Stability of Optimal Controls for the Stationary Boussinesq Equations

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.

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 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.

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 equations𝐮=𝐠−o𝜈nΔ𝐮Γ+,(𝐮⋅𝑇∇−=)𝜆𝐮𝜓Δ+𝑇+∇on𝑝𝐮=Γ⋅∇_𝐷𝐟𝑇−=𝛽̃𝐆𝑓𝑇,indΩiv,𝐮=0inΩ,((22..12)),𝜆𝜕𝑇𝜕𝑛+𝛼𝑇=𝜒onΓ^𝑁.(2.3) 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, 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 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.

We assume that the following conditions are satisfied:(i)Ω is a bounded domain in ℝ^𝑑, 𝑑=2,3, with Lipschitz boundary Γ∈𝐶^0,1, consisting of coupled components Γ^(𝑖), 𝑖=1,2,…,𝑁; Γ=Γ_𝐷∪Γ_𝑁 and meas Γ_𝐷>0.

Below we shall use the Sobolev spaces 𝐻^𝑠(𝐷) and 𝐿²(𝐷), where 𝑠∈ℝ, or 𝐇^𝑠(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), 𝐇^1/2(Γ), 𝐻^1/2(Γ_𝐷) and their subspaces𝐇𝒯=𝜃∈𝐻¹(Ω)∶𝜃∣_Γ𝐷=0,𝐿²⁰(Ω)=𝑟∈𝐿²(Ω)∶_Ω𝑟𝑑𝑥=0_1div,𝐕=𝐯∈𝐇(₁₀Ω)=𝐯∈𝐇¹(Ω)∶div𝐯=0,𝐇¹⁰(Ω)=𝐯∈𝐻¹(Ω)∶𝐯∣_Γ=0,𝐇(Ω)∶div𝐯=0,𝐇¹(Ω)=𝐯∈𝐇¹(Ω)∶(𝐯⋅𝐧,1)_Γ^(𝑖)=0,𝐯⋅𝐧∣_Γ𝑁=0_1/2,(Γ)=𝐯∣_Γ∶𝐯∈𝐇¹(Ω)⊂𝐇^1/2(Γ),𝐿²⁺Γ_𝑁=𝜙∈𝐿²Γ_𝑁∶𝜙≥0a.e.onΓ_𝑁.(2.4)

The inner products and norms in 𝐿²(Ω), 𝐿²(𝑄), or 𝐿²(Γ_𝑁) 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 𝐇^1/2(Γ) or 𝐻^1/2(Γ_𝐷) are denoted by ‖⋅‖_1/2,Γ or ‖⋅‖_1/2,Γ𝐷; the norm in the dual space 𝐇^1/2(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(Γ_𝑁).

The following technical lemma holds (see [2, 20]).

Lemma 2.1. Under conditions (i) there exist constants 𝛿_𝑖>0, 𝛾_𝑖>0, 𝐶_𝑑, 𝐶_𝑟, and 𝛽₁>0 such that ||(((𝐮∇⋅𝐯∇,∇)𝐯𝐯,)𝐰≥)||𝛿≤₀‖𝛾𝐯₀‖²¹∀𝐯∈𝐇¹⁰(Ω),(∇𝑇,∇𝑇)≥𝛿₁‖𝑇‖²¹∀𝑇∈𝒯,(2.5)||(𝐛𝑇,𝐯)||≤‖𝐮𝛽‖₁₁‖𝐯‖₁‖𝐰‖₁,||(𝐮⋅∇𝑇,𝜂)||≤𝛾₁‖𝐮‖₁‖𝑇‖₁‖𝜂‖₁,(2.6)||(𝜒,𝑇)_Γ𝑁‖𝑇‖₁‖𝐯‖₁∀𝑇∈𝐻¹(Ω),𝐯∈𝐇¹(Ω),(2.7)||≤𝛾‖₂‖𝑇𝜒‖‖_Γ𝑁𝑄‖𝑇‖₁,||(𝛼𝑇,𝜂)_Γ𝑁||≤𝛾₃‖𝛼‖_Γ𝑁‖𝑇‖₁‖𝜂‖₁,‖rot𝐯‖≤≤𝐶𝛾_4𝑟‖𝑇‖₁,‖𝐯‖_𝑄≤𝛾₄‖𝐯‖₁,(2.8)‖𝐯‖₁,‖div𝐯‖≤𝐶_𝑑‖𝐯‖₁.(2.9) Bilinear form −(div⋅,⋅) satisfies the inf-sup condition inf_{𝑟∈𝐿20(Ω)𝑟≠0}sup_{𝐯∈𝐇10(Ω)𝐯≠0}−(div𝐯,𝑟)‖𝐯‖₁‖𝑟‖≥𝛽=const>0.(2.10) Besides the following identities take place: (𝐮⋅∇((𝑇𝐮,⋅𝑇∇))=𝐯,0𝐯)∀=𝐮0∈∀𝐇𝐮_1div∈𝐇^1div(Ω),𝐯∈𝐇¹⁰(Ω),(2.11)(Ω)∩𝐇¹(Ω),𝑇∈𝐻¹(Ω).(2.12)

Let 𝐠∈𝐇^1/2(Γ), 𝜒∈𝐿²(Γ_𝑁), 𝜓∈𝐻^1/2(Γ_𝐷), 𝑓∈𝐿²(Ω) in addition to (i), (ii). We multiply the equations in (2.1), (2.2) by test functions 𝐯∈𝐇¹⁰(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)∈𝐇¹(Ω)×𝐿²⁰(Ω)×𝐻¹(Ω) satisfying the relations 𝜈(∇𝐮,𝜆∇(∇𝐯)𝑇+,∇((𝑆𝐮)⋅+∇𝜆)(𝐮𝛼,𝑇𝐯,)𝑆−)(_Γ𝑁𝑝,div𝐯)=⟨𝐟,𝐯⟩−(𝐛𝑇,𝐯)∀𝐯∈𝐇¹⁰(Ω),𝐛≡𝛽̃𝐆,(2.13)div𝐮=0inΩ,+(𝐮𝐮⋅=∇𝐠𝑇,𝑆o)n=Γ(,𝑓,𝑆)𝑇+=(𝜒𝜓,𝑆o)n_Γ𝑁Γ_𝐷∀𝑆∈𝒯,(2.14).(2.15)

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 quadruple 𝐠∈𝐇^1/2(Γ), 𝜒∈𝐿²(Γ_𝑁), 𝑓∈𝐿²(Ω), 𝜓∈𝐻^1/2(Γ_𝐷) a weak solution (𝐮,𝑝,𝑇) that satisfies the estimates ‖𝐮‖₁≤𝑀_𝐮,‖𝑝‖≤𝑀_𝑝,‖𝑇‖₁≤𝑀_𝑇.(2.16) Here 𝑀_𝐮, 𝑀_𝑝 and 𝑀_𝑇 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 𝛾₀𝛿₀𝜈𝑀^𝐮+𝛾¹𝛿₀𝜈𝛿𝛽¹₁𝜆𝑀^𝑇<1,(2.17) 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:𝐼₁(𝐯)=‖‖𝐯−𝐯_𝑑‖‖^2𝑄,𝐼₂(𝐯)=‖‖𝐯−𝐯_𝑑‖‖^21,𝑄,𝐼₃(𝐯)=‖‖rot𝐯−𝜁_𝑑‖‖^2𝑄.(3.1) Here 𝑄 is a subdomain of Ω. The functionals 𝐼₁, 𝐼₂, and 𝐼₃ where functions 𝐮_𝑑∈𝐋²(𝑄) (or 𝐮_𝑑∈𝐇¹(𝑄)) and 𝜁_𝑑∈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(Γ_𝑁), 𝜓∈𝐻^1/2(Γ_𝐷), 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 𝐇^1/2(Γ).

Let 𝑋=𝐇¹(Ω)×𝐿²⁰(Ω)×𝐻¹(Ω), 𝑌=𝐇⁻¹(Ω)×𝐿²⁰(Ω)×𝐇^1/2(Γ)×𝒯^∗×𝐻^1/2(Γ_𝐷). 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 𝐾₁⊂𝐿²(Γ_𝑁), 𝐾₂⊂𝐻^1/2(Γ_𝐷), 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 formula𝐽(𝐱,𝑢)=𝜇⁰2𝐼(𝐮)+𝜇2¹‖𝜒‖^2Γ𝑁+𝜇²2‖𝜓‖^21/2,Γ𝐷+𝜇³2‖𝑓‖².(3.2) 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 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)𝐾₁⊂𝐿²(Γ_𝑁), 𝐾₂⊂𝐻^1/2(Γ_𝐷), 𝐾₃⊂𝐿²(Ω) are nonempty closed convex sets;(iv)𝜇₀>0, 𝜇_𝑙>0 or 𝜇₀>0, 𝜇_𝑙≥0 and 𝐾_𝑙 is a bounded set, 𝑙=1,2,3.

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:𝐹(𝐱,𝑢,𝐠)=𝐹(𝐮,𝑝,𝑇,𝜒,𝜓,𝑓,𝐠)=0.(3.3) Here 𝐹=(𝐹₁,𝐹₂,𝐹₃,𝐹₄,𝐹₅)∶𝑋×𝐾×𝐇^1/2(Γ)→𝑌 is the operator acting by formulas⟨𝐹𝐹₁(𝐱),𝐯⟩=𝜈(∇𝐮,∇𝐯)+((𝐮⋅∇)𝐮,𝐯)−(𝑝,div𝐯)−⟨𝐟,𝐯⟩+(𝐛𝑇,𝐯),₂⟨𝐹₄(𝐱)=div𝐮,𝐹₃(𝐱,𝐠)=𝐮∣_Γ−𝐠,𝐹₅(𝐱,𝜓)=𝑇∣_Γ𝐷−𝜓,(𝐱,𝑓,𝜒),𝑆⟩=𝜆(∇𝑇,∇𝑆)+𝜆(𝛼𝑇,𝑆)_Γ𝑁+(𝐮⋅∇𝑇,𝑆)−(𝑓,𝑆)−(𝜒,𝑆)_Γ𝑁.(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)∈𝐾₁×𝐾₂×𝐾₃=𝐾 such that𝐽(𝐱,𝑢)≡𝜇⁰2𝐼(𝐮)+𝜇2¹‖𝜒‖^2Γ𝑁+𝜇²2‖𝜓‖^21/2,Γ𝐷+𝜇³2‖𝑓‖²𝐹(𝐱,𝑢,𝐠)=0,(𝐱,𝑢)∈𝑋×𝐾.⟶inf,(3.5)

Let 𝑋^∗≡𝐇¹(Ω)^∗×𝐿²⁰(Ω)×𝐻¹(Ω)^∗ and 𝑌^∗≡𝐇¹⁰(Ω)×𝐿²⁰(Ω)×𝐇^1/2(Γ)^∗×𝒯×𝐻^1/2(Γ_𝐷)^∗ be the duals of the spaces 𝑋 and 𝑌. Let 𝐹^𝐱(𝐱̂,𝑢,̂𝐠)∶𝑋→𝑌 denotes the Fréchet derivative of 𝐹 with respect to 𝐱 at the point (𝐱̂,𝑢,̂𝐠). By 𝐹^𝐱(𝐱̂,𝑢,̂𝐠)^∗∶𝑌^∗→𝑋^∗ we denote the adjoint operator of 𝐹^𝐱(𝐱̂,𝑢,̂𝐠) which is determined by the relation𝐹^𝐱(𝐱̂,𝑢,̂𝐠)^∗𝐲^∗,𝐱_{𝑋∗×𝑋}=𝐲^∗,𝐹^𝐱(𝐱̂,𝑢,̂𝐠)𝐱_{𝑌∗×𝑌}∀𝐱∈𝑋,𝐲^∗∈𝑌^∗.(3.6) According to the general theory of extremum problems (see [21]) we introduce an element 𝐲^∗=(𝜉,𝜎,𝜁,𝜃,𝜁^𝑡)∈𝑌^∗ which is referred to as the adjoint state and define the Lagrangian ℒ∶𝑋×𝐾×ℝ⁺×𝑌^∗×𝐇^1/2(Γ)→ℝ, where ℝ⁺={𝑥∈ℝ∶𝑥≥0}, byℒ𝐱,𝑢,𝜆₀,𝐲^∗,𝐠=+𝜆₀⟨𝐽𝐹₁(𝐱,𝑢)+⟨𝐲^∗,𝐹(𝐱,𝑢)⟩≡𝜆₀𝐽(𝐱,𝑢)(𝐱),𝜉⟩+𝐹₂(𝐱),𝜎+⟨𝜁,𝐹₃(𝐱,𝐠)⟩_Γ+𝜅⟨𝐹₄(𝐱,𝑓,𝜒),𝜃⟩+𝜅𝜁^𝑡,𝐹₅(𝐱,𝜓)_Γ𝐷.(3.7) Here and below ⟨𝜁,⋅⟩_Γ≡⟨𝜁,⋅⟩𝐇^1/2_(Γ)∗×𝐇^1/2_(Γ), ⟨𝜁^𝑡,⋅⟩_Γ𝐷≡⟨𝜁^𝑡,⋅⟩_𝐻^1/2_(Γ𝐷)^∗_×𝐻^1/2_(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, [𝜉]=[𝐮]=𝐿₀𝑇⁰⁻¹,[𝜃]=[𝑇]=𝐾₀,[𝜎]=[𝑝]=𝐿²⁰𝑇⁰⁻².(3.8) 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]=𝐿²⁰𝑇⁰⁻²𝐾⁰⁻².

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 and 𝐠∈𝐇^1/2(Γ). Then there exists at least one solution (𝐱̂,𝑢)̂=(𝐮̂,𝑝,̂𝑇,𝜒,𝜓,𝑓) to problem (3.5) for 𝐼=𝐼_𝑘, 𝑘=1,2,3.

Theorem 3.2. Let under conditions of Theorem 3.1 a pair (𝐱̂,𝑢)̂≡(𝐮̂,𝑝,̂𝑇,𝜒,𝜓,𝑓)∈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,𝜁^𝑡)∈ℝ⁺×𝑌^∗ such that the Euler-Lagrange equation 𝐹^𝐱(𝐱̂,𝑢,̂𝐠)^∗𝐲^∗=−𝜆₀𝐽^𝐱(𝐱̂,𝑢)̂in𝑋^∗(3.9) for the adjoint state 𝐲^∗ is satisfied and the minimum principle holds which is equivalent to the inequality ℒ𝐱̂,𝑢,̂𝜆₀,𝐲^∗,𝐠≤ℒ𝐱̂,𝑢,𝜆₀,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𝐽^𝜒(𝐱̂,ℒ𝑢)̂,𝜒=𝜇₁𝜒,𝜒_Γ𝑁,𝐽^𝜓(𝐱̂,𝑢)̂,𝜓=𝜇₂𝜓,𝜓_1/2,Γ𝐷,𝐽^𝑓(𝐱̂,𝑢)̂,𝑓=𝜇₃𝑓,𝑓_𝜒,ℒ_𝜓𝐱̂,𝑢,̂𝜆₀,𝐲^∗,𝐠,𝜒=𝜆₀𝜇₁𝜒,𝜒_Γ𝑁−𝜅(𝜃,𝜒)_Γ𝑁≡𝜆₀𝜇₁𝜒−𝜅𝜃,𝜒_Γ𝑁∀𝜒∈𝐾₁,𝐱̂,ℒ𝑢,̂_𝑓𝜆₀,𝐲^∗,𝐠,𝜓=𝜆₀𝜇₂𝜓,𝜓_1/2,Γ𝐷−𝜅𝜁^𝑡,𝜓_Γ𝐷≡𝜆₀𝜇₂𝜓−𝜅𝜁^𝑡,𝜓_Γ𝐷∀𝜓∈𝐾₂,𝐱̂,𝑢,̂𝜆₀,𝐲^∗,𝐠,𝑓=𝜆₀𝜇₃𝑓,𝑓−𝜅(𝜃,𝑓)≡𝜆⁰𝜇₃𝑓−𝜅𝜃,𝑓∀𝑓∈𝐾₃.(3.11) Here for example ℒ^𝜒(𝐱̂,𝑢,̂𝜆₀,𝐲^∗,𝐠) is the Gateaux derivative with respect to 𝜒 at the point (𝐱̂,𝑢,̂𝜆₀,𝐲^∗,𝐠)∈𝑋×𝐾×ℝ⁺×𝑌^∗×𝐇^1/2(Γ). Since 𝐾₁,𝐾₂,𝐾₃ are convex sets, at the minimum point 𝑢̂=(𝜒,𝜓,𝑓) of the functional ℒ(𝐱̂,⋅,𝜆₀,𝐲^∗,𝐠) the following conditions are satisfied (see [22]):ℒ_𝜓ℒ^𝜒𝐱̂,𝑢,̂𝜆₀,𝐲^∗,𝐠,𝜒−𝜒≡𝜆₀𝜇₁𝜒−𝜅𝜃,𝜒−𝜒_Γ𝑁≥0∀𝜒∈𝐾₁,ℒ𝐱_𝑓̂,𝑢,̂𝜆₀,𝐲^∗,𝐠,𝜓−𝜓=𝜆₀𝜇₂𝜓−𝜅𝜁^𝑡,𝜓−𝜓_1/2,Γ𝐷≥0∀𝜓∈𝐾₂,𝐱̂,𝑢,̂𝜆₀,𝐲^∗,𝐠,𝑓−𝑓=𝜆₀𝜇₃𝑓−𝜅𝜃,𝑓−𝑓≥0∀𝑓∈𝐾₃.(3.12)

We also note that the Euler-Lagrange equation (3.9) is equivalent to identities𝜈−((∇𝜎𝐰,d,i∇v𝜉𝐰))++((⟨𝐮𝜁̂,⋅𝐰∇⟩)_Γ𝐰,𝜉)+((𝐰⋅∇)𝐮̂,𝜉)+𝜅𝐰⋅∇𝑇,𝜃(𝑟,div𝜉+)=𝜆₀0𝐽∀^𝐮𝑟(𝐱∈̂,𝑢𝐿)̂₂₀,𝐰=0∀𝐰∈𝐇¹(Ω),𝜅𝜆(∇𝜏,∇𝜃)+𝜆(𝛼𝜏,𝜃)_Γ𝑁(Ω),+(𝐮̂⋅∇𝜏,𝜃)+𝜁^𝑡,𝜏_Γ𝐷+(𝐛𝜏,𝜉)=0∀𝜏∈𝐻¹(Ω).(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. 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⊂𝐇^1/2(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 problem𝐽(𝐱,𝑢)≡𝜇⁰2𝐼(𝐮)+𝜇2¹‖𝜒‖^2Γ𝑁+𝜇²2‖𝜓‖^21/2,Γ𝐷+𝜇³2‖𝑓‖²⟶inf,𝐹(𝐱,𝑢,𝐠̃)=0,(𝐱,𝑢)∈𝑋×𝐾.(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 𝐠̃∈𝐺.

By Theorem 3.1 the following estimates hold for triples (𝐮_𝑖,𝑝_𝑖,𝑇_𝑖): ‖‖𝐮_𝑖‖‖₁≤𝑀^0𝐮,‖‖𝑝_𝑖‖‖≤𝑀^0𝑝,‖‖𝑇_𝑖‖‖₁≤𝑀^0𝑇.(3.15) Here𝑀^0𝐮=sup_{𝑢∈𝐾,𝐠∈𝐺}𝑀_𝐮𝑢₀,𝑢,𝐠,𝑀^0𝑝=sup_{𝑢∈𝐾,𝐠∈𝐺}𝑀_𝑝𝑢₀,𝑢,𝐠,𝑀^0𝑇=sup_{𝑢∈𝐾,𝐠∈𝐺}𝑀_𝑇𝑢₀,𝑢,𝐠,(3.16) 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 byℛ𝑒=𝛾⁰𝑀^0𝐮𝛿₀𝜈,ℛ𝑎=𝛿𝛾¹₀𝜈𝛽¹𝑀^0𝑇𝛿₁𝜆,𝒫=𝛿⁰𝜈𝛿₁𝜆.(3.17) 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‖𝑢‖²=_Ω𝑢²𝑑𝑥,|𝑢|²¹=_Ω||∇𝑢||²𝑑𝑥,‖𝑢‖²¹=𝑙⁻²‖𝑢‖²+|𝑢|²¹.(3.18) Here 𝑙 is a dimensional factor of dimension [𝑙]=𝐿₀ whose value is equal to 1.

Assume that the following condition takes place:ℛ𝑒+ℛ𝑎≡𝛾⁰𝑀^0𝐮𝛿₀𝜈+𝛿𝛾¹₀𝜈𝛽¹𝑀^0𝑇𝛿₁𝜆<12.(3.19)

Let us denote by (1,𝐲^∗𝑖), where 𝐲^∗𝑖≡(𝜉_𝑖,𝜎_𝑖,𝜁_𝑖,𝜃_𝑖,𝜁^𝑡𝑖)∈𝐇¹⁰(Ω)×𝐿²⁰(Ω)×𝐇^1/2(Γ)^∗×𝒯×𝐻^1/2(Γ_𝐷)^∗, 𝑖=1,2, Lagrange multipliers corresponding to solutions (𝐱_𝑖,𝑢_𝑖). By Theorems 3.2 and 3.3 and (3.12) they satisfy relations𝜈∇+𝐰⟨,𝜁∇_𝑖𝜉_𝑖+𝐮_𝑖⋅∇𝐰,𝜉_𝑖+(𝐰⋅∇)𝐮_𝑖,𝜉_𝑖+𝜅𝐰⋅∇𝑇_𝑖,𝜃_𝑖−𝜎_𝑖,div𝐰,𝐰⟩_Γ+𝜇⁰2𝐼^𝑖^𝐮_𝑖,𝐰=0∀𝐰∈𝐇¹div𝜉_𝑖(Ω),𝑖=1,2,(3.20)𝜅𝜆∇𝜏,∇𝜃_𝑖,𝑟=0∀𝑟∈𝐿²⁰(Ω),(3.21)𝜇+𝜆(𝛼𝜏,𝜃)_Γ𝑁+𝐮_𝑖⋅∇𝜏,𝜃_𝑖+𝜁^𝑡𝑖,𝜏_Γ𝐷+𝐛𝜏,𝜉^𝑖=0∀𝜏∈𝐻¹(Ω),(3.22)₁𝜒_𝑖−𝜅𝜃_𝑖,𝜒−𝜒_𝑖_Γ𝑁+𝜇₂𝜓_𝑖−𝜅𝜁^𝑡𝑖,𝜓−𝜓_𝑖_Γ𝐷+𝜇₃𝑓_𝑖−𝜅𝜃_𝑖,𝑓−𝑓_𝑖≥0∀(𝜒,𝜓,𝑓)∈𝐾.(3.23) We renamed 𝐼¹≡𝐼, 𝐼²≡𝐼 in (3.20). Set 𝜉=𝜉₁−𝜉₂, 𝜎=𝜎₁−𝜎₂, 𝜁=𝜁₁−𝜁₂, 𝜃=𝜃₁−𝜃₂, 𝜁^𝑡=𝜁^𝑡1−𝜁^𝑡2, 𝐠=𝐠₁−𝐠₂, and𝐮=𝐮₁−𝐮₂,𝑝=𝑝₁−𝑝₂,𝑇=𝑇₁−𝑇₂,𝜒=𝜒₁−𝜒₂,𝜓=𝜓₁−𝜓₂,𝑓=𝑓₁−𝑓₂.(3.24) 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 obtain𝜆(∇𝜈𝑇(∇,∇𝐮,𝑆∇)𝐯+)+𝜆(𝛼(𝑇𝐮,𝑆⋅∇))_Γ𝑁𝐮₁+𝐮₂⋅∇𝐮,𝐯−(div𝐯,𝑝)+(𝐛𝑇,𝐯)=0∀𝐯∈𝐇¹⁰(Ω),(3.25)div+𝐮=𝐮0⋅∇in𝑇₁Ω,𝑆,+𝐮∣𝐮_2Γ⋅∇𝑇,𝑆=(𝑓,𝑆)+(𝜒,𝑆)_Γ𝑁∀𝑆∈𝒯,(3.26)=𝐠,𝑇|_Γ𝐷=𝜓.(3.27) 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−𝜅(𝜒,𝜃)_Γ𝑁+𝜁^𝑡,𝜓_Γ𝐷+(𝑓,𝜃)≤−𝜇₁‖𝜒‖^2Γ_𝑁−𝜇₂‖𝜓‖^21/2,Γ_𝐷−𝜇₃‖𝑓‖².(3.28)

Subtract the identities (3.20)–(3.22), written for (𝐱₂,𝑢₂,𝐲^∗2,𝐠₂) from the corresponding identities for (𝐱₁,𝑢₁,𝐲^∗1,𝐠₁), set 𝐰=𝐮, 𝜏=𝑇 and add. Using (3.27) we obtain𝜈(∇𝐮+,∇⟨𝜁𝜉,)𝐠+⟩_Γ𝐮₁⋅∇𝐮+(𝐮⋅∇)𝐮₁,𝜉+2(𝐮⋅∇)𝐮,𝜉₂+𝜅𝐮⋅∇𝑇₁,𝜃+𝜅(𝐮⋅∇)𝑇,𝜃₂+𝜇+𝜅𝜆(∇𝑇,∇𝜃)+𝜆(𝛼𝑇,𝜃)_Γ𝑁+𝐮₁⋅∇𝑇,𝜃+𝐮⋅∇𝑇,𝜃₂+𝜁^𝑡,𝜓_Γ𝐷+(𝐛𝑇,𝜉)₀2𝐼^𝐮₁−𝐼^𝐮₂,𝐮=0.(3.29) 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(𝐮⋅≤∇)−𝐮⟨,𝜁𝜉,₁𝐠+⟩_Γ𝜉₂+𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂+𝜇2⁰𝐼^𝐮₁−𝐼^𝐮₂,𝐮−𝜇₁‖𝜒‖^2Γ_𝑁−𝜇₂‖𝜓‖^21/2,Γ_𝐷−𝜇₃‖𝑓‖².(3.30) Thus we have proved the following result.

Theorem 3.4. Let under conditions of Theorem 3.2 for functionals 𝐼 and 𝐼 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, 𝐲^∗𝑖=(𝜉_𝑖,𝜎_𝑖,𝜁_𝑖,𝜃_𝑖,𝜁^𝑡𝑖), 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 differences 𝐮=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 obtain𝜈(∇𝐮̃,∇𝐮̃)=−𝜈∇𝐮₀,∇𝐮̃−𝐮₀⋅∇𝐮₁,𝐮̃−(𝐮̃⋅∇)𝐮₁,𝐮̃−𝐮₂⋅∇𝐮₀,𝐮̃−(𝐛𝑇,𝐮̃).(3.31) Using estimates (2.5), (2.6), (2.7), and (3.15), we deduce from (3.31) that𝛿₀𝜈‖𝐮̃‖²¹≤𝜈‖‖𝐮₀‖‖₁‖𝐮̃‖₁+𝛾₀𝑀^0𝐮‖𝐮̃‖²¹+2𝛾₀𝑀^0𝐮‖‖𝐮₀‖‖₁‖𝐮̃‖₁+𝛽₁‖𝑇‖₁‖𝐮̃‖₁.(3.32)

It follows from (3.19) that𝛿₀𝜈2<𝛿⁰𝜈−𝛾₀𝑀^0𝐮−𝛽¹𝛾₁𝛿₁𝜆𝑀^0𝑇≤𝛿₀𝜈−𝛾₀𝑀^0𝐮.(3.33) Rewriting the inequality (3.32) by (3.33) as𝛿⁰𝜈2‖𝐮̃‖²¹≤𝛿₀𝜈−𝛾₀𝑀^0𝐮‖𝐮̃‖²¹≤𝜈+2𝛾₀𝑀^0𝐮‖‖𝐮₀‖‖₁‖𝐮̃‖₁+𝛽₁‖𝑇‖₁‖𝐮̃‖₁,(3.34) we obtain that‖𝐮̃‖₁≤2𝛿₀𝜈𝜈+2𝛾⁰𝑀^0𝐮‖‖𝐮₀‖‖₁+2𝛽¹𝛿₀𝜈‖𝑇‖¹≤2𝛿⁰⁻¹+4ℛ𝑒‖‖𝐮₀‖‖₁+2𝛽¹𝛿₀𝜈‖𝑇‖¹≤2ℛ‖‖𝐮₀‖‖₁+2𝛽¹𝛿₀𝜈‖𝑇‖¹,ℛ≡𝛿⁰⁻¹+2ℛ𝑒.(3.35) Taking into account the relation 𝐮=𝐮₀+𝐮̃, we come to the following estimate ‖𝐮‖₁ via ‖𝐠‖_1/2,Γ and ‖𝑇‖₁:‖𝐮‖₁≤‖‖𝐮₀‖‖₁+‖𝐮̃‖₁≤(2ℛ+1)‖‖𝐮₀‖‖₁+2𝛽¹𝛿₀𝜈‖𝑇‖¹≤𝐶₀(2ℛ+1)‖𝐠‖_1/2,Γ+2𝛽¹𝛿₀𝜈‖𝑇‖¹.(3.36)

Denote by 𝑇₀∈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+𝑇, where 𝑇∈𝒯 is a new unknown function. Set 𝑇=𝑇₀+𝑇, 𝑆=𝑇 in (3.26). We obtain𝜆∇=𝑇,−∇𝜆𝑇∇+𝑇₀𝜆𝛼𝑇,𝑇_Γ𝑁+𝐮₂⋅∇𝑇,𝑇,∇𝑇−𝜆𝛼𝑇₀,𝑇_Γ𝑁−𝐮₂⋅∇𝑇₀,𝑇−𝐮⋅∇𝑇₁,𝑇+𝑓,𝑇+𝜒,𝑇_Γ𝑁.(3.37) Using estimates (2.5)–(2.8) and (3.15) we deduce that𝛿₁𝜆‖‖𝑇‖‖²¹≤𝜆‖‖𝑇₀‖‖₁‖‖𝑇‖‖₁+𝛾₁𝑀^0𝐮‖‖𝑇₀‖‖₁‖‖𝑇‖‖+𝜆𝛾₃‖𝛼‖_Γ𝑁‖‖𝑇₀‖‖₁‖‖𝑇‖‖₁+𝛾₁𝑀^0𝑇‖𝐮‖₁‖‖𝑇‖‖₁+𝛾₂‖𝜒‖_Γ𝑁+𝛾₄‖𝑓‖‖‖𝑇‖‖₁(3.38) or‖‖𝑇‖‖₁≤1𝛿₁𝜆𝜆+𝛾¹𝑀^0𝐮+𝜆𝛾₃‖𝛼‖_Γ𝑁‖‖𝑇₀‖‖₁+𝛾¹𝑀^0𝑇𝛿₁𝜆‖𝐮‖¹+1𝛿₁𝜆𝛾²‖𝜒‖_Γ𝑁+𝛾₄‖𝑓‖.(3.39) Taking into account the relation 𝑇=𝑇₀+𝑇, we obtain from this estimate that‖𝑇‖₁≤𝐶₁(𝒩+1)‖𝜓‖_1/2,Γ𝐷+𝛾²‖𝜒‖_Γ𝑁+𝛾₄‖𝑓‖𝛿₁𝜆+𝛾¹𝑀^0𝑇𝛿₁𝜆‖𝐮‖¹,𝒩=𝜆+𝛾¹𝑀^0𝐮+𝜆𝛾₃‖𝛼‖_Γ𝑁𝛿₁𝜆.(3.40)

Using further the estimate (3.36) for 𝐮, we deduce from (3.40) that‖𝑇‖₁≤𝐶₁(𝒩+1)‖𝜓‖_1/2,Γ𝐷+𝛾²‖𝜒‖_Γ𝑁+𝛾₄‖𝑓‖+𝐶₀𝛿₁𝜆(2ℛ+1)𝛾¹𝑀^0𝑇𝛿₁𝜆‖𝐠‖^1/2,Γ+2𝛽¹𝛿₀𝜈𝛾¹𝑀^0𝑇𝛿₁𝜆‖𝑇‖¹.(3.41) From this inequality and (3.17), (3.19) we come to the following estimate:‖𝑇‖₁≤𝐶¹(𝒩+1)1−2ℛ𝑎‖𝜓‖^1/2,Γ𝐷+𝛾²‖𝜒‖_Γ𝑁+𝛾₄‖𝑓‖𝛿₁𝜆(1−2ℛ𝑎)+𝐶⁰(2ℛ+1)1−2ℛ𝑎𝛾¹𝑀^0𝑇𝛿₁𝜆‖𝐠‖^1/2,Γ.(3.42)

Using (3.42), we deduce from (3.36) that‖𝐮‖₁≤𝐶₀(2ℛ+1)1+11−2ℛ𝑎2𝛿𝛽₀𝜈¹𝛾¹𝑀^0𝑇𝛿₁𝜆‖𝐠‖^1/2,Γ+2𝛽¹𝛿₀𝜈𝐶¹(𝒩+1)1−2ℛ𝑎‖𝜓‖^1/2,Γ𝐷+𝛾²‖𝜒‖_Γ𝑁+𝛾₄‖𝑓‖𝛿₁𝜆(1−2ℛ𝑎).(3.43) Taking into account (3.17) we come to the following estimate for ‖𝐮‖₁:‖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.(3.44)

An analogous estimate holds and for the pressure difference 𝑝=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∈𝐇¹⁰(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 have𝛽₀‖‖𝐯₀‖‖₁‖𝑝‖≤−div𝐯₀,𝑝≤𝜈+2𝛾₀𝑀^0𝐮‖‖𝐯₀‖‖₁‖𝐮‖₁+𝛽₁‖𝑇‖₁‖‖𝐯₀‖‖₁.(3.45) Dividing to ‖𝐯₀‖₁≠0, we deduce that‖𝑝‖≤𝜈+2𝛾⁰𝑀^0𝐮𝛽₀‖𝐮‖₁+𝛽¹𝛽₀‖𝑇‖¹=𝛿⁰𝜈𝛽₀ℛ‖𝐮‖₁+𝛽¹𝛽₀‖𝑇‖₁.(3.46) Using (3.42) and (3.44), we come to the following final estimate for ‖𝑝‖:‖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(3.4.7)

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∈𝐻^1/2(Γ_𝐷) 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 ‖𝑇‖₁≤𝛾²‖𝜒‖_Γ𝑁𝛿₁𝜆(1−2ℛ𝑎)+𝐶⁰(2ℛ+1)1−2ℛ𝑎𝛾¹𝑀^0𝑇𝛿₁𝜆‖𝐠‖^1/2,Γ‖𝐮‖₁,(3.48)≤2𝛽¹𝛾₂‖𝜒‖_Γ𝑁𝛿₀𝜈𝛿₁𝜆(1−2ℛ𝑎)+𝐶⁰(2ℛ+1)‖𝐠‖_1/2,Γ‖𝑝‖≤2ℛ+11−2ℛ𝑎,(3.49)𝛽₀(1−2ℛ𝑎)𝛽¹𝛾₂‖𝜒‖_Γ𝑁𝛿₁𝜆+𝛿⁰𝜈𝐶₀(ℛ+ℛ𝑎)‖𝐠‖_1/2,Γ(𝐮⋅∇)𝐮,𝜉₁,(3.50)+𝜉₂+𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂+𝜇⁰2𝐼^𝐮₁−𝐼^𝐮₂,𝐮≤−⟨𝜁,𝐠⟩_Γ−𝜇₁‖𝜒‖^2Γ_𝑁.(3.51)

4. 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=𝐼₁ and the heat flux 𝜒 on the part Γ_𝑁 of Γ is a unique control; that is, we consider one-parametric control problem𝐽(𝐯,𝜒)≡𝜇⁰2‖‖𝐯−𝐯^𝑑‖‖^2𝑄+𝜇¹2‖𝜒‖^2Γ𝑁⟶inf,𝐹(𝐱,𝜒,𝐠)=0,𝐱=(𝐯,𝑝,𝑇)∈𝑋,𝜒∈𝐾₁.(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 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 𝐯_𝑑≡𝐮^𝑑(1)∈𝐋²(𝑄) and 𝐠=𝐠₁∈𝐺⊂𝐇^1/2(Γ), and let (𝐱₂,𝑢₂)≡(𝐮₂,𝑝₂,𝑇₂,𝜒₂) be a solution to problem (4.1), that corresponds to perturbed functions 𝐯_𝑑̃≡𝐮^𝑑(2)∈𝐋²(𝑄) and 𝐠̃=𝐠₂∈𝐺⊂𝐇^1/2(Γ). Setting 𝐮_𝑑=𝐮^𝑑(1)−𝐮^𝑑(2) in addition to (3.24) we note that under conditions of problem (4.1) we have𝐼^1𝐮_𝑖,𝐰=2𝐮_𝑖−𝐮^𝑑(𝑖),𝐰_𝑄,𝐼^1𝐮₁−𝐼^1𝐮₂,𝐮=2‖𝐮‖^2𝑄−𝐮,𝐮_𝑑_𝑄.(4.2) 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 form𝜈∇𝐰,∇𝜉_𝑖++𝐮⟨_𝑖𝜁⋅_𝑖∇𝐰,𝜉_𝑖+(𝐰⋅∇)𝐮_𝑖,𝜉_𝑖+𝜅𝐰⋅∇𝑇_𝑖,𝜃_𝑖−𝜎_𝑖,div𝐰,𝐰⟩_Γdi+v𝜇𝜉_0𝑖𝐮^𝑖−𝐮^𝑑(𝑖),𝐰_𝑄=0∀𝐰∈𝐇¹(Ω),(4.3)(𝐮⋅∇)𝐮,𝜉₁,𝑟=0∀𝑟∈𝐿²⁰(Ω),(4.4)+𝜉₂+𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂+𝜇₀‖𝐮‖^2𝑄−𝐮,𝐮_𝑑_𝑄≤−⟨𝜁,𝐠⟩^Γ−𝜇₁‖𝜒‖^2Γ_𝑁.(4.5)

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 obtain𝜈∇𝜉_𝑖,∇𝜅𝜉𝜆_𝑖=∇−𝜃_𝑖𝜉_𝑖⋅∇𝐮_𝑖,𝜉_𝑖−𝜅𝜉_𝑖⋅∇𝑇_𝑖,𝜃_𝑖−𝜇₀𝐮^𝑖−𝐮^𝑑(𝑖),𝜉_𝑖_𝑄,(4.6),∇𝜃_𝑖+𝛼𝜃_𝑖,𝜃_𝑖_Γ𝑁=−𝐛𝜃^𝑖,𝜉_𝑖,𝑖=1,2.(4.7) Using estimates (2.5)–(2.8) and (3.15) we have||∇𝜉𝜉_𝑖𝑖,∇𝜉_𝑖≥𝛿₀‖‖𝜉_𝑖‖‖²¹,∇𝜃_𝑖,∇𝜃_𝑖≥𝛿₁‖‖𝜃_𝑖‖‖²¹,(4.8)𝜅||𝜉⋅∇𝐮_𝑖,𝜉_𝑖||≤𝛾₀‖‖𝐮_𝑖‖‖₁‖‖𝜉_𝑖‖‖²¹≤𝛾₀𝑀^0𝐮‖‖𝜉_𝑖‖‖²¹,(4.9)_𝑖‖‖𝐮⋅∇𝑇_𝑖,𝜃_𝑖||≤𝜅𝛾₁𝑀^0𝑇‖‖𝜉_𝑖‖‖₁‖‖𝜃_𝑖‖‖₁,||𝐛𝜃_𝑖,𝜉_𝑖||≤𝛽₁‖‖𝜃_𝑖‖‖₁‖‖𝜉_𝑖‖‖₁,(4.10)_𝑖−𝐮|||𝐮^𝑑(𝑖)‖‖_𝑄≤‖‖𝐮_𝑖‖‖_𝑄+‖‖𝐮^𝑑(𝑖)‖‖_𝑄≤𝛾₄𝑀^0𝐮+‖‖𝐮^𝑑(𝑖)‖‖_𝑄≤𝛿₀𝜈𝛾₄𝛾⁰⁻¹ℛ𝑒+ℛ𝑒⁰,(4.11)_𝑖−𝐮^𝑑(𝑖),𝜉_𝑖_𝑄|||≤‖‖𝐮_𝑖−𝐮^𝑑(𝑖)‖‖_𝑄‖‖𝜉_𝑖‖‖_𝑄≤𝛿₀𝜈𝛾ℛ𝑒+ℛ𝑒⁰‖‖𝜉_𝑖‖‖₁,(4.12) where𝛾=𝛾²⁴𝛾⁰⁻¹,ℛ𝑒⁰=𝛾⁰𝛿₀𝜈𝛾₄max‖‖𝐮^𝑑(1)‖‖_𝑄,‖‖𝐮^𝑑(2)‖‖_𝑄.(4.13)

By virtue of (4.8)–(4.10) and (4.12), we deduce from (4.7) and (4.6) that𝜅‖‖𝜃_𝑖‖‖₁≤𝛽¹𝛿₁𝜆‖‖𝜉^𝑖𝛿‖‖₁,(4.14)₀𝜈‖‖𝜉_𝑖‖‖²¹≤𝛾₀𝑀^0𝐮‖‖𝜉_𝑖‖‖²¹+𝜅𝛾₁𝑀^0𝑇‖‖𝜃_𝑖‖‖₁+𝜇₀𝛿₀𝜈𝛾ℛ𝑒+ℛ𝑒⁰‖‖𝜉_𝑖‖‖₁.(4.15) Taking into account (4.14), we obtain from (4.15) that𝛿⁰𝜈−𝛾₀𝑀^0𝐮−𝛽¹𝛾₁𝛿₁𝜆𝑀^0𝑇‖‖𝜉^𝑖‖‖²¹≤𝜇₀𝛿₀𝜈𝛾ℛ𝑒+ℛ𝑒⁰‖‖𝜉_𝑖‖‖₁.(4.16) Using (3.33) we deduce successfully from (4.16), (4.14) that‖‖𝜉_𝑖‖‖₁≤2𝜇₀𝛾ℛ𝑒+ℛ𝑒⁰,𝜅‖‖𝜃_𝑖‖‖₁≤2𝜇₀𝛾𝛽¹𝛿₁𝜆ℛ𝑒+ℛ𝑒⁰.(4.17)

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∈𝐿²⁰(Ω) and any small number 𝛿>0 there exists a function 𝐯_𝑖∈𝐇¹⁰(Ω), 𝐯_𝑖≠0, such that the inequality−div𝐯_𝑖,𝜎_𝑖≥𝛽₀‖‖𝐯_𝑖‖‖₁‖‖𝜎_𝑖‖‖,𝑖=1,2,𝛽₀=𝛽−𝛿(4.18) holds. Setting in (4.3) 𝐰=𝐯_𝑖 and using this estimate together with estimates (2.6), (3.15), (4.11), we have𝛽₀‖‖𝐯_𝑖‖‖₁‖‖𝜎_𝑖‖‖≤≤𝜈−‖‖𝐯_𝑖div𝐯_𝑖,𝜎_𝑖‖‖₁‖‖𝜉_𝑖‖‖₁+2𝛾₀𝑀^0𝐮‖‖𝐯_𝑖‖‖₁‖‖𝜉_𝑖‖‖₁+𝜅𝛾₁𝑀^0𝑇‖‖𝐯_𝑖‖‖₁‖‖𝜃_𝑖‖‖₁+𝜇₀𝛿₀𝜈𝛾ℛ𝑒+ℛ𝑒⁰‖‖𝐯_𝑖‖‖₁.(4.19) From this inequality we deduce by (4.17) that‖‖𝜎_𝑖‖‖≤1𝛽₀𝜈+2𝛾₀≤𝛿𝑀^0𝐮‖‖𝜉_𝑖‖‖₁+𝛾₁𝑀^0𝑇𝜅‖‖𝜃_𝑖‖‖₁+𝜇₀𝛿₀𝜈𝛾ℛ𝑒+ℛ𝑒⁰₀𝜈𝛽₀ℛ‖‖𝜉^𝑖‖‖₁+𝛾¹𝑀^0𝑇𝛿₀𝜈𝜅‖‖𝜃^𝑖‖‖₁+𝜇₀𝛾ℛ𝑒+ℛ𝑒⁰.(4.20) Taking into account (4.17), we come from (4.20) to the estimate‖‖𝜎_𝑖‖‖≤𝜇₀𝛾𝛿⁰𝜈𝛽₀ℛ𝑒+ℛ𝑒⁰(2ℛ+2ℛ𝑎+1).(4.21)

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||⟨𝜁_𝑖,𝐰⟩_Γ||≤𝜈+2𝛾₀𝑀^0𝐮‖‖𝜉_𝑖‖‖₁+𝛾₁𝑀^0𝑇𝜅‖‖𝜃_𝑖‖‖₁+𝐶_𝑑‖‖𝜎_𝑖‖‖+𝜇₀𝛿₀𝜈𝛾ℛ𝑒+ℛ𝑒⁰‖𝐰‖₁≤𝜇₀𝛿₀𝜈𝛾1+𝐶_𝑑𝛽⁰⁻¹ℛ𝑒+ℛ𝑒⁰(2ℛ+2ℛ𝑎+1)‖𝐰‖₁∀𝐰∈𝐇¹(Ω).(4.22) As 𝜁=𝜁₁−𝜁₂ we obtain from this inequality that‖𝜁‖_−1/2,Γ≤𝜇₀𝑎,𝑎=2𝛿₀𝜈𝛾1+𝐶_𝑑𝛽⁰⁻¹ℛ𝑒+ℛ𝑒⁰(2ℛ+2ℛ𝑎+1).(4.23)

Taking into account (2.6), (3.48), (3.49), and estimates (4.17) for 𝜉_𝑖, 𝜃_𝑖, we have ||(𝐮⋅∇)𝐮,𝜉₁+𝜉₂||≤≤𝛾4𝜇₀‖₀𝐮‖²¹‖‖𝜉₁‖‖₁+‖‖𝜉₂‖‖₁𝛾₀𝛾ℛ𝑒+ℛ𝑒⁰2𝛽¹𝛾₂‖𝜒‖_Γ𝑁𝛿₀𝜈𝛿₁𝜆(1−2ℛ𝑎)+𝐶⁰(2ℛ+1)‖𝐠‖_1/2,Γ1−2ℛ𝑎²𝜅||𝐮⋅∇𝑇,𝜃₁,+𝜃₂||≤4𝜇⁰𝛾₁𝛾𝛽₁ℛ𝑒+ℛ𝑒⁰𝛿₁𝜆2𝛽¹𝛾₂‖𝜒‖_Γ𝑁𝛿₀𝜈𝛿₁𝜆(1−2ℛ𝑎)+𝐶⁰(2ℛ+1)‖𝐠‖_1/2,Γ×𝛾1−2ℛ𝑎₂‖𝜒‖_Γ𝑁𝛿₁𝜆(1−2ℛ𝑎)+𝐶⁰(2ℛ+1)1−2ℛ𝑎𝛾¹𝑀^0𝑇𝛿₁𝜆‖𝐠‖^1/2,Γ.(4.24) It follows from (4.24) that||(𝐮⋅∇)𝐮,𝜉₁+𝜉₂+𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂||≤𝜇₀𝑏‖𝐠‖^21/2,Γ+𝑐‖𝜒‖^2Γ_𝑁.(4.25) Here constants 𝑏 and 𝑐 are given by𝑏=4𝛾𝛾₀𝐶²⁰(2ℛ+1)²ℛ𝑒+ℛ𝑒⁰(1−2ℛ𝑎)²3+𝛾𝛾¹₀²𝒫²ℛ𝑎²𝑐=4𝛾𝛾₀,𝛿𝛽¹₀𝜈𝛿𝛾²₁𝜆²ℛ𝑒+ℛ𝑒⁰(1−2ℛ𝑎)²12+𝛾𝛾¹₀²𝒫².(4.26)

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:(1−𝜀)𝜇₁≥𝜇₀𝑐,𝜀=const>0.(4.27) Under condition (4.27) we deduce from (4.25) that||(𝐮⋅∇)𝐮,𝜉₁+𝜉₂+𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂||≤𝜇₀𝑏‖𝐠‖^21/2,Γ+(1−𝜀)𝜇₁‖𝜒‖^2Γ_𝑁.(4.28) 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 inequality𝜇₀‖𝐮‖^2𝑄−𝐮,𝐮_𝑑_𝑄≤−(𝐮⋅∇)𝐮,𝜉¹+𝜉₂−𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂−⟨𝜁,𝐠⟩_Γ−𝜇₁‖𝜒‖^2Γ_𝑁≤−𝜀𝜇₁‖𝜒‖^2Γ_𝑁+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ.(4.29) It follows from this inequality that𝜇₀‖𝐮‖^2𝑄≤𝜇₀𝐮,𝐮_𝑑_𝑄−𝜀𝜇₁‖𝜒‖^2Γ_𝑁+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ.(4.30) Excluding nonpositive term −𝜀𝜇₁‖𝜒‖^2Γ_𝑁 from the right-hand side of (4.30), we deduce from (4.30) that‖𝐮‖^2𝑄≤‖‖𝐮_𝑑‖‖_𝑄‖𝐮‖_𝑄+𝑎‖𝐠‖_1/2,Γ+𝑏‖𝐠‖^21/2,Γ.(4.31)

Equation (4.31) is a quadratic inequality for ‖𝐮‖_𝑄. Solving it we come to the following estimate for ‖𝐮‖_𝑄:‖𝐮‖_𝑄≤‖‖𝐮_𝑑‖‖_𝑄+𝑎‖𝐠‖_1/2,Γ+𝑏‖𝐠‖^21/2,Γ^1/2.(4.32) As 𝐮=𝐮₁−𝐮₂, 𝐮_𝑑=𝐮^𝑑(1)−𝐮^𝑑(2), 𝐠=𝐠₁−𝐠₂, the estimate (4.32) is equivalent to the following estimate for the velocity difference 𝐮₁−𝐮₂:‖‖𝐮₁−𝐮₂‖‖_𝑄≤‖‖𝐮^𝑑(1)−𝐮^𝑑(2)‖‖_𝑄+𝑎‖‖𝐠₁−𝐠₂‖‖_1/2,Γ+𝑏‖‖𝐠₁−𝐠₂‖‖^21/2,Γ^1/2.(4.33) This estimate under 𝑄=Ω has the sense of the stability estimate in 𝐋²(Ω) of the component 𝐮̂ of the solution (𝐮̂,𝑝,̂𝑇,𝜒) to problem (4.1) relative to small perturbations of functions 𝐯_𝑑∈𝐋²(Ω) and 𝐠∈𝐺 in the norms of 𝐋²(Ω) and 𝐇^1/2(Γ), respectively. In particular case where 𝐠₁=𝐠₂ the estimate (4.33) transforms to “exact” a priori estimate ‖𝐮₁−𝐮₂‖_𝑄≤‖𝐮^𝑑(1)−𝐮^𝑑(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 𝐮^𝑑(1)=𝐮^𝑑(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) that𝜀𝜇₁‖𝜒‖^2Γ_𝑁≤−𝜇₀‖𝐮‖^2𝑄+𝜇₀‖‖𝐮_𝑑‖‖_𝑄‖𝐮‖_𝑄+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ≤𝜇₀‖‖𝐮_𝑑‖‖^2𝑄+𝜇₀‖‖𝐮_𝑑‖‖_𝑄𝑎‖𝐠‖_1/2,Γ+𝑏‖𝐠‖^21/2,Γ^1/2+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ≤𝜇₀‖‖𝐮^𝑑‖‖_𝑄+𝑎‖𝐠‖_1/2,Γ+𝑏‖𝐠‖^21/2,Γ^1/2².(4.34) From (4.34) and (3.48)–(3.50) we come to the following stability estimates:‖‖𝜒₁−𝜒₂‖‖_Γ𝑁≤𝜇₀𝜀𝜇₁‖‖𝐮Δ,(4.35)₁−𝐮₂‖‖₁≤2𝛽¹𝛾₂𝛿₀𝜈𝛿₁𝜆(1−2ℛ𝑎)𝜇₀𝜀𝜇₁Δ+𝐶⁰(2ℛ+1)‖‖𝐠₁−𝐠₂‖‖_1/2,Γ‖‖𝑇1−2ℛ𝑎,(4.36)₁−𝑇₂‖‖₁≤𝛾²𝛿₁𝜆(1−2ℛ𝑎)𝜇₀𝜀𝜇₁Δ+𝐶⁰(2ℛ+1)1−2ℛ𝑎𝛾¹𝑀^0𝑇𝛿₁𝜆‖‖𝐠¹‖‖𝑝−𝐠₂‖‖_1/2,Γ,(4.37)₁−𝑝₂‖‖≤(2ℛ+1)𝛽¹𝛾₂𝛽₀𝛿₁𝜆(1−2ℛ𝑎)𝜇₀𝜀𝜇₁Δ+𝐶⁰(2ℛ+1)(ℛ+ℛ𝑎)1−2ℛ𝑎𝛿⁰𝜈𝛽₀‖‖𝐠₁−𝐠₂‖‖_1/2,Γ,(4.38) whereΔ=‖‖𝐮^𝑑(1)−𝐮^𝑑(2)‖‖_𝑄+𝑎‖‖𝐠₁−𝐠₂‖‖_1/2,Γ+𝑏‖‖𝐠₁−𝐠₂‖‖^21/2,Γ^1/2.(4.39) Thus we have proved the theorem.

Theorem 4.1. Let, under conditions (i), (ii), (iii) for 𝐾₁ and (3.19), the quadruple (𝐮_𝑖,𝑝_𝑖,𝑇_𝑖,𝜒_𝑖) be a solution to problem (4.1) corresponding to given functions 𝐯_𝑑=𝐮^𝑑(𝑖)∈𝐋²(𝑄) and 𝐠_𝑖∈𝐺⊂𝐇^1/2(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 problem𝐽(𝐯,𝜒,𝜓,𝑓)≡𝜇⁰2‖‖𝐯−𝐯^𝑑‖‖^2𝑄+𝜇¹2‖𝜒‖^2Γ𝑁+𝜇²2‖𝜓‖^21/2,Γ𝐷+𝜇³2‖𝑓‖²𝐹(𝐱,𝑢,𝐠)=0,𝐱=(𝐯,𝑝,𝑇)∈𝑋,𝑢=(𝜒,𝜓,𝑓)∈𝐾⟶inf,(4.40) corresponding to the cost functional 𝐼₁(𝐯)=‖𝐯−𝐯_𝑑‖^2𝑄. Let (𝐱₁,𝑢₁)≡(𝐮₁,𝑝₁,𝑇₁,𝜒₁,𝜓₁,𝑓₁) be a solution to problem (4.40) corresponding to given functions 𝐯_𝑑≡𝐮^𝑑(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 𝐯_𝑑̃≡𝐮^𝑑(2)∈𝐋²(𝑄) and 𝐠̃=𝐠₂∈𝐺. Setting 𝐮_𝑑=𝐮^𝑑(1)−𝐮^𝑑(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(𝐮⋅≤∇)−𝐮⟨,𝜁𝜉,¹𝐠+⟩_Γ𝜉₂+𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂+𝜇₀‖𝐮‖^2𝑄−𝐮,𝐮_𝑑_𝑄−𝜇₁‖𝜒‖^2Γ_𝑁−𝜇₂‖𝜓‖^21/2,Γ_𝐷−𝜇₃‖𝑓‖².(4.41)

From (4.3), (4.4), and (3.22) we come to the same estimates (4.17), (4.21), and (4.23) for norms ‖𝜉_𝑖‖₁, ‖𝜃_𝑖‖₁, ‖𝜎_𝑖‖ and ‖𝜁‖_−1/2,Γ. Taking into account these estimates and estimates (3.42), (3.44) for ‖𝑇‖₁, ‖𝐮‖₁, we deduce that||(𝐮≤⋅∇𝛾)₀𝐮,𝜉₁+𝜉₂||×‖2𝐮𝛽‖²¹₁‖‖𝜉₁‖‖₁+‖‖𝜉₂‖‖₁≤4𝜇₀𝛾₀𝛾ℛ𝑒+ℛ𝑒⁰𝐶₁(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,≤4𝜇₀+𝜃₂||𝛾₁𝛾𝛽₁ℛ𝑒+ℛ𝑒⁰×2𝛽₁𝛿₁𝜆𝐶₁(𝒩+1)‖𝜓‖_1/2,Γ𝐷𝛿₀𝜈(1−2ℛ𝑎)+2𝛽¹𝛾₂‖𝜒‖_Γ𝑁+𝛾₄‖𝑓‖𝛿₀𝜈𝛿₁𝜆(1−2ℛ𝑎)+𝐶⁰(2ℛ+1)‖𝐠‖_1/2,Γ×𝐶1−2ℛ𝑎₁(𝒩+1)1−2ℛ𝑎‖𝜓‖^1/2,Γ𝐷+𝛾²‖𝜒‖_Γ𝑁+𝛾₄‖𝑓‖𝛿₁𝜆(1−2ℛ𝑎)+𝐶⁰(2ℛ+1)1−2ℛ𝑎𝛾¹𝑀^0𝑇𝛿₁𝜆‖𝐠‖^1/2,Γ.(4.42) Here parameters 𝛾 and ℛ𝑒⁰ are given by (4.13). From (4.42) we obtain that||(𝐮⋅∇)𝐮,𝜉₁+𝜉₂+𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂||≤𝜇₀𝑏‖𝐠‖^21/2,Γ+𝑐₁‖𝜒‖^2Γ_𝑁+𝑐₂‖𝜓‖^21/2,Γ_𝐷+𝑐₃‖𝑓‖².(4.43) Here constants 𝑏, 𝑐₁, 𝑐₂, and 𝑐₃ are given by relations𝑏=8𝛾₀𝛾𝐶²⁰(2ℛ+1)²ℛ𝑒+ℛ𝑒⁰(1−2ℛ𝑎)²3+𝛾𝛾¹₀²𝒫²ℛ𝑎²𝑐₁,=8𝛾₀𝛾𝛽¹𝛿₀𝜈𝛿𝛾²₁𝜆²ℛ𝑒+ℛ𝑒⁰(1−2ℛ𝑎)²12+𝛾𝛾¹₀²𝒫²𝑐₂,=8𝛾₀𝛾𝐶²¹(𝒩+1)²𝛿𝛽¹₀𝜈²ℛ𝑒+ℛ𝑒⁰(1−2ℛ𝑎)²12+𝛾𝛾¹₀²𝒫²𝑐₃,=8𝛾₀𝛾𝛽¹𝛿₀𝜈𝛿𝛾⁴₁𝜆²ℛ𝑒+ℛ𝑒⁰(1−2ℛ𝑎)²12+𝛾𝛾¹₀²𝒫².(4.44)

Let the data for problem (4.40) and parameters 𝜇₀, 𝜇₁, 𝜇₂, and 𝜇₃ be such that𝜇₁(1−𝜀)≥𝜇₀𝑐₁,𝜇₂(1−𝜀)≥𝜇₀𝑐₂,𝜇₃(1−𝜀)≥𝜇₀𝑐₃,𝜀=const>0.(4.45) Under condition (4.45) we deduce from (4.43) that||(𝐮≤⋅∇()1𝐮−,𝜉𝜀₁)+𝜇₁𝜉₂+𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂||‖𝜒‖^2Γ_𝑁+(1−𝜀)𝜇₂‖𝜓‖^21/2,Γ_𝐷+(1−𝜀)𝜇₃‖𝑓‖²+𝜇₀𝑏‖𝐠‖^21/2,Γ.(4.46) Taking into account (4.46) and (4.23), we come from (4.41) to the inequality𝜇₀‖𝐮‖^2𝑄−𝐮,𝐮_𝑑_𝑄≤−−⟨𝜁(,𝐮𝐠⟩⋅_Γ∇)𝐮,𝜉¹+𝜉₂−𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂−𝜇₁‖𝜒‖^2Γ_𝑁−𝜇₂‖𝜓‖^21/2,Γ_𝐷−𝜇₃‖𝑓‖²≤−𝜀𝜇₁‖𝜒‖^2Γ_𝑁−𝜀𝜇₂‖𝜓‖^21/2,Γ_𝐷−𝜀𝜇₃‖𝑓‖²+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ.(4.47) It follows from this inequality that𝜇₀‖𝐮‖^2𝑄≤𝜇₀𝐮,𝐮_𝑑_𝑄−𝜀𝜇₁‖𝜒‖^2Γ_𝑁−𝜀𝜇₂‖𝜓‖^21/2,Γ_𝐷−𝜀𝜇₃‖𝑓‖²+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ.(4.48) 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 𝐯_𝑑∈𝐋²(Ω) and 𝐠∈𝐺 in the norms of 𝐋²(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.

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) as𝜀𝜇₁‖𝜒‖^2Γ_𝑁+𝜀𝜇₂‖𝜓‖^21/2,Γ_𝐷+𝜀𝜇₃‖𝑓‖²≤−𝜇₀‖𝐮‖^2𝑄+𝜇₀‖𝐮‖_𝑄‖‖𝐮_𝑑‖‖_𝑄+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ≤𝜇₀‖‖𝐮^𝑑‖‖_𝑄+𝑎‖𝐠‖_1/2,Γ+𝑏‖𝐠‖^21/2,Γ^1/2².(4.49) From this inequality and from (3.42)–(3.47) we come to the following stability estimates:‖‖𝜒₁−𝜒₂‖‖_Γ𝑁≤𝜇₀𝜀𝜇₁Δ,‖‖𝜓₁−𝜓₂‖‖_1/2,Γ𝐷≤𝜇₀𝜀𝜇₂Δ,‖‖𝑓₁−𝑓₂‖‖≤𝜇₀𝜀𝜇₃‖‖𝐮Δ,(4.50)₁−𝐮₂‖‖₁≤2𝛽¹𝑑𝛿₀𝜈(1−2ℛ𝑎)𝜇₀𝜀Δ+𝐶⁰(2ℛ+1)‖‖𝐠₁−𝐠₂‖‖_1/2,Γ‖‖𝑇1−2ℛ𝑎,(4.51)₁−𝑇₂‖‖₁≤𝑑1−2ℛ𝑎𝜇₀𝜀Δ+𝐶⁰(2ℛ+1)1−2ℛ𝑎𝛾¹𝑀^0𝑇𝛿₁𝜆‖‖𝐠¹‖‖𝑝−𝐠₂‖‖_1/2,Γ,(4.52)₁−𝑝₂‖‖≤(2ℛ+1)𝛽₀(1−2ℛ𝑎)𝛽¹𝑑𝜇₀𝜀Δ+𝛿⁰𝜈𝐶₀(ℛ+ℛ𝑎)‖‖𝐠₁−𝐠₂‖‖_1/2,Γ.(4.53) Here a constant 𝑑 depending on 𝜇₁, 𝜇₂, and 𝜇₃ is given by𝑑=𝛾²𝛿₁𝜆√𝜇₁+𝐶¹(𝒩+1)√𝜇₂+𝛾⁴𝛿₁𝜆√𝜇₃,(4.54) 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 (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^𝑑(𝑖)∈𝐋²(𝑄) and 𝐠_𝑖∈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 𝐽(𝐯,𝜒)≡𝜇⁰2‖‖𝐯−𝐯^𝑑‖‖^21,𝑄+𝜇¹2‖𝜒‖^2Γ𝑁⟶inf,𝐹(𝐱,𝜒,𝐠)=0,𝐱∈𝑋,𝜒∈𝐾₁(4.55) corresponding to the cost functional 𝐼₂(𝐯)=‖𝐯−𝐯_𝑑‖^21,𝑄. Let us denote by (𝐱₁,𝑢₁)≡(𝐮₁,𝑝₁,𝑇₁,𝜒₁) a solution to problem (4.55) which corresponds to given functions 𝐯_𝑑≡𝐮^𝑑(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 𝐯_𝑑̃≡𝐮^𝑑(2)∈𝐋²(𝑄) and 𝐠̃=𝐠₂∈𝐺. Setting 𝐮_𝑑=𝐮^𝑑(1)−𝐮^𝑑(2) in addition to (3.24) we note that under conditions of problem (4.55) we have𝐼^2𝐮_𝑖,𝐰=2𝐮_𝑖−𝐮^𝑑(𝑖),𝐰_1,𝑄,𝐼^2𝐮₁−𝐼^2𝐮₂,𝐮=2‖𝐮‖^21,𝑄−𝐮,𝐮_𝑑_1,𝑄.(4.56) 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 relations𝜈∇𝐰,∇𝜉_𝑖+𝐮_𝑖⋅∇𝐰,𝜉_𝑖+(𝐰⋅∇)𝐮_𝑖,𝜉_𝑖+𝜅𝐰⋅∇𝑇_𝑖,𝜃_𝑖−𝜎_𝑖,div𝐰+⟨𝜁_𝑖,𝐰⟩_Γ(𝐮⋅∇=)𝐮−,𝜉𝜇₀₁𝐮^𝑖−𝐮^𝑑(𝑖),𝐰_1,𝑄∀𝐰∈𝐇¹(Ω),𝑖=1,2,(4.57)+𝜉₂+𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂+𝜇₀‖𝐮‖^21,𝑄−𝐮,𝐮_𝑑_1,𝑄≤−⟨𝜁,𝐠⟩^Γ−𝜇₁‖𝜒‖^2Γ_𝑁.(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 relation𝜈∇𝜉_𝑖,∇𝜉_𝑖=−𝜉_𝑖⋅∇𝐮_𝑖,𝜉_𝑖−𝜅𝜉_𝑖⋅∇𝑇_𝑖,𝜃_𝑖−𝜇₀𝐮^𝑖−𝐮^𝑑(𝑖),𝜉_𝑖_1,𝑄.(4.59) Using estimates (3.15) we deduce in addition to (4.8)–(4.10) that|||𝐮_𝑖‖‖𝐮_𝑖−𝐮^𝑑(𝑖)‖‖_1,𝑄≤‖‖𝐮_𝑖‖‖_1,𝑄+‖‖𝐮^𝑑(𝑖)‖‖_1,𝑄≤𝑀^0𝐮+‖‖𝐮^𝑑(𝑖)‖‖_𝑄≤𝛿₀𝜈𝛾⁰⁻¹ℛ𝑒+ℛ𝑒⁰,−𝐮^𝑑(𝑖),𝜉_𝑖_1,𝑄|||≤‖‖𝐮_𝑖−𝐮^𝑑(𝑖)‖‖_1,𝑄‖‖𝜉_𝑖‖‖_1,𝑄≤𝑀^0𝐮+‖‖𝐮^𝑑(𝑖)‖‖_1,𝑄‖‖𝜉^𝑖‖‖₁≤𝛿₀𝜈𝛾ℛ𝑒+ℛ𝑒⁰‖‖𝜉_𝑖‖‖₁,(4.60) where𝛾=𝛾⁰⁻¹,ℛ𝑒⁰=𝛾⁰𝛿₀𝜈max‖‖𝐮^𝑑(1)‖‖_1,𝑄,‖‖𝐮^𝑑(2)‖‖_1,𝑄.(4.61) Proceeding further as above in study of problem (4.1) we come to the estimates for 𝜉_𝑖, 𝜃_𝑖, 𝜎_𝑖 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 inequality𝜇₀‖𝐮‖^21,𝑄−𝐮,𝐮_𝑑_1,𝑄≤−(𝐮⋅∇)𝐮,𝜉¹+𝜉₂−𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂−‖𝜁‖_−1/2,Γ‖𝐠‖_1/2,Γ−𝜇₁‖𝜒‖^2Γ_𝑁≤−𝜀𝜇₁‖𝜒‖^2Γ_𝑁+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ.(4.62) It follows from this inequality that𝜇₀‖𝐮‖^21,𝑄≤𝜇₀𝐮,𝐮_𝑑_1,𝑄−𝜀𝜇₁‖𝜒‖^2Γ_𝑁+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ.(4.63) Excluding nonpositive term −𝜀𝜇₁‖𝜒‖^2Γ_𝑁, we deduce from (4.63) that‖𝐮‖^21,𝑄≤‖‖𝐮_𝑑‖‖_1,𝑄‖𝐮‖_1,𝑄+𝑎‖𝐠‖_1/2,Γ+𝑏‖𝐠‖^21/2,Γ.(4.64) Equation (4.31) is a quadratic inequality relative to ‖𝐮‖_1,𝑄. By solving it we come to the estimate‖𝐮‖_1,𝑄≤‖‖𝐮_𝑑‖‖_1,𝑄+𝑎‖𝐠‖_1/2,Γ+𝑏‖𝐠‖^21/2,Γ^1/2(4.65) which is equivalent to the following estimate for 𝐮₁−𝐮₂: ‖‖𝐮₁−𝐮₂‖‖_1,𝑄≤‖‖𝐮^𝑑(1)−𝐮^𝑑(2)‖‖_1,𝑄+𝑎‖‖𝐠₁−𝐠₂‖‖_1/2,Γ+𝑏‖‖𝐠₁−𝐠₂‖‖^21/2,Γ^1/2.(4.66) 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,𝑝,̂𝑇,𝜒) to problem (4.55) relative to small perturbations of functions 𝐯_𝑑∈𝐇¹(Ω) and 𝐠∈𝐺 in the norms of 𝐇¹(Ω) and 𝐇^1/2(Γ) respectively. In the case where 𝐮^𝑑(1)=𝐮^𝑑(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) that𝜀𝜇₁‖𝜒‖^2Γ_𝑁≤𝜇₀‖𝐮‖_1,𝑄‖‖𝐮_𝑑‖‖_1,𝑄+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ≤𝜇₀‖‖𝐮^𝑑‖‖_1,𝑄+𝑎‖𝐠‖_1/2,Γ+𝑏‖𝐠‖^21/2,Γ^1/2².(4.67) From (4.67) and (3.48)–(3.50) we come to the estimates (4.35)–(4.38) where one should setΔ=‖‖𝐮^𝑑(1)−𝐮^𝑑(2)‖‖_1,𝑄+𝑎‖‖𝐠₁−𝐠₂‖‖_1/2,Γ+𝑏‖‖𝐠₁−𝐠₂‖‖^21/2,Γ^1/2.(4.68)

Thus we have proved the following theorem.

Theorem 4.3. Let, under conditions (i), (ii), (iii) for 𝐾₁ and (3.19), the quadruple (𝐮_𝑖,𝑝_𝑖,𝑇,𝜒_𝑖) be a solution to problem (4.55) corresponding to given functions 𝐯_𝑑=𝐮^𝑑(𝑖)∈𝐇¹(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 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 problem𝐽(𝐯,𝜒,𝜓,𝑓)≡𝜇⁰2‖‖𝐯−𝐯^𝑑‖‖^21,𝑄+𝜇¹2‖𝜒‖^2Γ𝑁+𝜇²2‖𝜓‖^21/2,Γ𝐷+𝜇³2‖𝑓‖²𝐹(𝐱,𝜒,𝜓,𝑓,𝐠)=0,(𝐱,𝜒,𝜓,𝑓)∈𝑋×𝐾₁⟶inf,×𝐾₂×𝐾₃.(4.69) 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 (𝐮_𝑖,𝑝_𝑖,𝑇_𝑖,𝜒_𝑖,𝜓_𝑖,𝑓_𝑖) be a solution to problem (4.69) corresponding to given functions 𝐮_𝑑=𝐮^𝑑(𝑖)∈𝐇¹(𝑄) and 𝐠_𝑖∈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 problem𝐽(𝐯,𝜒)≡𝜇⁰2‖‖rot𝐯−𝜁^𝑑‖‖^2𝑄+𝜇¹2‖𝜒‖^2Γ𝑁⟶inf,𝐹(𝐱,𝜒,𝐠)=0,𝐱∈𝑋,𝜒∈𝐾₁,(5.1) which corresponds to the cost functional 𝐼₃(𝐯)=‖rot𝐯−𝜁_𝑑‖^2𝑄. Let (𝐱₁,𝑢₁)≡(𝐮₁,𝑝₁,𝑇₁,𝜒₁) be a solution to problem (5.1) corresponding to given functions 𝜁_𝑑≡𝜁^𝑑(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 𝜁̃_𝑑≡𝜁^𝑑(2)∈𝐋²(𝑄) and 𝐠̃=𝐠₂∈𝐺. Setting 𝐮_𝑑=𝐮^𝑑(1)−𝐮^𝑑(2) in addition to (3.24), we have under conditions of problem (4.1) 𝐼^3𝐮_𝑖,𝐰=2rot𝐮_𝑖−𝜁^𝑑(𝑖),rot𝐰_𝑄,𝐼^3𝐮₁−𝐼^3𝐮₂,𝐮=2‖rot𝐮‖^2𝑄−rot𝐮,𝜁_𝑑_𝑄.(5.2) 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 relations𝜈∇𝐰,∇𝜉_𝑖+𝐮_𝑖⋅∇𝐰,𝜉_𝑖+(𝐰⋅∇)𝐮_𝑖,𝜉_𝑖+𝜅𝐰⋅∇𝑇_𝑖,𝜃_𝑖−𝜎_𝑖,div𝐰+⟨𝜁_𝑖,𝐰⟩_Γ(𝐮⋅∇)𝐮,𝜉₁=−𝜇₀rot𝐮^𝑖−𝜁^𝑑(𝑖),𝐰_𝑄∀𝐰∈𝐇¹(Ω),(5.3)+𝜉₂+𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂+𝜇₀‖rot𝐮‖^2𝑄−rot𝐮,𝜁_𝑑_𝑄≤−⟨𝜁,𝐠⟩^Γ−𝜇₁‖𝜒‖^2Γ_𝑁.(5.4)

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 relation𝜈∇𝜉_𝑖,∇𝜉_𝑖=−𝜉_𝑖⋅∇𝐮_𝑖,𝜉_𝑖−𝜅𝜉_𝑖⋅∇𝑇_𝑖,𝜃_𝑖−𝜇₀rot𝐮^𝑖−𝜁^𝑑(𝑖),𝜉_𝑖_𝑄.(5.5) Using (2.9), (3.15) we deduce in addition to (4.8)–(4.10) that‖‖rot𝐮_𝑖−𝜁^𝑑(𝑖)‖‖_𝑄≤‖‖rot𝐮_𝑖‖‖_𝑄+‖‖𝜁^𝑑(𝑖)‖‖_𝑄≤𝐶_𝑟𝑀^0𝐮+‖‖𝜁^𝑑(𝑖)‖‖_𝑄≤𝛿⁰𝜈𝐶^2𝑟|||rot𝐮_𝑖𝛾₀ℛ𝑒+ℛ𝑒⁰,−𝜁^𝑑(𝑖),rot𝜉_𝑖_𝑄|||≤‖‖rot𝐮_𝑖−𝜁^𝑑(𝑖)‖‖_𝑄‖‖𝜉_𝑖‖‖_𝑄≤𝐶_𝑟𝐶^𝑟𝑀^0𝐮+‖‖𝜁^𝑑(𝑖)‖‖_𝑄‖‖𝜉^𝑖‖‖₁≤𝛿₀𝜈𝛾ℛ𝑒+ℛ𝑒⁰‖‖𝜉_𝑖‖‖₁,(5.6) where𝛾=𝐶^2𝑟,ℛ𝑒⁰=𝛾⁰𝛿₀𝜈𝐶_𝑟max‖‖𝜁^𝑑(1)‖‖_𝑄,‖‖𝜁^𝑑(2)‖‖_𝑄.(5.7) Arguing as above in analysis of problem (4.1) we come to the same estimates (4.17), (4.21), and (4.23) for ‖𝜉_𝑖‖₁, ‖𝜃_𝑖‖₁, ‖𝜎_𝑖‖, 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 inequality𝜇₀‖≤ro−t𝐮(‖𝐮^2𝑄⋅−∇)𝐮ro,t𝜉₁𝐮,𝜁_𝑑_𝑄+𝜉₂−𝜅𝐮⋅∇𝑇,𝜃₁+𝜃₂+‖𝜁‖_−1/2,Γ‖𝐠‖_1/2,Γ−𝜇₁‖𝜒‖^2Γ_𝑁≤−𝜀𝜇₁‖𝜒‖^2Γ_𝑁+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ.(5.8) It follows from this inequality that𝜇₀‖rot𝐮‖^2𝑄≤𝜇₀rot𝐮,𝜁_𝑑_𝑄−𝜀𝜇₁‖𝜒‖^2Γ_𝑁+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ.(5.9) Excluding nonpositive term −𝜀𝜇₁‖𝜒‖^2Γ_𝑁, we deduce from (5.9) that‖rot𝐮‖^2𝑄≤‖‖𝜁_𝑑‖‖_𝑄‖rot𝐮‖_𝑄+𝑎‖𝐠‖_1/2,Γ+𝑏‖𝐠‖^21/2,Γ.(5.10) Equation (5.10) is a quadratic inequality relative to ‖rot𝐮‖_𝑄. Solving it we come to the estimate‖rot𝐮‖_𝑄≤‖‖𝜁_𝑑‖‖_𝑄+𝑎‖𝐠‖_1/2,Γ+𝑏‖𝐠‖^21/2,Γ^1/2,(5.11) which is equivalent to the following estimate for the difference rot𝐮₁−rot𝐮₂:‖‖rot𝐮₁−rot𝐮₂‖‖_𝑄≤‖‖𝜁^𝑑(1)−𝜁^𝑑(2)‖‖_𝑄+𝑎‖‖𝐠₁−𝐠₂‖‖_1/2,Γ+𝑏‖‖𝐠₁−𝐠₂‖‖^21/2,Γ^1/2.(5.12)

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,𝑝,̂𝑇,𝜒) to problem (5.1) relative to small perturbations of functions 𝜁_𝑑∈𝐋²(Ω) and 𝐠∈𝐇^1/2(Γ) in the norms of 𝐋²(Ω) and 𝐇^1/2(Γ), respectively. In particular case where 𝜁^𝑑(1)=𝜁^𝑑(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.

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) that𝜀𝜇₁‖𝜒‖^2Γ_𝑁≤−𝜇₀‖rot𝐮‖^2𝑄+𝜇₀‖‖𝜁_𝑑‖‖_𝑄‖rot𝐮‖_𝑄+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ≤𝜇₀‖‖𝜁_𝑑‖‖^2𝑄+𝜇₀‖‖𝜁_𝑑‖‖_𝑄𝑎‖𝐠‖_1/2,Γ+𝑏‖𝐠‖^21/2,Γ^1/2+𝜇₀𝑎‖𝐠‖_1/2,Γ+𝜇₀𝑏‖𝐠‖^21/2,Γ≤𝜇₀‖‖𝜁^𝑑‖‖_𝑄+𝑎‖𝐠‖_1/2,Γ+𝑏‖𝐠‖^21/2,Γ^1/2².(5.13) From (5.13) and (3.48)–(3.50) we come to the estimates (4.35)–(4.38) whereΔ=‖‖𝜁^𝑑(1)−𝜁^𝑑(2)‖‖_𝑄+𝑎‖‖𝐠₁−𝐠₂‖‖_1/2,Γ+𝑏‖‖𝐠₁−𝐠₂‖‖^21/2,Γ^1/2.(5.14) Thus the following theorem is proved.

Theorem 5.1. Let, under conditions (i), (ii), (iii) for 𝐾₁ 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^𝑑(𝑖)∈𝐋²(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 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 problem𝐽(𝐯,𝜒,𝜓,𝑓)≡𝜇⁰2‖‖rot𝐯−𝜁^𝑑‖‖^2𝑄+𝜇¹2‖𝜒‖^2Γ𝑁+𝜇²2‖𝜓‖^21/2,Γ𝐷+𝜇³2‖𝑓‖²𝐹(𝐱,𝜒,𝜓,𝑓,𝐠)=0,(𝐱,𝜒,𝜓,𝑓)∈𝑋×𝐾₁⟶inf,×𝐾₂×𝐾₃.(5.15) 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 (𝐮_𝑖,𝑝_𝑖,𝑇_𝑖,𝜒_𝑖,𝜓_𝑖,𝑓_𝑖) be a solution to problem (5.15) corresponding to given functions 𝜁_𝑑=𝜁^𝑑(𝑖)∈𝐋²(𝑄) and 𝐠_𝑖∈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 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 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‖^2Γ_𝑁 in the expression for minimized functional in (4.1) or the terms (𝜇₁/2)‖𝜒‖^2Γ_𝑁, (𝜇₂/2)‖𝜓‖^21/2,Γ_𝐷 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.

Acknowledgments

This work was supported by the Russian Foundation for Basic Research (Project no. 10-01-00219-a) and the Far East Branch of the Russian Academy of Sciences (Project no. 09-I-P29-01).