=ΓD∪ΓN and meas ΓD>0.Below we shall use the Sobolev spacesHs(D) and L2(D), where s∈ℝ, or Hs(D) and L2(D) for the vector functions where D denotes Ω, its subset Q, Γ or a part Γ0 of the boundary Γ. In particularly we need the function spaces H1(Ω), L2(Ω), H1(Ω), H1/2(Γ), H1/2(ΓD) and their subspaces(2.4)T={θ∈H1(Ω):θ∣ΓD=0},L02(Ω)={r∈L2(Ω):∫Ωrdx=0},Hdiv⁡1(Ω)={v∈H1(Ω):div⁡v=0},H01(Ω)={v∈H1(Ω):v∣Γ=0},V={v∈H01(Ω):div⁡v=0},H̃1(Ω)={v∈H1(Ω):(v⋅n,1)Γ(i)=0,v⋅n∣ΓN=0},H̃1/2(Γ)={v∣Γ:v∈H̃1(Ω)}⊂H1/2(Γ),L+2(ΓN)={ϕ∈L2(ΓN):ϕ≥0a.e.on⁡ΓN}.The inner products and norms inL2(Ω), L2(Q), or L2(ΓN) are denoted by (·,·), ∥·∥, (·,·)Q, ∥·∥Q, or (·,·)ΓN, ∥·∥ΓN. The inner products, norms and seminorms in H1(Q) and H1(Q) are denoted by (·,·)1,Q, ∥·∥1,Q, and |·|1,Q or (·,·)1, ∥·∥1 and |·|1 if Q=Ω. The norms in H1/2(Γ) or H1/2(ΓD) are denoted by ∥·∥1/2,Γ or ∥·∥1/2,ΓD; the norm in the dual space H̃1/2(Γ)* is denoted by ∥·∥-1/2,Γ. Set b≡β̃G. Let in addition to condition (i) the following conditions hold:(ii)
f∈H-1(Ω), b≡β̃G∈L2(Ω), α∈L2(ΓN).The following technical lemma holds (see [2, 20]).Lemma 2.1.
Under conditions (i) there exist constantsδi>0,  γi>0,  Cd,  Cr, and  β1>0 such that
(2.5)(∇v,∇v)≥δ0‖v‖12∀v∈H01(Ω),(∇T,∇T)≥δ1‖T‖12∀T∈T,(2.6)|((u⋅∇)v,w)|≤γ0‖u‖1‖v‖1‖w‖1,|(u⋅∇T,η)|≤γ1‖u‖1‖T‖1‖η‖1,(2.7)|(bT,v)|≤β1‖T‖1‖v‖1∀T∈H1(Ω),v∈H1(Ω),(2.8)|(χ,T)ΓN|≤γ2‖χ‖ΓN‖T‖1,|(αT,η)ΓN|≤γ3‖α‖ΓN‖T‖1‖η‖1,‖T‖Q≤γ4‖T‖1,‖v‖Q≤γ4‖v‖1,(2.9)‖rot⁡v‖≤Cr‖v‖1,‖div⁡v‖≤Cd‖v‖1.
Bilinear form -(div·,·) satisfies the inf-sup condition
(2.10)inf⁡r∈L02(Ω)r≠0sup⁡v∈H01(Ω)v≠0-(div⁡v,r)‖v‖1‖r‖≥β=const>0.
Besides the following identities take place:
(2.11)((u⋅∇)v,v)=0∀u∈Hdiv⁡1(Ω),v∈H01(Ω),(2.12)(u⋅∇T,T)=0∀u∈Hdiv⁡1(Ω)∩H̃1(Ω),T∈H1(Ω).Letg∈H̃1/2(Γ), χ∈L2(ΓN), ψ∈H1/2(ΓD), f∈L2(Ω) in addition to (i), (ii). We multiply the equations in (2.1), (2.2) by test functions v∈H01(Ω) and S∈𝒯 and integrate the results over Ω with use of Green's formulas to obtain the weak formulation for the model (2.1)–(2.3). It consists of finding a triple x≡(u,p,T)∈H̃1(Ω)×L02(Ω)×H1(Ω) satisfying the relations
(2.13)ν(∇u,∇v)+((u⋅∇)u,v)-(p,div⁡v)=〈f,v〉-(bT,v)∀v∈H01(Ω),b≡β̃G,(2.14)λ(∇T,∇S)+λ(αT,S)ΓN+(u⋅∇T,S)=(f,S)+(χ,S)ΓN∀S∈T,(2.15)div⁡u=0inΩ,u=gon⁡Γ,T=ψon⁡ΓD.Following theorem (see [2]) establishes the solvability of Problem 1 and gives a priori estimates for its solution.Theorem 2.2.
Let conditions (i), (ii) be satisfied. Then Problem 1 has for every quadrupleg∈H̃1/2(Γ), χ∈L2(ΓN), f∈L2(Ω), ψ∈H1/2(ΓD) a weak solution (u,p,T) that satisfies the estimates
(2.16)‖u‖1≤Mu,‖p‖≤Mp,‖T‖1≤MT.
Here Mu, Mp and MT are nondecreasing continuous functions of the norms ∥f∥-1, ∥b∥, ∥g∥1/2,Γ, ∥χ∥ΓN, ∥f∥, ∥ψ∥1/2,ΓD, ∥α∥ΓN. If, additionally, f,g,χ,f,ψ,α are small in the sense that
(2.17)γ0δ0νMu+γ1δ0νβ1δ1λMT<1,
where δ0, δ1, γ0, γ1 and β1 are constants entering into (2.5)–(2.7), then the weak solution to Problem 1 is unique.


## 3. 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:(3.1)I1(v)=‖v-vd‖Q2,I2(v)=‖v-vd‖1,Q2,I3(v)=‖rot⁡v-ζd‖Q2.
Here Q is a subdomain of Ω. The functionals I1, I2, and I3 where functions ud∈L2(Q) (or ud∈H1(Q)) and ζd∈L2(Q) 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∈L2(ΓN), ψ∈H1/2(ΓD), and f∈L2(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 H1/2(Γ).LetX=H̃1(Ω)×L02(Ω)×H1(Ω), Y=H-1(Ω)×L02(Ω)×H̃1/2(Γ)×𝒯*×H1/2(ΓD). Denote by I:H̃1(Ω)→ℝ a weakly lower semicontinuous functional. We assume that the controls χ, ψ, and f vary in some sets K1⊂L2(ΓN), K2⊂H1/2(ΓD), K3⊂L2(Ω). Setting K=K1×K2×K3, x=(u,p,T), u0=(f,b,α), u=(χ,ψ,f) we introduce the functional J:X×K→ℝ by the formula(3.2)J(x,u)=μ02I(u)+μ12‖χ‖ΓN2+μ22‖ψ‖1/2,ΓD2+μ32‖f‖2.
Here μ0,μ1,μ2,μ3 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 μi 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)
K1⊂L2(ΓN), K2⊂H1/2(ΓD), K3⊂L2(Ω) are nonempty closed convex sets;(iv)
μ0>0, μl>0 or μ0>0, μl≥0 and Kl is a bounded set, l=1,2,3.Considering the functionalJ 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:(3.3)F(x,u,g)=F(u,p,T,χ,ψ,f,g)=0.
Here F=(F1,F2,F3,F4,F5):X×K×H̃1/2(Γ)→Y is the operator acting by formulas(3.4)〈F1(x),v〉=ν(∇u,∇v)+((u⋅∇)u,v)-(p,div⁡v)-〈f,v〉+(bT,v),F2(x)=div⁡u,F3(x,g)=u∣Γ-g,F5(x,ψ)=T∣ΓD-ψ,〈F4(x,f,χ),S〉=λ(∇T,∇S)+λ(αT,S)ΓN+(u⋅∇T,S)-(f,S)-(χ,S)ΓN.
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)∈K1×K2×K3=K such that(3.5)J(x,u)≡μ02I(u)+μ12‖χ‖ΓN2+μ22‖ψ‖1/2,ΓD2+μ32‖f‖2⟶inf⁡,F(x,u,g)=0,(x,u)∈X×K.LetX*≡H̃1(Ω)*×L02(Ω)×H1(Ω)* and Y*≡H01(Ω)×L02(Ω)×H̃1/2(Γ)*×𝒯×H1/2(ΓD)* be the duals of the spaces X and Y. Let Fx′(x̂,û,g):X→Y denotes the Fréchet derivative of F with respect to x at the point (x̂,û,g). By Fx′(x̂,û,g)*:Y*→X* we denote the adjoint operator of Fx′(x̂,û,g) which is determined by the relation(3.6)〈Fx′(x̂,û,g)*y*,x〉X*×X=〈y*,Fx′(x̂,û,g)x〉Y*×Y∀x∈X,y*∈Y*.
According to the general theory of extremum problems (see [21]) we introduce an element y*=(ξ,σ,ζ,θ,ζt)∈Y* which is referred to as the adjoint state and define the Lagrangian ℒ:X×K×ℝ+×Y*×H̃1/2(Γ)→ℝ, where ℝ+={x∈ℝ:x≥0}, by(3.7)L(x,u,λ0,y*,g)=λ0J(x,u)+〈y*,F(x,u)〉≡λ0J(x,u)+〈F1(x),ξ〉+(F2(x),σ)+〈ζ,F3(x,g)〉Γ+κ〈F4(x,f,χ),θ〉+κ〈ζt,F5(x,ψ)〉ΓD.
Here and below 〈ζ,·〉Γ≡〈ζ,·〉H̃1/2(Γ)*×H̃1/2(Γ), 〈ζt,·〉ΓD≡〈ζt,·〉H1/2(ΓD)*×H1/2(ΓD) 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, (3.8)[ξ]=[u]=L0T0-1,[θ]=[T]=K0,[σ]=[p]=L02T0-2.
Here L0,T0,M0,K0 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]=L02T0-2K0-2.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.Theorem 3.1.
Let conditions (i)–(iv) hold andg∈H̃1/2(Γ). Then there exists at least one solution (x̂,û)=(û,p̂,T̂,χ̂,ψ̂,f̂) to problem (3.5) for I=Ik, k=1,2,3.Theorem 3.2.
Let under conditions of Theorem3.1 a pair (x̂,û)≡(û,p̂,T̂,χ̂,ψ̂,f̂)∈X×K be a local minimizer in problem (3.5) and let the cost functional I be continuously differentiable with respect to u at the point x̂. Then there exists a nonzero Lagrange multiplier (λ0,y*)=(λ0,ξ,σ,ζ,θ,ζt)∈ℝ+×Y* such that the Euler-Lagrange equation
(3.9)Fx′(x̂,û,g)*y*=-λ0Jx′(x̂,û)inX*
for the adjoint state y* is satisfied and the minimum principle holds which is equivalent to the inequality
(3.10)L(x̂,û,λ0,y*,g)≤L(x̂,u,λ0,y*,g)∀u∈K.Theorem 3.3.
Let the assumptions of Theorem3.2 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 functionalJ 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)〈Jχ′(x̂,û),χ〉=μ1(χ̂,χ)ΓN,〈Jψ′(x̂,û),ψ〉=μ2(ψ̂,ψ)1/2,ΓD,〈Jf′(x̂,û),f〉=μ3(f̂,f),〈Lχ′(x̂,û,λ0,y*,g),χ〉=λ0μ1(χ̂,χ)ΓN-κ(θ,χ)ΓN≡(λ0μ1χ̂-κθ,χ)ΓN∀χ∈K1,〈Lψ′(x̂,û,λ0,y*,g),ψ〉=λ0μ2(ψ̂,ψ)1/2,ΓD-κ〈ζt,ψ〉ΓD≡〈λ0μ2ψ̂-κζt,ψ〉ΓD∀ψ∈K2,〈Lf′(x̂,û,λ0,y*,g),f〉=λ0μ3(f̂,f)-κ(θ,f)≡(λ0μ3f̂-κθ,f)∀f∈K3.
Here for example ℒχ′(x̂,û,λ0,y*,g) is the Gateaux derivative with respect to χ at the point (x̂,û,λ0,y*,g)∈X×K×ℝ+×Y*×H̃1/2(Γ). Since K1,K2,K3 are convex sets, at the minimum point û=(χ̂,ψ̂,f̂) of the functional ℒ(x̂,·,λ0,y*,g) the following conditions are satisfied (see [22]):(3.12)〈Lχ′(x̂,û,λ0,y*,g),χ-χ̂〉≡(λ0μ1χ̂-κθ,χ-χ̂)ΓN≥0∀χ∈K1,〈Lψ′(x̂,û,λ0,y*,g),ψ-ψ̂〉=〈λ0μ2ψ̂-κζt,ψ-ψ̂〉1/2,ΓD≥0∀ψ∈K2,〈Lf′(x̂,û,λ0,y*,g),f-f̂〉=(λ0μ3f̂-κθ,f-f̂)≥0∀f∈K3.We also note that the Euler-Lagrange equation (3.9) is equivalent to identities(3.13)ν(∇w,∇ξ)+((û⋅∇)w,ξ)+((w⋅∇)û,ξ)+κ(w⋅∇T̂,θ)-(σ,div⁡w)+〈ζ,w〉Γ+λ0〈Ju′(x̂,û),w〉=0∀w∈H̃1(Ω),(r,div⁡ξ)=0∀r∈L02(Ω),κ[λ(∇τ,∇θ)+λ(ατ,θ)ΓN+(û⋅∇τ,θ)+〈ζt,τ〉ΓD]+(bτ,ξ)=0∀τ∈H1(Ω).
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).Theorems3.1 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. 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 G⊂H̃1/2(Γ). Let (x1,u1)≡(u1,p1,T1,χ1,ψ1,f1)∈X×K be an arbitrary solution to problem (3.5) for a given function g=g1∈G. By (x2,u2)≡(u2,p2,T2,χ2,ψ2,f2)∈X×K we denote a solution to problem(3.14)J̃(x,u)≡μ02Ĩ(u)+μ12‖χ‖ΓN2+μ22‖ψ‖1/2,ΓD2+μ32‖f‖2⟶inf⁡,F(x,u,g̃)=0,(x,u)∈X×K.
It is obtained by replacing the functional I in (3.5) by a close functional Ĩ depending on u and by replacing a function g∈G by a close function g̃∈G.By Theorem3.1 the following estimates hold for triples (ui,pi,Ti): (3.15)‖ui‖1≤Mu0,‖pi‖≤Mp0,‖Ti‖1≤MT0.
Here(3.16)Mu0=sup⁡u∈K,g∈GMu(u0,u,g),Mp0=sup⁡u∈K,g∈GMp(u0,u,g),MT0=sup⁡u∈K,g∈GMT(u0,u,g),
where Mu, Mp, and MT are introduced in Theorem 3.1. We introduce “model” Reynolds number ℛe, Raley number ℛa, and Prandtl number 𝒫 by(3.17)Re=γ0Mu0δ0ν,Ra=γ1δ0νβ1MT0δ1λ,P=δ0νδ1λ.
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(3.18)‖u‖2=∫Ωu2dx,|u|12=∫Ω|∇u|2dx,‖u‖12=l-2‖u‖2+|u|12.
Here l is a dimensional factor of dimension [l]=L0 whose value is equal to 1.Assume that the following condition takes place:(3.19)Re+Ra≡γ0Mu0δ0ν+γ1δ0νβ1MT0δ1λ<12.Let us denote by(1,yi*), where yi*≡(ξi,σi,ζi,θi,ζit)∈H01(Ω)×L02(Ω)×H̃1/2(Γ)*×𝒯×H1/2(ΓD)*, i=1,2, Lagrange multipliers corresponding to solutions (xi,ui). By Theorems 3.2 and 3.3 and (3.12) they satisfy relations(3.20)ν(∇w,∇ξi)+((ui⋅∇)w,ξi)+((w⋅∇)ui,ξi)+κ(w⋅∇Ti,θi)-(σi,div⁡w)+〈ζi,w〉Γ+(μ02)〈(Ii)′(ui),w〉=0∀w∈H̃1(Ω),i=1,2,(3.21)(div⁡ξi,r)=0∀r∈L02(Ω),(3.22)κ[λ(∇τ,∇θi)+λ(ατ,θ)ΓN+(ui⋅∇τ,θi)+〈ζit,τ〉ΓD]+(bτ,ξi)=0∀τ∈H1(Ω),(3.23)(μ1χi-κθi,χ-χi)ΓN+〈μ2ψi-κζit,ψ-ψi〉ΓD+(μ3fi-κθi,f-fi)≥0∀(χ,ψ,f)∈K.
We renamed I1≡I, I2≡Ĩ in (3.20). Set ξ=ξ1-ξ2, σ=σ1-σ2, ζ=ζ1-ζ2, θ=θ1-θ2, ζt=ζ1t-ζ2t, g=g1-g2, and(3.24)u=u1-u2,p=p1-p2,T=T1-T2,χ=χ1-χ2,ψ=ψ1-ψ2,f=f1-f2.
Let us subtract (2.13)–(2.15), written for u2,p2,T2,u2,g2 from (2.13)–(2.15) for u1, p1, T1, u1, g1. We obtain(3.25)ν(∇u,∇v)+((u⋅∇)u1+(u2⋅∇)u,v)-(div⁡v,p)+(bT,v)=0∀v∈H01(Ω),(3.26)λ(∇T,∇S)+λ(αT,S)ΓN+(u⋅∇T1,S)+(u2⋅∇T,S)=(f,S)+(χ,S)ΓN∀S∈T,(3.27)div⁡u=0inΩ,u∣Γ=g,T|ΓD=ψ.
We set χ=χ1, ψ=ψ1, f=f1 in the inequality (3.23) under i=2 and χ=χ2, ψ=ψ2, f=f2 in the same inequality under i=1 and add. We obtain(3.28)-κ[(χ,θ)ΓN+〈ζt,ψ〉ΓD+(f,θ)]≤-μ1‖χ‖ΓN2-μ2‖ψ‖1/2,ΓD2-μ3‖f‖2.Subtract the identities (3.20)–(3.22), written for (x2,u2,y2*,g2) from the corresponding identities for (x1,u1,y1*,g1), set w=u, τ=T and add. Using (3.27) we obtain(3.29)ν(∇u,∇ξ)+((u1⋅∇)u+(u⋅∇)u1,ξ)+2((u⋅∇)u,ξ2)+κ(u⋅∇T1,θ)+κ((u⋅∇)T,θ2)+〈ζ,g〉Γ+κ[λ(∇T,∇θ)+λ(αT,θ)ΓN+(u1⋅∇T,θ)+(u⋅∇T,θ2)+〈ζt,ψ〉ΓD]+(bT,ξ)+(μ02)〈I′(u1)-Ĩ′(u2),u〉=0.
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)((u⋅∇)u,ξ1+ξ2)+κ(u⋅∇T,θ1+θ2)+(μ02)〈I′(u1)-Ĩ′(u2),u〉≤-〈ζ,g〉Γ-μ1‖χ‖ΓN2-μ2‖ψ‖1/2,ΓD2-μ3‖f‖2.
Thus we have proved the following result.Theorem 3.4.
Let under conditions of Theorem3.2 for functionals I and Ĩ and condition (3.19) quadruples (u1,p1,T1,u1) and (u2,p2,T2,u2) be solutions to problem (3.5) under g=g1 and problem (3.14) under g=g2, respectively, yi*=(ξi,σi,ζi,θi,ζit), i=1,2 be corresponding Lagrange multipliers. Then the inequality (3.30) holds for differences u,p,T,χ,ψ,f, defined in (3.24), where g=g1-g2, ζ=ζ1-ζ2.Below we shall need the estimates of differencesu=u1-u2, p=p1-p2, T=T1-T2 entering into (3.25)–(3.27) by differences χ=χ1-χ2, ψ=ψ1-ψ2, f=f1-f2, and g=g1-g2. Denote by u0∈H1(Ω) a vector such that divu0=0 in Ω, u0∣Γ=g, ∥u0∥1≤C0∥g∥1/2,Γ. Here C0 is a constant depending on Ω. The existence of u0 follows from [20, page 24]. We present the difference u≡u1-u2 as u=u0+ũ, where ũ∈V is a new unknown function. Set u=u0+ũ, v=ũ in (3.25). Taking into account (2.9) we obtain(3.31)ν(∇ũ,∇ũ)=-ν(∇u0,∇ũ)-((u0⋅∇)u1,ũ)-((ũ⋅∇)u1,ũ)-((u2⋅∇)u0,ũ)-(bT,ũ).
Using estimates (2.5), (2.6), (2.7), and (3.15), we deduce from (3.31) that(3.32)δ0ν‖ũ‖12≤ν‖u0‖1‖ũ‖1+γ0Mu0‖ũ‖12+2γ0Mu0‖u0‖1‖ũ‖1+β1‖T‖1‖ũ‖1.It follows from (3.19) that(3.33)δ0ν2<δ0ν-γ0Mu0-β1γ1δ1λMT0≤δ0ν-γ0Mu0.
Rewriting the inequality (3.32) by (3.33) as(3.34)(δ0ν2)‖ũ‖12≤(δ0ν-γ0Mu0)‖ũ‖12≤(ν+2γ0Mu0)‖u0‖1‖ũ‖1+β1‖T‖1‖ũ‖1,
we obtain that(3.35)‖ũ‖1≤(2δ0ν)(ν+2γ0Mu0)‖u0‖1+(2β1δ0ν)‖T‖1≤(2δ0-1+4Re)‖u0‖1+(2β1δ0ν)‖T‖1≤2R‖u0‖1+(2β1δ0ν)‖T‖1,R≡δ0-1+2Re.
Taking into account the relation u=u0+ũ, we come to the following estimate ∥u∥1 via ∥g∥1/2,Γ and ∥T∥1:(3.36)‖u‖1≤‖u0‖1+‖ũ‖1≤(2R+1)‖u0‖1+(2β1δ0ν)‖T‖1≤C0(2R+1)‖g‖1/2,Γ+(2β1δ0ν)‖T‖1.Denote byT0∈H1(Ω) a function such that T0∣ΓD=ψ and the estimate ∥T0∥1≤C1∥ψ∥1/2,ΓD holds with a certain constant C1, which does not depend on ψ. Let us present the difference T=T1-T2 as T=T0+T̃, where T̃∈𝒯 is a new unknown function. Set T=T0+T̃, S=T̃ in (3.26). We obtain(3.37)λ(∇T̃,∇T̃)+λ(αT̃,T̃)ΓN+(u2⋅∇T̃,T̃)=-λ(∇T0,∇T̃)-λ(αT0,T̃)ΓN-(u2⋅∇T0,T̃)-(u⋅∇T1,T̃)+(f,T̃)+(χ,T̃)ΓN.
Using estimates (2.5)–(2.8) and (3.15) we deduce that(3.38)δ1λ‖T̃‖12≤λ‖T0‖1‖T̃‖1+γ1Mu0‖T0‖1‖T̃‖+λγ3‖α‖ΓN‖T0‖1‖T̃‖1+γ1MT0‖u‖1‖T̃‖1+(γ2‖χ‖ΓN+γ4‖f‖)‖T̃‖1
or(3.39)‖T̃‖1≤1δ1λ(λ+γ1Mu0+λγ3‖α‖ΓN)‖T0‖1+γ1MT0δ1λ‖u‖1+1δ1λ(γ2‖χ‖ΓN+γ4‖f‖).
Taking into account the relation T=T0+T̃, we obtain from this estimate that(3.40)‖T‖1≤C1(N+1)‖ψ‖1/2,ΓD+γ2‖χ‖ΓN+γ4‖f‖δ1λ+γ1MT0δ1λ‖u‖1,N=λ+γ1Mu0+λγ3‖α‖ΓNδ1λ.Using further the estimate (3.36) for u, we deduce from (3.40) that(3.41)‖T‖1≤C1(N+1)‖ψ‖1/2,ΓD+γ2‖χ‖ΓN+γ4‖f‖δ1λ+C0(2R+1)γ1MT0δ1λ‖g‖1/2,Γ+2β1δ0νγ1MT0δ1λ‖T‖1.
From this inequality and (3.17), (3.19) we come to the following estimate:(3.42)‖T‖1≤C1(N+1)1-2Ra‖ψ‖1/2,ΓD+γ2‖χ‖ΓN+γ4‖f‖δ1λ(1-2Ra)+C0(2R+1)1-2Raγ1MT0δ1λ‖g‖1/2,Γ.Using (3.42), we deduce from (3.36) that(3.43)‖u‖1≤C0(2R+1)(1+11-2Ra2β1δ0νγ1MT0δ1λ)‖g‖1/2,Γ+2β1δ0ν[C1(N+1)1-2Ra‖ψ‖1/2,ΓD+γ2‖χ‖ΓN+γ4‖f‖δ1λ(1-2Ra)].
Taking into account (3.17) we come to the following estimate for ∥u∥1:(3.44)‖u‖1≤2β1δ0ν[C1(N+1)1-2Ra‖ψ‖1/2,ΓD+γ2‖χ‖ΓN+γ4‖f‖δ1λ(1-2Ra)]+C0(2R+1)‖g‖1/2,Γ1-2Ra.An analogous estimate holds and for the pressure differencep=p1-p2. In order to establish this estimate we make use of inf-sup condition (2.10). By (2.10) for the function p=p1-p2 and any (small) number δ>0 there exists a function v0∈H01(Ω), v0≠0, such that -(divv0,p)≥β0∥v0∥1∥p∥ where β0=(β-δ)>0. Set v=v0 in the identity for u in (3.25) and make of this estimate and estimates (2.6), (2.7), (3.15). We shall have(3.45)β0‖v0‖1‖p‖≤-(div⁡v0,p)≤(ν+2γ0Mu0)‖v0‖1‖u‖1+β1‖T‖1‖v0‖1.
Dividing to ∥v0∥1≠0, we deduce that(3.46)‖p‖≤ν+2γ0Mu0β0‖u‖1+(β1β0)‖T‖1=δ0νβ0R‖u‖1+β1β0‖T‖1.
Using (3.42) and (3.44), we come to the following final estimate for ∥p∥:(3.47)‖p‖≤2R+1β0(1-2Ra)×[β1C1(N+1)‖ψ‖1/2,ΓD+β1(γ2‖χ‖ΓN+γ4‖f‖)δ1λ+δ0νC0(R+Ra)‖g‖1/2,Γ].Remark 3.5.
Along with three-parametric control problem (3.5) we shall consider and one-parametric 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 K2 consists of one element ψ0∈H1/2(ΓD) and the set K3 consists of one element f0∈L2(Ω).For this case the conditions f≡f1-f2=0,ψ≡ψ1-ψ2=0 take place, and the estimates (3.42)–(3.47) and inequality (3.30) take the form
(3.48)‖T‖1≤γ2‖χ‖ΓNδ1λ(1-2Ra)+C0(2R+1)1-2Raγ1MT0δ1λ‖g‖1/2,Γ,(3.49)‖u‖1≤2β1γ2‖χ‖ΓNδ0νδ1λ(1-2Ra)+C0(2R+1)‖g‖1/2,Γ1-2Ra,(3.50)‖p‖≤2R+1β0(1-2Ra)[β1γ2‖χ‖ΓNδ1λ+δ0νC0(R+Ra)‖g‖1/2,Γ],(3.51)((u⋅∇)u,ξ1+ξ2)+κ(u⋅∇T,θ1+θ2)+(μ02)〈I′(u1)-Ĩ′(u2),u〉≤-〈ζ,g〉Γ-μ1‖χ‖ΓN2.


## 4. Control Problems for Velocity Tracking-Type Cost Functionals

Based on Theorem3.4 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=I1 and the heat flux χ on the part ΓN of Γ is a unique control; that is, we consider one-parametric control problem(4.1)J(v,χ)≡μ02‖v-vd‖Q2+μ12‖χ‖ΓN2⟶inf⁡,F(x,χ,g)=0,x=(v,p,T)∈X,χ∈K1.
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 K2 and K3 consists of one element.Let(x1,u1)≡(u1,p1,T1,χ1) be a solution to problem (4.1), that corresponds to given functions vd≡ud(1)∈L2(Q) and g=g1∈G⊂H̃1/2(Γ), and let (x2,u2)≡(u2,p2,T2,χ2) be a solution to problem (4.1), that corresponds to perturbed functions ṽd≡ud(2)∈L2(Q) and g̃=g2∈G⊂H̃1/2(Γ). Setting ud=ud(1)-ud(2) in addition to (3.24) we note that under conditions of problem (4.1) we have(4.2)〈I1′(ui),w〉=2(ui-ud(i),w)Q,(I1′(u1)-Ĩ1′(u2),u)=2(‖u‖Q2-(u,ud)Q).
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(4.3)ν(∇w,∇ξi)+((ui⋅∇)w,ξi)+((w⋅∇)ui,ξi)+κ(w⋅∇Ti,θi)-(σi,div⁡w)+〈ζi,w〉Γ+μ0(ui-ud(i),w)Q=0∀w∈H̃1(Ω),(4.4)(div⁡ξi,r)=0∀r∈L02(Ω),(4.5)((u⋅∇)u,ξ1+ξ2)+κ(u⋅∇T,θ1+θ2)+μ0(‖u‖Q2-(u,ud)Q)≤-〈ζ,g〉Γ-μ1‖χ‖ΓN2.Using identities (4.3), (4.4), (3.22) we estimate parameters ξi, θi, σi and ζi. Firstly we deduce estimates for norms ∥ξi∥1 and ∥θi∥1. To this end we set w=ξi, τ=θi in (4.3), (3.22). Taking into account (2.11), (2.12), and condition ξi∈V, which follows from (4.4), we obtain(4.6)ν(∇ξi,∇ξi)=-((ξi⋅∇)ui,ξi)-κ(ξi⋅∇Ti,θi)-μ0(ui-ud(i),ξi)Q,(4.7)κλ[(∇θi,∇θi)+(αθi,θi)ΓN]=-(bθi,ξi),i=1,2.
Using estimates (2.5)–(2.8) and (3.15) we have(4.8)(∇ξi,∇ξi)≥δ0‖ξi‖12,(∇θi,∇θi)≥δ1‖θi‖12,(4.9)|((ξi⋅∇)ui,ξi)|≤γ0‖ui‖1‖ξi‖12≤γ0Mu0‖ξi‖12,(4.10)κ|(ξi⋅∇Ti,θi)|≤κγ1MT0‖ξi‖1‖θi‖1,|(bθi,ξi)|≤β1‖θi‖1‖ξi‖1,(4.11)‖ui-ud(i)‖Q≤‖ui‖Q+‖ud(i)‖Q≤γ4Mu0+‖ud(i)‖Q≤δ0νγ4γ0-1(Re+Re0),(4.12)|(ui-ud(i),ξi)Q|≤‖ui-ud(i)‖Q‖ξi‖Q≤δ0νγ(Re+Re0)‖ξi‖1,
where(4.13)γ=γ42γ0-1,Re0=γ0δ0νγ4max⁡(‖ud(1)‖Q,‖ud(2)‖Q).By virtue of (4.8)–(4.10) and (4.12), we deduce from (4.7) and (4.6) that(4.14)κ‖θi‖1≤β1δ1λ‖ξi‖1,(4.15)δ0ν‖ξi‖12≤γ0Mu0‖ξi‖12+κγ1MT0‖θi‖1+μ0δ0νγ(Re+Re0)‖ξi‖1.
Taking into account (4.14), we obtain from (4.15) that(4.16)(δ0ν-γ0Mu0-β1γ1δ1λMT0)‖ξi‖12≤μ0δ0νγ(Re+Re0)‖ξi‖1.
Using (3.33) we deduce successfully from (4.16), (4.14) that(4.17)‖ξi‖1≤2μ0γ(Re+Re0),κ‖θi‖1≤2μ0γβ1δ1λ(Re+Re0).Let us estimate further the norms∥σi∥ and ∥ζi∥-1/2,Γ from (4.3). In order to estimate ∥σi∥ we make use of inf-sup condition (2.10). By (2.10) for a function σi∈L02(Ω) and any small number δ>0 there exists a function vi∈H01(Ω), vi≠0, such that the inequality(4.18)-(div⁡vi,σi)≥β0‖vi‖1‖σi‖,i=1,2,β0=β-δ
holds. Setting in (4.3) w=vi and using this estimate together with estimates (2.6), (3.15), (4.11), we have(4.19)β0‖vi‖1‖σi‖≤-(div⁡vi,σi)≤ν‖vi‖1‖ξi‖1+2γ0Mu0‖vi‖1‖ξi‖1+κγ1MT0‖vi‖1‖θi‖1+μ0δ0νγ(Re+Re0)‖vi‖1.
From this inequality we deduce by (4.17) that(4.20)‖σi‖≤1β0[(ν+2γ0Mu0)‖ξi‖1+γ1MT0κ‖θi‖1+μ0δ0νγ(Re+Re0)]≤δ0νβ0[R‖ξi‖1+γ1MT0δ0νκ‖θi‖1+μ0γ(Re+Re0)].
Taking into account (4.17), we come from (4.20) to the estimate(4.21)‖σi‖≤μ0γδ0νβ0(Re+Re0)(2R+2Ra+1).It remains to estimate∥ζ∥-1/2,Γ. To this end we make again use of identity (4.3). Using estimates (2.6), (2.9) and (3.15), (4.11), (4.17), (4.21) as well we have(4.22)|〈ζi,w〉Γ|≤[(ν+2γ0Mu0)‖ξi‖1+γ1MT0κ‖θi‖1+Cd‖σi‖+μ0δ0νγ(Re+Re0)]‖w‖1≤μ0δ0νγ(1+Cdβ0-1)(Re+Re0)(2R+2Ra+1)‖w‖1∀w∈H̃1(Ω).
As ζ=ζ1-ζ2 we obtain from this inequality that(4.23)‖ζ‖-1/2,Γ≤μ0a,a=2δ0νγ(1+Cdβ0-1)(Re+Re0)(2R+2Ra+1).Taking into account (2.6), (3.48), (3.49), and estimates (4.17) for ξi, θi, we have (4.24)|((u⋅∇)u,ξ1+ξ2)|≤γ0‖u‖12(‖ξ1‖1+‖ξ2‖1)≤4μ0γ0γ(Re+Re0)[2β1γ2‖χ‖ΓNδ0νδ1λ(1-2Ra)+C0(2R+1)‖g‖1/2,Γ1-2Ra]2,κ|(u⋅∇T,θ1+θ2)|≤4μ0γ1γβ1(Re+Re0)δ1λ[2β1γ2‖χ‖ΓNδ0νδ1λ(1-2Ra)+C0(2R+1)‖g‖1/2,Γ1-2Ra]×[γ2‖χ‖ΓNδ1λ(1-2Ra)+C0(2R+1)1-2Raγ1MT0δ1λ‖g‖1/2,Γ].
It follows from (4.24) that(4.25)|((u⋅∇)u,ξ1+ξ2)+κ(u⋅∇T,θ1+θ2)|≤μ0(b‖g‖1/2,Γ2+c‖χ‖ΓN2).
Here constants b and c are given by(4.26)b=4γγ0C02(2R+1)2(Re+Re0)(1-2Ra)2[3+(γ1γ0)2P2Ra2],c=4γγ0(β1δ0νγ2δ1λ)2(Re+Re0)(1-2Ra)2[12+(γ1γ0)2P2].Let the data for problem (4.1) and parameters μ0, μ1 be such that with a certain constant ε>0 the following condition takes place:(4.27)(1-ε)μ1≥μ0c,ε=const>0.
Under condition (4.27) we deduce from (4.25) that(4.28)|((u⋅∇)u,ξ1+ξ2)+κ(u⋅∇T,θ1+θ2)|≤μ0b‖g‖1/2,Γ2+(1-ε)μ1‖χ‖ΓN2.
Taking into account (4.28) and the estimate |〈ζ,g〉Γ|≤∥ζ∥-1/2,Γ∥g∥1/2,Γ≤μ0a∥g∥1/2,Γ which follows from (4.23), we come from (4.5) to the inequality(4.29)μ0(‖u‖Q2-(u,ud)Q)≤-((u⋅∇)u,ξ1+ξ2)-κ(u⋅∇T,θ1+θ2)-〈ζ,g〉Γ-μ1‖χ‖ΓN2≤-εμ1‖χ‖ΓN2+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2.
It follows from this inequality that(4.30)μ0‖u‖Q2≤μ0(u,ud)Q-εμ1‖χ‖ΓN2+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2.
Excluding nonpositive term -εμ1∥χ∥ΓN2 from the right-hand side of (4.30), we deduce from (4.30) that(4.31)‖u‖Q2≤‖ud‖Q‖u‖Q+a‖g‖1/2,Γ+b‖g‖1/2,Γ2.Equation (4.31) is a quadratic inequality for ∥u∥Q. Solving it we come to the following estimate for ∥u∥Q:(4.32)‖u‖Q≤‖ud‖Q+(a‖g‖1/2,Γ+b‖g‖1/2,Γ2)1/2.
As u=u1-u2, ud=ud(1)-ud(2), g=g1-g2, the estimate (4.32) is equivalent to the following estimate for the velocity difference u1-u2:(4.33)‖u1-u2‖Q≤‖ud(1)-ud(2)‖Q+(a‖g1-g2‖1/2,Γ+b‖g1-g2‖1/2,Γ2)1/2.
This estimate under Q=Ω has the sense of the stability estimate in L2(Ω) of the component û of the solution (û,p̂,T̂,χ̂) to problem (4.1) relative to small perturbations of functions vd∈L2(Ω) and g∈G in the norms of L2(Ω) and H1/2(Γ), respectively. In particular case where g1=g2 the estimate (4.33) transforms to “exact” a priori estimate ∥u1-u2∥Q≤∥ud(1)-ud(2)∥Q. It was obtained when studying control problems for Navier-Stokes and in [18] when studying control problems for heat convection equations. If besides ud(1)=ud(2) it follows from (4.33) that u1=u2 in Ω, if Q=Ω. This yields together with (4.30), (3.48), (3.50) that χ1=χ2, T1=T2, p1=p2. 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)εμ1‖χ‖ΓN2≤-μ0‖u‖Q2+μ0‖ud‖Q‖u‖Q+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2≤μ0‖ud‖Q2+μ0‖ud‖Q(a‖g‖1/2,Γ+b‖g‖1/2,Γ2)1/2+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2≤μ0[‖ud‖Q+(a‖g‖1/2,Γ+b‖g‖1/2,Γ2)1/2]2.
From (4.34) and (3.48)–(3.50) we come to the following stability estimates:(4.35)‖χ1-χ2‖ΓN≤μ0εμ1Δ,(4.36)‖u1-u2‖1≤2β1γ2δ0νδ1λ(1-2Ra)μ0εμ1Δ+C0(2R+1)‖g1-g2‖1/2,Γ1-2Ra,(4.37)‖T1-T2‖1≤γ2δ1λ(1-2Ra)μ0εμ1Δ+C0(2R+1)1-2Raγ1MT0δ1λ‖g1-g2‖1/2,Γ,(4.38)‖p1-p2‖≤(2R+1)β1γ2β0δ1λ(1-2Ra)μ0εμ1Δ+C0(2R+1)(R+Ra)1-2Raδ0νβ0‖g1-g2‖1/2,Γ,
where(4.39)Δ=‖ud(1)-ud(2)‖Q+(a‖g1-g2‖1/2,Γ+b‖g1-g2‖1/2,Γ2)1/2.
Thus we have proved the theorem.Theorem 4.1.
Let, under conditions (i), (ii), (iii) forK1 and (3.19), the quadruple (ui,pi,Ti,χi) be a solution to problem (4.1) corresponding to given functions vd=ud(i)∈L2(Q) and gi∈G⊂H̃1/2(Γ), i=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 ℛe0 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(4.40)J(v,χ,ψ,f)≡μ02‖v-vd‖Q2+μ12‖χ‖ΓN2+μ22‖ψ‖1/2,ΓD2+μ32‖f‖2⟶inf⁡,F(x,u,g)=0,x=(v,p,T)∈X,u=(χ,ψ,f)∈K
corresponding to the cost functional I1(v)=∥v-vd∥Q2. Let (x1,u1)≡(u1,p1,T1,χ1,ψ1,f1) be a solution to problem (4.40) corresponding to given functions vd≡ud(1)∈L2(Q) and g=g1∈G, and let (x2,u2)≡(u2,p2,T2,χ2,ψ2,f2) be a solution to problem (4.40) corresponding to perturbed functions ṽd≡ud(2)∈L2(Q) and g̃=g2∈G. Setting ud=ud(1)-ud(2) 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(4.41)((u⋅∇)u,ξ1+ξ2)+κ(u⋅∇T,θ1+θ2)+μ0(‖u‖Q2-(u,ud)Q)≤-〈ζ,g〉Γ-μ1‖χ‖ΓN2-μ2‖ψ‖1/2,ΓD2-μ3‖f‖2.From (4.3), (4.4), and (3.22) we come to the same estimates (4.17), (4.21), and (4.23) for norms ∥ξi∥1, ∥θi∥1, ∥σi∥ and ∥ζ∥-1/2,Γ. Taking into account these estimates and estimates (3.42), (3.44) for ∥T∥1, ∥u∥1, we deduce that(4.42)|((u⋅∇)u,ξ1+ξ2)|≤γ0‖u‖12(‖ξ1‖1+‖ξ2‖1)≤4μ0γ0γ(Re+Re0)×[2β1C1(N+1)‖ψ‖1/2,ΓDδ0ν(1-2Ra)+2β1(γ2‖χ‖ΓN+γ4‖f‖)δ0νδ1λ(1-2Ra)+C0(2R+1)‖g‖1/2,Γ1-2Ra]2,|(u⋅∇T,θ1+θ2)|≤4μ0γ1γβ1(Re+Re0)δ1λ×[2β1C1(N+1)‖ψ‖1/2,ΓDδ0ν(1-2Ra)+2β1(γ2‖χ‖ΓN+γ4‖f‖)δ0νδ1λ(1-2Ra)+C0(2R+1)‖g‖1/2,Γ1-2Ra]×[C1(N+1)1-2Ra‖ψ‖1/2,ΓD+γ2‖χ‖ΓN+γ4‖f‖δ1λ(1-2Ra)+C0(2R+1)1-2Raγ1MT0δ1λ‖g‖1/2,Γ].
Here parameters γ and ℛe0 are given by (4.13). From (4.42) we obtain that(4.43)|((u⋅∇)u,ξ1+ξ2)+κ(u⋅∇T,θ1+θ2)|≤μ0(b‖g‖1/2,Γ2+c1‖χ‖ΓN2+c2‖ψ‖1/2,ΓD2+c3‖f‖2).
Here constants b, c1, c2, and c3 are given by relations(4.44)b=8γ0γC02(2R+1)2(Re+Re0)(1-2Ra)2[3+(γ1γ0)2P2Ra2],c1=8γ0γ(β1δ0νγ2δ1λ)2(Re+Re0)(1-2Ra)2[12+(γ1γ0)2P2],c2=8γ0γC12(N+1)2(β1δ0ν)2(Re+Re0)(1-2Ra)2[12+(γ1γ0)2P2],c3=8γ0γ(β1δ0νγ4δ1λ)2(Re+Re0)(1-2Ra)2[12+(γ1γ0)2P2].Let the data for problem (4.40) and parameters μ0, μ1, μ2, and μ3 be such that(4.45)μ1(1-ε)≥μ0c1,μ2(1-ε)≥μ0c2,μ3(1-ε)≥μ0c3,ε=const>0.
Under condition (4.45) we deduce from (4.43) that(4.46)|((u⋅∇)u,ξ1+ξ2)+κ(u⋅∇T,θ1+θ2)|≤(1-ε)μ1‖χ‖ΓN2+(1-ε)μ2‖ψ‖1/2,ΓD2+(1-ε)μ3‖f‖2+μ0b‖g‖1/2,Γ2.
Taking into account (4.46) and (4.23), we come from (4.41) to the inequality(4.47)μ0(‖u‖Q2-(u,ud)Q)≤-((u⋅∇)u,ξ1+ξ2)-κ(u⋅∇T,θ1+θ2)-〈ζ,g〉Γ-μ1‖χ‖ΓN2-μ2‖ψ‖1/2,ΓD2-μ3‖f‖2≤-εμ1‖χ‖ΓN2-εμ2‖ψ‖1/2,ΓD2-εμ3‖f‖2+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2.
It follows from this inequality that(4.48)μ0‖u‖Q2≤μ0(u,ud)Q-εμ1‖χ‖ΓN2-εμ2‖ψ‖1/2,ΓD2-εμ3‖f‖2+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2.
Excluding 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 L2(Ω) of the component û of the solution to problem (4.40) relative to small perturbations of functions vd∈L2(Ω) and g∈G in the norms of L2(Ω) and H1/2(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.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)εμ1‖χ‖ΓN2+εμ2‖ψ‖1/2,ΓD2+εμ3‖f‖2≤-μ0‖u‖Q2+μ0‖u‖Q‖ud‖Q+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2≤μ0[‖ud‖Q+(a‖g‖1/2,Γ+b‖g‖1/2,Γ2)1/2]2.
From this inequality and from (3.42)–(3.47) we come to the following stability estimates:(4.50)‖χ1-χ2‖ΓN≤μ0εμ1Δ,‖ψ1-ψ2‖1/2,ΓD≤μ0εμ2Δ,‖f1-f2‖≤μ0εμ3Δ,(4.51)‖u1-u2‖1≤2β1dδ0ν(1-2Ra)μ0εΔ+C0(2R+1)‖g1-g2‖1/2,Γ1-2Ra,(4.52)‖T1-T2‖1≤d1-2Raμ0εΔ+C0(2R+1)1-2Raγ1MT0δ1λ‖g1-g2‖1/2,Γ,(4.53)‖p1-p2‖≤(2R+1)β0(1-2Ra)[β1dμ0εΔ+δ0νC0(R+Ra)‖g1-g2‖1/2,Γ].
Here a constant d depending on μ1, μ2, and μ3 is given by(4.54)d=γ2δ1λμ1+C1(N+1)μ2+γ4δ1λμ3,
and a quantity Δ is defined in (4.39). Thus the following theorem is proved.Theorem 4.2.
Let, under conditions (i), (ii), (iii), and (3.19), an element (ui,pi,Ti, χi,ψi,fi) be a solution to problem (4.40) corresponding to given functions vd=ud(i)∈L2(Q) and gi∈G,i=1,2, where Q is an arbitrary open subset, and let parameters a and b,c1,c2,c3 be defined in (4.23) and (4.44), where γ and ℛe0are 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).In the same manner one can study control problem(4.55)J(v,χ)≡μ02‖v-vd‖1,Q2+μ12‖χ‖ΓN2⟶inf⁡,F(x,χ,g)=0,x∈X,χ∈K1
corresponding to the cost functional I2(v)=∥v-vd∥1,Q2. Let us denote by (x1,u1)≡(u1,p1,T1,χ1) a solution to problem (4.55) which corresponds to given functions vd≡ud(1)∈L2(Q) and g=g1∈G; by (x2,u2)≡(u2,p2,T2,χ2) we denote a solution to problem (4.1) which corresponds to perturbed functions ṽd≡ud(2)∈L2(Q) and g̃=g2∈G. Setting ud=ud(1)-ud(2) in addition to (3.24) we note that under conditions of problem (4.55) we have(4.56)〈I2′(ui),w〉=2(ui-ud(i),w)1,Q,(I2′(u1)-Ĩ2′(u2),u)=2(‖u‖1,Q2-(u,ud)1,Q).
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(4.57)ν(∇w,∇ξi)+((ui⋅∇)w,ξi)+((w⋅∇)ui,ξi)+κ(w⋅∇Ti,θi)-(σi,div⁡w)+〈ζi,w〉Γ=-μ0(ui-ud(i),w)1,Q∀w∈H̃1(Ω),i=1,2,(4.58)((u⋅∇)u,ξ1+ξ2)+κ(u⋅∇T,θ1+θ2)+μ0(‖u‖1,Q2-(u,ud)1,Q)≤-〈ζ,g〉Γ-μ1‖χ‖ΓN2.Using identities (4.57), (4.4), and (3.22) we estimate parameters ξi, θi, σi and ζi. To this end we set w=ξi, τ=θi in (4.57), (3.22). Taking into account (2.11), (2.12) and condition ξi∈V which follows from (4.4) we obtain (4.7) and relation(4.59)ν(∇ξi,∇ξi)=-((ξi⋅∇)ui,ξi)-κ(ξi⋅∇Ti,θi)-μ0(ui-ud(i),ξi)1,Q.
Using estimates (3.15) we deduce in addition to (4.8)–(4.10) that(4.60)‖ui-ud(i)‖1,Q≤‖ui‖1,Q+‖ud(i)‖1,Q≤Mu0+‖ud(i)‖Q≤δ0νγ0-1(Re+Re0),|(ui-ud(i),ξi)1,Q|≤‖ui-ud(i)‖1,Q‖ξi‖1,Q≤(Mu0+‖ud(i)‖1,Q)‖ξi‖1≤δ0νγ(Re+Re0)‖ξi‖1,
where(4.61)γ=γ0-1,Re0=γ0δ0νmax⁡(‖ud(1)‖1,Q,‖ud(2)‖1,Q).
Proceeding further as above in study of problem (4.1) we come to the estimates for ξi, θi, σi and ζ=ζ1-ζ2. They have a form (4.17), (4.21), and (4.23), where parameters γ and ℛe0 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(4.62)μ0(‖u‖1,Q2-(u,ud)1,Q)≤-((u⋅∇)u,ξ1+ξ2)-κ(u⋅∇T,θ1+θ2)-‖ζ‖-1/2,Γ‖g‖1/2,Γ-μ1‖χ‖ΓN2≤-εμ1‖χ‖ΓN2+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2.
It follows from this inequality that(4.63)μ0‖u‖1,Q2≤μ0(u,ud)1,Q-εμ1‖χ‖ΓN2+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2.
Excluding nonpositive term -εμ1∥χ∥ΓN2, we deduce from (4.63) that(4.64)‖u‖1,Q2≤‖ud‖1,Q‖u‖1,Q+a‖g‖1/2,Γ+b‖g‖1/2,Γ2.
Equation (4.31) is a quadratic inequality relative to ∥u∥1,Q. By solving it we come to the estimate(4.65)‖u‖1,Q≤‖ud‖1,Q+(a‖g‖1/2,Γ+b‖g‖1/2,Γ2)1/2
which is equivalent to the following estimate for u1-u2: (4.66)‖u1-u2‖1,Q≤‖ud(1)-ud(2)‖1,Q+(a‖g1-g2‖1/2,Γ+b‖g1-g2‖1/2,Γ2)1/2.
The estimate (4.66) under Q=Ω has the sense of the stability estimate in the norm H1(Ω) of the component û of the solution (û,p̂,T̂,χ̂) to problem (4.55) relative to small perturbations of functions vd∈H1(Ω) and g∈G in the norms of H1(Ω) and H1/2(Γ) respectively. In the case where ud(1)=ud(2) and g1=g2 it follows from (4.66) that u1=u2 in Ω, if Q=Ω. This yields together with (4.63), (3.48), (3.50) that χ1=χ2, T1=T2, p1=p2. The latter means the uniqueness of the solution to problem (4.55) when Q=Ω and (4.27) holds.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)εμ1‖χ‖ΓN2≤μ0‖u‖1,Q‖ud‖1,Q+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2≤μ0[‖ud‖1,Q+(a‖g‖1/2,Γ+b‖g‖1/2,Γ2)1/2]2.
From (4.67) and (3.48)–(3.50) we come to the estimates (4.35)–(4.38) where one should set(4.68)Δ=‖ud(1)-ud(2)‖1,Q+(a‖g1-g2‖1/2,Γ+b‖g1-g2‖1/2,Γ2)1/2.Thus we have proved the following theorem.Theorem 4.3.
Let, under conditions (i), (ii), (iii) forK1 and (3.19), the quadruple (ui,pi,T,χi) be a solution to problem (4.55) corresponding to given functions vd=ud(i)∈H1(Q) and gi∈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 ℛe0 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).In the similar way one can study three-parametric control problem(4.69)J(v,χ,ψ,f)≡μ02‖v-vd‖1,Q2+μ12‖χ‖ΓN2+μ22‖ψ‖1/2,ΓD2+μ32‖f‖2⟶inf⁡,F(x,χ,ψ,f,g)=0,(x,χ,ψ,f)∈X×K1×K2×K3.
It is obtained from (4.40) by replacing of the cost functional I1(v) by I2(v). Analogous analysis shows that the following theorem holds.Theorem 4.4.
Let, under conditions (i), (ii), (iii) and (3.19), an element (ui,pi,Ti,χi,ψi,fi) be a solution to problem (4.69) corresponding to given functions ud=ud(i)∈H1(Q) and gi∈G, i=1,2, where Q⊂Ω is an arbitrary open subset and let parameters a and b,c1,c2,c3 are defined in (4.23) and (4.26), in which γ and ℛe0 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).


## 5. Control Problem for Vorticity Tracking-Type Cost Functional

Consider now one-parametric control problem(5.1)J(v,χ)≡μ02‖rot⁡v-ζd‖Q2+μ12‖χ‖ΓN2⟶inf⁡,F(x,χ,g)=0,x∈X,χ∈K1,
which corresponds to the cost functional I3(v)=∥rotv-ζd∥Q2. Let (x1,u1)≡(u1,p1,T1,χ1) be a solution to problem (5.1) corresponding to given functions ζd≡ζd(1)∈L2(Q) and g=g1∈G, and let (x2,u2)≡(u2,p2,T2,χ2) be a solution to problem (4.1) corresponding to perturbed functions ζ̃d≡ζd(2)∈L2(Q) and g̃=g2∈G. Setting ud=ud(1)-ud(2) in addition to (3.24), we have under conditions of problem (4.1) (5.2)〈I3′(ui),w〉=2(rot⁡ui-ζd(i),rot⁡w)Q,(I3′(u1)-Ĩ3′(u2),u)=2(‖rot⁡u‖Q2-(rot⁡u,ζd)Q).
Identity (3.22) for problem (5.1) does not change, while identities (3.20), (3.21) and inequality (3.51) transform due to (5.2) to (4.4) and relations(5.3)ν(∇w,∇ξi)+((ui⋅∇)w,ξi)+((w⋅∇)ui,ξi)+κ(w⋅∇Ti,θi)-(σi,div⁡w)+〈ζi,w〉Γ=-μ0(rot⁡ui-ζd(i),w)Q∀w∈H̃1(Ω),(5.4)((u⋅∇)u,ξ1+ξ2)+κ(u⋅∇T,θ1+θ2)+μ0(‖rot⁡u‖Q2-(rot⁡u,ζd)Q)≤-〈ζ,g〉Γ-μ1‖χ‖ΓN2.Using identities (5.3), (3.22), (4.4) we estimate parameters ξi, θi, σi, and ζi. Firstly we deduce estimates of norms ∥ξi∥1 and ∥θi∥1. To this end we set w=ξi, τ=θi in (5.3), (3.22). Taking into account (2.11), (2.12) and condition ξi∈V, which follows from (4.4), we obtain (4.7) and relation(5.5)ν(∇ξi,∇ξi)=-((ξi⋅∇)ui,ξi)-κ(ξi⋅∇Ti,θi)-μ0(rot⁡ui-ζd(i),ξi)Q.
Using (2.9), (3.15) we deduce in addition to (4.8)–(4.10) that(5.6)‖rot⁡ui-ζd(i)‖Q≤‖rot⁡ui‖Q+‖ζd(i)‖Q≤CrMu0+‖ζd(i)‖Q≤δ0νCr2γ0(Re+Re0),|(rot⁡ui-ζd(i),rot⁡ξi)Q|≤‖rot⁡ui-ζd(i)‖Q‖ξi‖Q≤Cr(CrMu0+‖ζd(i)‖Q)‖ξi‖1≤δ0νγ(Re+Re0)‖ξi‖1,
where(5.7)γ=Cr2,Re0=γ0δ0νCrmax⁡(‖ζd(1)‖Q,‖ζd(2)‖Q).
Arguing as above in analysis of problem (4.1) we come to the same estimates (4.17), (4.21), and (4.23) for ∥ξi∥1, ∥θi∥1, ∥σi∥, and ∥ζ∥-1/2,Γ in which parameters γ and ℛe0 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(5.8)μ0(‖rot⁡u‖Q2-(rot⁡u,ζd)Q)≤-((u⋅∇)u,ξ1+ξ2)-κ(u⋅∇T,θ1+θ2)+‖ζ‖-1/2,Γ‖g‖1/2,Γ-μ1‖χ‖ΓN2≤-εμ1‖χ‖ΓN2+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2.
It follows from this inequality that(5.9)μ0‖rot⁡u‖Q2≤μ0(rot⁡u,ζd)Q-εμ1‖χ‖ΓN2+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2.
Excluding nonpositive term -εμ1∥χ∥ΓN2, we deduce from (5.9) that(5.10)‖rot⁡u‖Q2≤‖ζd‖Q‖rot⁡u‖Q+a‖g‖1/2,Γ+b‖g‖1/2,Γ2.
Equation (5.10) is a quadratic inequality relative to ∥rotu∥Q. Solving it we come to the estimate(5.11)‖rot⁡u‖Q≤‖ζd‖Q+(a‖g‖1/2,Γ+b‖g‖1/2,Γ2)1/2,
which is equivalent to the following estimate for the difference rotu1-rotu2:(5.12)‖rot⁡u1-rot⁡u2‖Q≤‖ζd(1)-ζd(2)‖Q+(a‖g1-g2‖1/2,Γ+b‖g1-g2‖1/2,Γ2)1/2.The estimate (5.12) under Q=Ω has the sense of the stability estimate in the norm L2(Ω) of the vorticity curlû of the component û of the solution (û,p̂,T̂,χ̂) to problem (5.1) relative to small perturbations of functions ζd∈L2(Ω) and g∈H̃1/2(Γ) in the norms of L2(Ω) and H1/2(Γ), respectively. In particular case where ζd(1)=ζd(2) and g1=g2 it follows from (5.11) that rotu1=rotu2 in Ω, if Q=Ω. From this relation and from (4.30), (3.48), (3.50) it follows that χ1=χ2, T1=T2, p1=p2. The latter means the uniqueness of the solution to problem (4.1) when Q=Ω and condition (4.27) holds.IfQ≠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)εμ1‖χ‖ΓN2≤-μ0‖rot⁡u‖Q2+μ0‖ζd‖Q‖rot⁡u‖Q+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2≤μ0‖ζd‖Q2+μ0‖ζd‖Q(a‖g‖1/2,Γ+b‖g‖1/2,Γ2)1/2+μ0a‖g‖1/2,Γ+μ0b‖g‖1/2,Γ2≤μ0[‖ζd‖Q+(a‖g‖1/2,Γ+b‖g‖1/2,Γ2)1/2]2.
From (5.13) and (3.48)–(3.50) we come to the estimates (4.35)–(4.38) where(5.14)Δ=‖ζd(1)-ζd(2)‖Q+(a‖g1-g2‖1/2,Γ+b‖g1-g2‖1/2,Γ2)1/2.
Thus the following theorem is proved.Theorem 5.1.
Let, under conditions (i), (ii), (iii) forK1 and (3.19), the quadruple (ui,pi,Ti,χi) be a solution to problem (5.1) corresponding to given functions ζd(i)∈L2(Q) and gi∈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 ℛe0 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)J(v,χ,ψ,f)≡μ02‖rot⁡v-ζd‖Q2+μ12‖χ‖ΓN2+μ22‖ψ‖1/2,ΓD2+μ32‖f‖2⟶inf⁡,F(x,χ,ψ,f,g)=0,(x,χ,ψ,f)∈X×K1×K2×K3.
It is obtained from (4.40) by replacing the cost functional I1(v) by I3(v). The following theorem holds.Theorem 5.2.
Let, under conditions (i), (ii), (iii), and (3.19), an element (ui,pi,Ti,χi,ψi,fi) be a solution to problem (5.15) corresponding to given functions ζd=ζd(i)∈L2(Q) and gi∈G, i=1,2, where Q⊂Ω is an arbitrary open subset, and let parameters a and b,c1,c2,c3 be given by relations (4.23) and (4.44), in which γ and ℛe0 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).


## 6. 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 I1(v) it has the form of estimates (4.27) of the parameters μ0 and μ1 included in (4.1), while for the three-parametric problem (4.40) it has the form of estimates (4.45) of the parameters μ0, μ1, μ2, and μ3 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), (2.2) in the form of the constant c defined in (4.26) for problem (4.1) or in the form of three constants c1, c2, c3 defined in (4.44) for problem (4.40). An analysis of the expressions for c or c1, c2, c3 shows that for fixed values of the parameters μl 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 μ1, μ2, and μ3 should be positive and exceed the constants on the right-hand sides of inequalities (4.27) and (4.45). This means that the term (μ1/2)∥χ∥ΓN2 in the expression for minimized functional in (4.1) or the terms (μ1/2)∥χ∥ΓN2, (μ2/2)∥ψ∥1/2,ΓD2 and (μ3/2)∥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.



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