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International Journal of Differential Equations
Volume 2011, Article ID 535736, 28 pages
http://dx.doi.org/10.1155/2011/535736
Research Article

Stability of Optimal Controls for the Stationary Boussinesq Equations

Computational Fluid Dynamics Laboratory, Institute of Applied Mathematics FEB RAS, 7 Radio Street, Vladivostok 690041, Russia

Received 26 May 2011; Accepted 3 August 2011

Academic Editor: Yuji Liu

Copyright © 2011 Gennady Alekseev and Dmitry Tereshko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [14]. A number of papers are devoted to theoretical study of control problems for stationary models of heat and mass transfer (see e.g., [519]). 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 [14]).

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 equations̃𝜈Δ𝐮+(𝐮)𝐮+𝑝=𝐟𝛽𝐆𝑇,div𝐮=0inΩ,(2.1)𝜆Δ𝑇+𝐮𝑇=𝑓inΩ,(2.2)𝐮=𝐠onΓ,𝑇=𝜓onΓ𝐷,𝜆𝜕𝑇𝜕𝑛+𝛼𝑇=𝜒onΓ𝑁.(2.3) Here Ω is a bounded domain in the space 𝑑, 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, ̃𝛽 is the volumetric thermal expansion coefficient, 𝜆 is the thermal conductivity coefficient, 𝐠 is a given vector-function on Γ, 𝜓 is a given function on a part Γ𝐷 of Γ, 𝜒 is a function given on another part Γ𝑁=ΓΓ𝐷 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 𝑑, 𝑑=2,3, with Lipschitz boundary Γ𝐶0,1, consisting of coupled components Γ(𝑖), 𝑖=1,2,,𝑁; Γ=Γ𝐷Γ𝑁 and meas Γ𝐷>0.

Below we shall use the Sobolev spaces 𝐻𝑠(𝐷) and 𝐿2(𝐷), where 𝑠, 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 subspaces𝒯=𝜃𝐻1(Ω)𝜃Γ𝐷=0,𝐿20(Ω)=𝑟𝐿2(Ω)Ω,𝐇𝑟𝑑𝑥=01div(Ω)=𝐯𝐇1(Ω)div𝐯=0,𝐇10(Ω)=𝐯𝐻1(Ω)𝐯Γ,=0𝐕=𝐯𝐇10,𝐇(Ω)div𝐯=01(Ω)=𝐯𝐇1(Ω)(𝐯𝐧,1)Γ(𝑖)=0,𝐯𝐧Γ𝑁,𝐇=01/2(Γ)=𝐯Γ𝐇𝐯1(Ω)𝐇1/2(Γ),𝐿2+Γ𝑁=𝜙𝐿2Γ𝑁𝜙0a.e.onΓ𝑁.(2.4)

The inner products and norms in 𝐿2(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 𝐇1/2(Γ) is denoted by 1/2,Γ. Set ̃𝐛𝛽𝐆. Let in addition to condition (i) the following conditions hold:(ii)𝐟𝐇1(Ω), ̃𝐛𝛽𝐆𝐋2(Ω), 𝛼𝐿2(Γ𝑁).

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

Lemma 2.1. Under conditions (i) there exist constants 𝛿𝑖>0, 𝛾𝑖>0, 𝐶𝑑, 𝐶𝑟, and 𝛽1>0 such that (𝐯,𝐯)𝛿0𝐯21𝐯𝐇10(Ω),(𝑇,𝑇)𝛿1𝑇21||||𝑇𝒯,(2.5)((𝐮)𝐯,𝐰)𝛾0𝐮1𝐯1𝐰1,||(||𝐮𝑇,𝜂)𝛾1𝐮1𝑇1𝜂1||||,(2.6)(𝐛𝑇,𝐯)𝛽1𝑇1𝐯1𝑇𝐻1(Ω),𝐯𝐇1||(Ω),(2.7)(𝜒,𝑇)Γ𝑁||𝛾2𝜒Γ𝑁𝑇1,||(𝛼𝑇,𝜂)Γ𝑁||𝛾3𝛼Γ𝑁𝑇1𝜂1,𝑇𝑄𝛾4𝑇1,𝐯𝑄𝛾4𝐯1,(2.8)rot𝐯𝐶𝑟𝐯1,div𝐯𝐶𝑑𝐯1.(2.9) Bilinear form (div,) satisfies the inf-sup condition inf𝑟𝐿20(Ω)𝑟0sup𝐯𝐇10(Ω)𝐯0(div𝐯,𝑟)𝐯1𝑟𝛽=const>0.(2.10) Besides the following identities take place: ((𝐮)𝐯,𝐯)=0𝐮𝐇1div(Ω),𝐯𝐇10((Ω),(2.11)𝐮𝑇,𝑇)=0𝐮𝐇1div(𝐇Ω)1(Ω),𝑇𝐻1(Ω).(2.12)

Let 𝐇𝐠1/2(Γ), 𝜒𝐿2(Γ𝑁), 𝜓𝐻1/2(Γ𝐷), 𝑓𝐿2(Ω) in addition to (i), (ii). We multiply the equations in (2.1), (2.2) by test functions 𝐯𝐇10(Ω) and 𝑆𝒯 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 𝐇𝐱(𝐮,𝑝,𝑇)1(Ω)×𝐿20(Ω)×𝐻1(Ω) satisfying the relations 𝜈(𝐮,𝐯)+((𝐮)𝐮,𝐯)(𝑝,div𝐯)=𝐟,𝐯(𝐛𝑇,𝐯)𝐯𝐇10̃(Ω),𝐛𝛽𝐆,(2.13)𝜆(𝑇,𝑆)+𝜆(𝛼𝑇,𝑆)Γ𝑁+(𝐮𝑇,𝑆)=(𝑓,𝑆)+(𝜒,𝑆)Γ𝑁𝑆𝒯,(2.14)div𝐮=0inΩ,𝐮=𝐠onΓ,𝑇=𝜓onΓ𝐷.(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 𝐇𝐠1/2(Γ), 𝜒𝐿2(Γ𝑁), 𝑓𝐿2(Ω), 𝜓𝐻1/2(Γ𝐷) a weak solution (𝐮,𝑝,𝑇) that satisfies the estimates 𝐮1𝑀𝐮,𝑝𝑀𝑝,𝑇1𝑀𝑇.(2.16) Here 𝑀𝐮, 𝑀𝑝 and 𝑀𝑇 are nondecreasing continuous functions of the norms 𝐟1, 𝐛, 𝐠1/2,Γ, 𝜒Γ𝑁, 𝑓, 𝜓1/2,Γ𝐷, 𝛼Γ𝑁. If, additionally, 𝐟,𝐠,𝜒,𝑓,𝜓,𝛼 are small in the sense that 𝛾0𝛿0𝜈𝑀𝐮+𝛾1𝛿0𝜈𝛽1𝛿1𝜆𝑀𝑇<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)=𝐯𝐯𝑑2𝑄,𝐼2(𝐯)=𝐯𝐯𝑑21,𝑄,𝐼3(𝐯)=rot𝐯𝜁𝑑2𝑄.(3.1) Here 𝑄 is a subdomain of Ω. The functionals 𝐼1, 𝐼2, and 𝐼3 where functions 𝐮𝑑𝐋2(𝑄) (or 𝐮𝑑𝐇1(𝑄)) and 𝜁𝑑𝐋2(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 𝜒𝐿2(Γ𝑁), 𝜓𝐻1/2(Γ𝐷), and 𝑓𝐿2(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).

Let 𝐇𝑋=1(Ω)×𝐿20(Ω)×𝐻1(Ω), 𝑌=𝐇1(Ω)×𝐿20𝐇(Ω)×1/2(Γ)×𝒯×𝐻1/2(Γ𝐷). Denote by 𝐇𝐼1(Ω) a weakly lower semicontinuous functional. We assume that the controls 𝜒, 𝜓, and 𝑓 vary in some sets 𝐾1𝐿2(Γ𝑁), 𝐾2𝐻1/2(Γ𝐷), 𝐾3𝐿2(Ω). Setting 𝐾=𝐾1×𝐾2×𝐾3, 𝐱=(𝐮,𝑝,𝑇), 𝑢0=(𝐟,𝐛,𝛼), 𝑢=(𝜒,𝜓,𝑓) we introduce the functional 𝐽𝑋×𝐾 by the formula𝜇𝐽(𝐱,𝑢)=02𝜇𝐼(𝐮)+12𝜒2Γ𝑁+𝜇22𝜓21/2,Γ𝐷+𝜇32𝑓2.(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 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)𝐾1𝐿2(Γ𝑁), 𝐾2𝐻1/2(Γ𝐷), 𝐾3𝐿2(Ω) are nonempty closed convex sets;(iv)𝜇0>0, 𝜇𝑙>0 or 𝜇0>0, 𝜇𝑙0 and 𝐾𝑙 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,𝐹5𝐇)𝑋×𝐾×1/2(Γ)𝑌 is the operator acting by formulas𝐹1𝐹(𝐱),𝐯=𝜈(𝐮,𝐯)+((𝐮)𝐮,𝐯)(𝑝,div𝐯)𝐟,𝐯+(𝐛𝑇,𝐯),2(𝐱)=div𝐮,𝐹3(𝐱,𝐠)=𝐮Γ𝐠,𝐹5(𝐱,𝜓)=𝑇Γ𝐷𝜓,𝐹4(𝐱,𝑓,𝜒),𝑆=𝜆(𝑇,𝑆)+𝜆(𝛼𝑇,𝑆)Γ𝑁+(𝐮𝑇,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 that𝜇𝐽(𝐱,𝑢)02𝜇𝐼(𝐮)+12𝜒2Γ𝑁+𝜇22𝜓21/2,Γ𝐷+𝜇32𝑓2inf,𝐹(𝐱,𝑢,𝐠)=0,(𝐱,𝑢)𝑋×𝐾.(3.5)

Let 𝑋𝐇1(Ω)×𝐿20(Ω)×𝐻1(Ω) and 𝑌𝐇10(Ω)×𝐿20𝐇(Ω)×1/2(Γ)×𝒯×𝐻1/2(Γ𝐷) be the duals of the spaces 𝑋 and 𝑌. Let 𝐹𝐱(̂𝐱,̂𝑢,𝐠)𝑋𝑌 denotes the Fréchet derivative of 𝐹 with respect to 𝐱 at the point (̂𝐱,̂𝑢,𝐠). By 𝐹𝐱(̂𝐱,̂𝑢,𝐠)𝑌𝑋 we denote the adjoint operator of 𝐹𝐱(̂𝐱,̂𝑢,𝐠) which is determined by the relation𝐹𝐱(̂)𝐱,̂𝑢,𝐠𝐲,𝐱𝑋×𝑋=𝐲,𝐹𝐱(̂𝐱,̂𝑢,𝐠)𝐱𝑌×𝑌𝐱𝑋,𝐲𝑌.(3.6) 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 𝑋×𝐾×+×𝑌×𝐇1/2(Γ), where +={𝑥𝑥0}, by𝐱,𝑢,𝜆0,𝐲,𝐠=𝜆0𝐽(𝐱,𝑢)+𝐲,𝐹(𝐱,𝑢)𝜆0𝐽(𝐱,𝑢)+𝐹1(𝐹𝐱),𝜉+2(𝐱),𝜎+𝜁,𝐹3(𝐱,𝐠)Γ+𝜅𝐹4𝜁(𝐱,𝑓,𝜒),𝜃+𝜅𝑡,𝐹5(𝐱,𝜓)Γ𝐷.(3.7) Here and below 𝜁,Γ𝐇𝜁,1/2(Γ)×𝐇1/2(Γ), 𝜁𝑡,Γ𝐷𝜁𝑡,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, [𝜉]=[𝐮]=𝐿0𝑇01,[𝜃]=[𝑇]=𝐾0,[𝜎]=[𝑝]=𝐿20𝑇02.(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]=𝐿20𝑇02𝐾02.

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 𝐇𝐠1/2(Γ). Then there exists at least one solution (̂̂𝐱,̂𝑢)=(𝐮,̂𝑝,𝑇,𝜒,𝜓,𝑓) to problem (3.5) for 𝐼=𝐼𝑘, 𝑘=1,2,3.

Theorem 3.2. Let under conditions of Theorem 3.1 a pair (̂̂𝐱,̂𝑢)(𝐮,̂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 ̂𝐱. Then there exists a nonzero Lagrange multiplier (𝜆0,𝐲)=(𝜆0,𝜉,𝜎,𝜁,𝜃,𝜁𝑡)+×𝑌 such that the Euler-Lagrange equation 𝐹𝐱(̂)𝐱,̂𝑢,𝐠𝐲=𝜆0𝐽𝐱(̂𝐱,̂𝑢)in𝑋(3.9) for the adjoint state 𝐲 is satisfied and the minimum principle holds which is equivalent to the inequality ̂𝐱,̂𝑢,𝜆0,𝐲̂,𝐠𝐱,𝑢,𝜆0,𝐲,𝐠𝑢𝐾.(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 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 by𝐽𝜒(̂𝐱,̂𝑢),𝜒=𝜇1𝜒,𝜒Γ𝑁,𝐽𝜓(̂𝐱,̂𝑢),𝜓=𝜇2𝜓,𝜓1/2,Γ𝐷,𝐽𝑓(̂𝐱,̂𝑢),𝑓=𝜇3,𝑓,𝑓𝜒̂𝐱,̂𝑢,𝜆0,𝐲,𝐠,𝜒=𝜆0𝜇1𝜒,𝜒Γ𝑁𝜅(𝜃,𝜒)Γ𝑁𝜆0𝜇1𝜒𝜅𝜃,𝜒Γ𝑁𝜒𝐾1,𝜓̂𝐱,̂𝑢,𝜆0,𝐲,𝐠,𝜓=𝜆0𝜇2𝜓,𝜓1/2,Γ𝐷𝜁𝜅𝑡,𝜓Γ𝐷𝜆0𝜇2𝜓𝜅𝜁𝑡,𝜓Γ𝐷𝜓𝐾2,𝑓̂𝐱,̂𝑢,𝜆0,𝐲,𝐠,𝑓=𝜆0𝜇3𝜆𝑓,𝑓𝜅(𝜃,𝑓)0𝜇3𝑓𝜅𝜃,𝑓𝑓𝐾3.(3.11) Here for example 𝜒(̂𝐱,̂𝑢,𝜆0,𝐲,𝐠) is the Gateaux derivative with respect to 𝜒 at the point (̂𝐱,̂𝑢,𝜆0,𝐲,𝐠)𝑋×𝐾×+×𝑌×𝐇1/2(Γ). Since 𝐾1,𝐾2,𝐾3 are convex sets, at the minimum point ̂𝑢=(𝜒,𝜓,𝑓) of the functional ̂(𝐱,,𝜆0,𝐲,𝐠) the following conditions are satisfied (see [22]):𝜒̂𝐱,̂𝑢,𝜆0,𝐲𝜆,𝐠,𝜒𝜒0𝜇1𝜒𝜅𝜃,𝜒𝜒Γ𝑁0𝜒𝐾1,𝜓̂𝐱,̂𝑢,𝜆0,𝐲=𝜆,𝐠,𝜓𝜓0𝜇2𝜓𝜅𝜁𝑡,𝜓𝜓1/2,Γ𝐷0𝜓𝐾2,𝑓̂𝐱,̂𝑢,𝜆0,𝐲𝑓=𝜆,𝐠,𝑓0𝜇3𝑓𝑓𝜅𝜃,𝑓0𝑓𝐾3.(3.12)

We also note that the Euler-Lagrange equation (3.9) is equivalent to identitieŝ̂𝜈(𝐰,𝜉)+((𝐮)𝐰,𝜉)+((𝐰)𝐮,𝜉)+𝜅𝐰𝑇,𝜃(𝜎,div𝐰)+𝜁,𝐰Γ+𝜆0𝐽𝐮(̂𝐇𝐱,̂𝑢),𝐰=0𝐰1(Ω),(𝑟,div𝜉)=0𝑟𝐿20𝜅(Ω),𝜆(𝜏,𝜃)+𝜆(𝛼𝜏,𝜃)Γ𝑁̂𝜁+(𝐮𝜏,𝜃)+𝑡,𝜏Γ𝐷+(𝐛𝜏,𝜉)=0𝜏𝐻1(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 𝐇𝐺1/2(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=𝐠1𝐺. By (𝐱2,𝑢2)(𝐮2,𝑝2,𝑇2,𝜒2,𝜓2,𝑓2)𝑋×𝐾 we denote a solution to problem𝜇𝐽(𝐱,𝑢)02𝜇𝐼(𝐮)+12𝜒2Γ𝑁+𝜇22𝜓21/2,Γ𝐷+𝜇32𝑓2̃inf,𝐹(𝐱,𝑢,𝐠)=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 𝐠𝐺 by a close function ̃𝐠𝐺.

By Theorem 3.1 the following estimates hold for triples (𝐮𝑖,𝑝𝑖,𝑇𝑖): 𝐮𝑖1𝑀0𝐮,𝑝𝑖𝑀0𝑝,𝑇𝑖1𝑀0𝑇.(3.15) Here𝑀0𝐮=sup𝑢𝐾,𝐠𝐺𝑀𝐮𝑢0,𝑢,𝐠,𝑀0𝑝=sup𝑢𝐾,𝐠𝐺𝑀𝑝𝑢0,𝑢,𝐠,𝑀0𝑇=sup𝑢𝐾,𝐠𝐺𝑀𝑇𝑢0,𝑢,𝐠,(3.16) where 𝑀𝐮, 𝑀𝑝, and 𝑀𝑇 are introduced in Theorem 3.1. We introduce “model” Reynolds number 𝑒, Raley number 𝑎, and Prandtl number 𝒫 by𝛾𝑒=0𝑀0𝐮𝛿0𝜈𝛾,𝑎=1𝛿0𝜈𝛽1𝑀0𝑇𝛿1𝜆𝛿,𝒫=0𝜈𝛿1𝜆.(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 𝑢1 (where 𝑢 is an arbitrary scalar) are defined as𝑢2=Ω𝑢2𝑑𝑥,|𝑢|21=Ω||||𝑢2𝑑𝑥,𝑢21=𝑙2𝑢2+|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:𝛾𝑒+𝑎0𝑀0𝐮𝛿0𝜈+𝛾1𝛿0𝜈𝛽1𝑀0𝑇𝛿1𝜆<12.(3.19)

Let us denote by (1,𝐲𝑖), where 𝐲𝑖(𝜉𝑖,𝜎𝑖,𝜁𝑖,𝜃𝑖,𝜁𝑡𝑖)𝐇10(Ω)×𝐿20𝐇(Ω)×1/2(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 relations𝜈𝐰,𝜉𝑖+𝐮𝑖𝐰,𝜉𝑖+(𝐰)𝐮𝑖,𝜉𝑖+𝜅𝐰𝑇𝑖,𝜃𝑖𝜎𝑖,div𝐰+𝜁𝑖,𝐰Γ+𝜇02𝐼𝑖𝐮𝑖𝐇,𝐰=0𝐰1(Ω),𝑖=1,2,(3.20)div𝜉𝑖,𝑟=0𝑟𝐿20𝜅𝜆(Ω),(3.21)𝜏,𝜃𝑖+𝜆(𝛼𝜏,𝜃)Γ𝑁+𝐮𝑖𝜏,𝜃𝑖+𝜁𝑡𝑖,𝜏Γ𝐷+𝐛𝜏,𝜉𝑖=0𝜏𝐻1𝜇(Ω),(3.22)1𝜒𝑖𝜅𝜃𝑖,𝜒𝜒𝑖Γ𝑁+𝜇2𝜓𝑖𝜅𝜁𝑡𝑖,𝜓𝜓𝑖Γ𝐷+𝜇3𝑓𝑖𝜅𝜃𝑖,𝑓𝑓𝑖0(𝜒,𝜓,𝑓)𝐾.(3.23) We renamed 𝐼1𝐼, 𝐼2𝐼 in (3.20). Set 𝜉=𝜉1𝜉2, 𝜎=𝜎1𝜎2, 𝜁=𝜁1𝜁2, 𝜃=𝜃1𝜃2, 𝜁𝑡=𝜁𝑡1𝜁𝑡2, 𝐠=𝐠1𝐠2, and𝐮=𝐮1𝐮2,𝑝=𝑝1𝑝2,𝑇=𝑇1𝑇2,𝜒=𝜒1𝜒2,𝜓=𝜓1𝜓2,𝑓=𝑓1𝑓2.(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 obtain𝜈(𝐮,𝐯)+(𝐮)𝐮1+𝐮2𝐮,𝐯(div𝐯,𝑝)+(𝐛𝑇,𝐯)=0𝐯𝐇10(Ω),(3.25)𝜆(𝑇,𝑆)+𝜆(𝛼𝑇,𝑆)Γ𝑁+𝐮𝑇1+𝐮,𝑆2𝑇,𝑆=(𝑓,𝑆)+(𝜒,𝑆)Γ𝑁𝑆𝒯,(3.26)div𝐮=0inΩ,𝐮Γ=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 obtain𝜅(𝜒,𝜃)Γ𝑁+𝜁𝑡,𝜓Γ𝐷+(𝑓,𝜃)𝜇1𝜒2Γ𝑁𝜇2𝜓21/2,Γ𝐷𝜇3𝑓2.(3.28)

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 obtain𝜈𝐮(𝐮,𝜉)+1𝐮+(𝐮)𝐮1,𝜉+2(𝐮)𝐮,𝜉2+𝜅𝐮𝑇1,𝜃+𝜅(𝐮)𝑇,𝜃2+𝜁,𝐠Γ+𝜅𝜆(𝑇,𝜃)+𝜆(𝛼𝑇,𝜃)Γ𝑁+𝐮1+𝑇,𝜃𝐮𝑇,𝜃2+𝜁𝑡,𝜓Γ𝐷+𝜇+(𝐛𝑇,𝜉)02𝐼𝐮1𝐼𝐮2,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,𝜉1+𝜉2+𝜅𝐮𝑇,𝜃1+𝜃2+𝜇02𝐼𝐮1𝐼𝐮2,𝐮𝜁,𝐠Γ𝜇1𝜒2Γ𝑁𝜇2𝜓21/2,Γ𝐷𝜇3𝑓2.(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, 𝐲𝑖=(𝜉𝑖,𝜎𝑖,𝜁𝑖,𝜃𝑖,𝜁𝑡𝑖), 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=𝐠1𝐠2, 𝜁=𝜁1𝜁2.

Below we shall need the estimates of differences 𝐮=𝐮1𝐮2, 𝑝=𝑝1𝑝2, 𝑇=𝑇1𝑇2 entering into (3.25)–(3.27) by differences 𝜒=𝜒1𝜒2, 𝜓=𝜓1𝜓2, 𝑓=𝑓1𝑓2, and 𝐠=𝐠1𝐠2. Denote by 𝐮0𝐇1(Ω) a vector such that div𝐮0=0 in Ω, 𝐮0Γ=𝐠, 𝐮01𝐶0𝐠1/2,Γ. Here 𝐶0 is a constant depending on Ω. The existence of 𝐮0 follows from [20, page 24]. We present the difference 𝐮𝐮1𝐮2 as 𝐮=𝐮0+̃𝐮, where ̃𝐮𝐕 is a new unknown function. Set 𝐮=𝐮0+̃𝐮, ̃𝐮𝐯= in (3.25). Taking into account (2.9) we obtain𝜈̃̃𝐮(𝐮,)=𝜈𝐮0̃𝐮𝐮,0𝐮1,̃𝐮(̃𝐮)𝐮1,̃𝐮𝐮2𝐮0,̃𝐮̃𝐮(𝐛𝑇,).(3.31) Using estimates (2.5), (2.6), (2.7), and (3.15), we deduce from (3.31) that𝛿0̃𝜈𝐮21𝐮𝜈01̃𝐮1+𝛾0𝑀0𝐮̃𝐮21+2𝛾0𝑀0𝐮𝐮01̃𝐮1+𝛽1𝑇1̃𝐮1.(3.32)

It follows from (3.19) that𝛿0𝜈2<𝛿0𝜈𝛾0𝑀0𝐮𝛽1𝛾1𝛿1𝜆𝑀0𝑇𝛿0𝜈𝛾0𝑀0𝐮.(3.33) Rewriting the inequality (3.32) by (3.33) as𝛿0𝜈2̃𝐮21𝛿0𝜈𝛾0𝑀0𝐮̃𝐮21𝜈+2𝛾0𝑀0𝐮𝐮01̃𝐮1+𝛽1𝑇1̃𝐮1,(3.34) we obtain that̃𝐮12𝛿0𝜈𝜈+2𝛾0𝑀0𝐮𝐮01+2𝛽1𝛿0𝜈𝑇12𝛿01𝐮+4𝑒01+2𝛽1𝛿0𝜈𝑇1𝐮201+2𝛽1𝛿0𝜈𝑇1,𝛿01+2𝑒.(3.35) Taking into account the relation 𝐮=𝐮0+̃𝐮, we come to the following estimate 𝐮1 via 𝐠1/2,Γ and 𝑇1:𝐮1𝐮01+̃𝐮1𝐮(2+1)01+2𝛽1𝛿0𝜈𝑇1𝐶0(2+1)𝐠1/2,Γ+2𝛽1𝛿0𝜈𝑇1.(3.36)

Denote by 𝑇0𝐻1(Ω) a function such that 𝑇0Γ𝐷=𝜓 and the estimate 𝑇01𝐶1𝜓1/2,Γ𝐷 holds with a certain constant 𝐶1, which does not depend on 𝜓. Let us present the difference 𝑇=𝑇1𝑇2 as 𝑇=𝑇0+𝑇, where 𝑇𝒯 is a new unknown function. Set 𝑇=𝑇0+𝑇, 𝑇𝑆= in (3.26). We obtain𝜆𝑇𝛼𝑇𝑇,+𝜆𝑇,Γ𝑁+𝐮2𝑇𝑇,=𝜆𝑇0𝑇,𝜆𝛼𝑇0,𝑇Γ𝑁𝐮2𝑇0,𝑇𝐮𝑇1,𝑇+𝑇+𝑇𝑓,𝜒,Γ𝑁.(3.37) Using estimates (2.5)–(2.8) and (3.15) we deduce that𝛿1𝜆𝑇21𝑇𝜆01𝑇1+𝛾1𝑀0𝐮𝑇01𝑇+𝜆𝛾3𝛼Γ𝑁𝑇01𝑇1+𝛾1𝑀0𝑇𝐮1𝑇1+𝛾2𝜒Γ𝑁+𝛾4𝑇𝑓1(3.38) or𝑇11𝛿1𝜆𝜆+𝛾1𝑀0𝐮+𝜆𝛾3𝛼Γ𝑁𝑇01+𝛾1𝑀0𝑇𝛿1𝜆𝐮1+1𝛿1𝜆𝛾2𝜒Γ𝑁+𝛾4𝑓.(3.39) Taking into account the relation 𝑇=𝑇0+𝑇, we obtain from this estimate that𝑇1𝐶1(𝒩+1)𝜓1/2,Γ𝐷+𝛾2𝜒Γ𝑁+𝛾4𝑓𝛿1𝜆+𝛾1𝑀0𝑇𝛿1𝜆𝐮1,𝒩=𝜆+𝛾1𝑀0𝐮+𝜆𝛾3𝛼Γ𝑁𝛿1𝜆.(3.40)

Using further the estimate (3.36) for 𝐮, we deduce from (3.40) that𝑇1𝐶1(𝒩+1)𝜓1/2,Γ𝐷+𝛾2𝜒Γ𝑁+𝛾4𝑓𝛿1𝜆+𝐶0𝛾(2+1)1𝑀0𝑇𝛿1𝜆𝐠1/2,Γ+2𝛽1𝛿0𝜈𝛾1𝑀0𝑇𝛿1𝜆𝑇1.(3.41) From this inequality and (3.17), (3.19) we come to the following estimate:𝑇1𝐶1(𝒩+1)12𝑎𝜓1/2,Γ𝐷+𝛾2𝜒Γ𝑁+𝛾4𝑓𝛿1+𝐶𝜆(12𝑎)0(2+1)𝛾12𝑎1𝑀0𝑇𝛿1𝜆𝐠1/2,Γ.(3.42)

Using (3.42), we deduce from (3.36) that𝐮1𝐶01(2+1)1+12𝑎2𝛽1𝛿0𝜈𝛾1𝑀0𝑇𝛿1𝜆𝐠1/2,Γ+2𝛽1𝛿0𝜈𝐶1(𝒩+1)12𝑎𝜓1/2,Γ𝐷+𝛾2𝜒Γ𝑁+𝛾4𝑓𝛿1.𝜆(12𝑎)(3.43) Taking into account (3.17) we come to the following estimate for 𝐮1:𝐮12𝛽1𝛿0𝜈𝐶1(𝒩+1)12𝑎𝜓1/2,Γ𝐷+𝛾2𝜒Γ𝑁+𝛾4𝑓𝛿1+𝐶𝜆(12𝑎)0(2+1)𝐠1/2,Γ12𝑎.(3.44)

An analogous estimate holds and for the pressure difference 𝑝=𝑝1𝑝2. In order to establish this estimate we make use of inf-sup condition (2.10). By (2.10) for the function 𝑝=𝑝1𝑝2 and any (small) number 𝛿>0 there exists a function 𝐯0𝐇10(Ω), 𝐯00, such that (div𝐯0,𝑝)𝛽0𝐯01𝑝 where 𝛽0=(𝛽𝛿)>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 have𝛽0𝐯01𝑝div𝐯0,𝑝𝜈+2𝛾0𝑀0𝐮𝐯01𝐮1+𝛽1𝑇1𝐯01.(3.45) Dividing to 𝐯010, we deduce that𝑝𝜈+2𝛾0𝑀0𝐮𝛽0𝐮1+𝛽1𝛽0𝑇1=𝛿0𝜈𝛽0𝐮1+𝛽1𝛽0𝑇1.(3.46) Using (3.42) and (3.44), we come to the following final estimate for 𝑝:𝑝2+1𝛽0×𝛽(12𝑎)1𝐶1(𝒩+1)𝜓1/2,Γ𝐷+𝛽1𝛾2𝜒Γ𝑁+𝛾4𝑓𝛿1𝜆+𝛿0𝜈𝐶0(+𝑎)𝐠1/2,Γ.(3.47)

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 𝜓0𝐻1/2(Γ𝐷) and the set 𝐾3 consists of one element 𝑓0𝐿2(Ω). For this case the conditions 𝑓𝑓1𝑓2=0, 𝜓𝜓1𝜓2=0 take place, and the estimates (3.42)–(3.47) and inequality (3.30) take the form 𝑇1𝛾2𝜒Γ𝑁𝛿1+𝐶𝜆(12𝑎)0(2+1)𝛾12𝑎1𝑀0𝑇𝛿1𝜆𝐠1/2,Γ,(3.48)𝐮12𝛽1𝛾2𝜒Γ𝑁𝛿0𝜈𝛿1+𝐶𝜆(12𝑎)0(2+1)𝐠1/2,Γ12𝑎,(3.49)𝑝2+1𝛽0(𝛽12𝑎)1𝛾2𝜒Γ𝑁𝛿1𝜆+𝛿0𝜈𝐶0(+𝑎)𝐠1/2,Γ,(3.50)(𝐮)𝐮,𝜉1+𝜉2+𝜅𝐮𝑇,𝜃1+𝜃2+𝜇02𝐼𝐮1𝐼𝐮2,𝐮𝜁,𝐠Γ𝜇1𝜒2Γ𝑁.(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 Γ𝑁 of Γ is a unique control; that is, we consider one-parametric control problem𝜇𝐽(𝐯,𝜒)02𝐯𝐯𝑑2𝑄+𝜇12𝜒2Γ𝑁inf,𝐹(𝐱,𝜒,𝐠)=0,𝐱=(𝐯,𝑝,𝑇)𝑋,𝜒𝐾1.(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 𝐯𝑑𝐮𝑑(1)𝐋2(𝑄) and 𝐠=𝐠1𝐇𝐺1/2(Γ), and let (𝐱2,𝑢2)(𝐮2,𝑝2,𝑇2,𝜒2) be a solution to problem (4.1), that corresponds to perturbed functions ̃𝐯𝑑𝐮𝑑(2)𝐋2(𝑄) and ̃𝐠=𝐠2𝐇𝐺1/2(Γ). Setting 𝐮𝑑=𝐮𝑑(1)𝐮𝑑(2) in addition to (3.24) we note that under conditions of problem (4.1) we have𝐼1𝐮𝑖𝐮,𝐰=2𝑖𝐮𝑑(𝑖),𝐰𝑄,𝐼1𝐮1𝐼1𝐮2,𝐮=2𝐮2𝑄𝐮,𝐮𝑑𝑄.(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 form𝜈𝐰,𝜉𝑖+𝐮𝑖𝐰,𝜉𝑖+(𝐰)𝐮𝑖,𝜉𝑖+𝜅𝐰𝑇𝑖,𝜃𝑖𝜎𝑖,div𝐰+𝜁𝑖,𝐰Γ+𝜇0𝐮𝑖𝐮𝑑(𝑖),𝐰𝑄𝐇=0𝐰1(Ω),(4.3)div𝜉𝑖,𝑟=0𝑟𝐿20(Ω),(4.4)(𝐮)𝐮,𝜉1+𝜉2+𝜅𝐮𝑇,𝜃1+𝜃2+𝜇0𝐮2𝑄𝐮,𝐮𝑑𝑄𝜁,𝐠Γ𝜇1𝜒2Γ𝑁.(4.5)

Using identities (4.3), (4.4), (3.22) we estimate parameters 𝜉𝑖, 𝜃𝑖, 𝜎𝑖 and 𝜁𝑖. Firstly we deduce estimates for norms 𝜉𝑖1 and 𝜃𝑖1. To this end we set 𝐰=𝜉𝑖, 𝜏=𝜃𝑖 in (4.3), (3.22). Taking into account (2.11), (2.12), and condition 𝜉𝑖𝐕, which follows from (4.4), we obtain𝜈𝜉𝑖,𝜉𝑖𝜉=𝑖𝐮𝑖,𝜉𝑖𝜉𝜅𝑖𝑇𝑖,𝜃𝑖𝜇0𝐮𝑖𝐮𝑑(𝑖),𝜉𝑖𝑄,(4.6)𝜅𝜆𝜃𝑖,𝜃𝑖+𝛼𝜃𝑖,𝜃𝑖Γ𝑁=𝐛𝜃𝑖,𝜉𝑖,𝑖=1,2.(4.7) Using estimates (2.5)–(2.8) and (3.15) we have𝜉𝑖,𝜉𝑖𝛿0𝜉𝑖21,𝜃𝑖,𝜃𝑖𝛿1𝜃𝑖21,||𝜉(4.8)𝑖𝐮𝑖,𝜉𝑖||𝛾0𝐮𝑖1𝜉𝑖21𝛾0𝑀0𝐮𝜉𝑖21,𝜅||𝜉(4.9)𝑖𝑇𝑖,𝜃𝑖||𝜅𝛾1𝑀0𝑇𝜉𝑖1𝜃𝑖1,||𝐛𝜃𝑖,𝜉𝑖||𝛽1𝜃𝑖1𝜉𝑖1𝐮,(4.10)𝑖𝐮𝑑(𝑖)𝑄𝐮𝑖𝑄+𝐮𝑑(𝑖)𝑄𝛾4𝑀0𝐮+𝐮𝑑(𝑖)𝑄𝛿0𝜈𝛾4𝛾01𝑒+𝑒0|||𝐮,(4.11)𝑖𝐮𝑑(𝑖),𝜉𝑖𝑄|||𝐮𝑖𝐮𝑑(𝑖)𝑄𝜉𝑖𝑄𝛿0𝜈𝛾𝑒+𝑒0𝜉𝑖1,(4.12) where𝛾=𝛾24𝛾01,𝑒0=𝛾0𝛿0𝜈𝛾4𝐮max𝑑(1)𝑄,𝐮𝑑(2)𝑄.(4.13)

By virtue of (4.8)–(4.10) and (4.12), we deduce from (4.7) and (4.6) that𝜅𝜃𝑖1𝛽1𝛿1𝜆𝜉𝑖1,𝛿(4.14)0𝜈𝜉𝑖21𝛾0𝑀0𝐮𝜉𝑖21+𝜅𝛾1𝑀0𝑇𝜃𝑖1+𝜇0𝛿0𝜈𝛾𝑒+𝑒0𝜉𝑖1.(4.15) Taking into account (4.14), we obtain from (4.15) that𝛿0𝜈𝛾0𝑀0𝐮𝛽1𝛾1𝛿1𝜆𝑀0𝑇𝜉𝑖21𝜇0𝛿0𝜈𝛾𝑒+𝑒0𝜉𝑖1.(4.16) Using (3.33) we deduce successfully from (4.16), (4.14) that𝜉𝑖12𝜇0𝛾𝑒+𝑒0𝜃,𝜅𝑖12𝜇0𝛾𝛽1𝛿1𝜆𝑒+𝑒0.(4.17)

Let us estimate further the norms 𝜎𝑖 and 𝜁𝑖1/2,Γ from (4.3). In order to estimate 𝜎𝑖 we make use of inf-sup condition (2.10). By (2.10) for a function 𝜎𝑖𝐿20(Ω) and any small number 𝛿>0 there exists a function 𝐯𝑖𝐇10(Ω), 𝐯𝑖0, such that the inequalitydiv𝐯𝑖,𝜎𝑖𝛽0𝐯𝑖1𝜎𝑖,𝑖=1,2,𝛽0=𝛽𝛿(4.18) holds. Setting in (4.3) 𝐰=𝐯𝑖 and using this estimate together with estimates (2.6), (3.15), (4.11), we have𝛽0𝐯𝑖1𝜎𝑖div𝐯𝑖,𝜎𝑖𝐯𝜈𝑖1𝜉𝑖1+2𝛾0𝑀0𝐮𝐯𝑖1𝜉𝑖1+𝜅𝛾1𝑀0𝑇𝐯𝑖1𝜃𝑖1+𝜇0𝛿0𝜈𝛾𝑒+𝑒0𝐯𝑖1.(4.19) From this inequality we deduce by (4.17) that𝜎𝑖1𝛽0𝜈+2𝛾0𝑀0𝐮𝜉𝑖1+𝛾1𝑀0𝑇𝜅𝜃𝑖1+𝜇0𝛿0𝜈𝛾𝑒+𝑒0𝛿0𝜈𝛽0𝜉𝑖1+𝛾1𝑀0𝑇𝛿0𝜈𝜅𝜃𝑖1+𝜇0𝛾𝑒+𝑒0.(4.20) Taking into account (4.17), we come from (4.20) to the estimate𝜎𝑖𝜇0𝛾𝛿0𝜈𝛽0𝑒+𝑒0(2+2𝑎+1).(4.21)

It remains to estimate 𝜁1/2,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||𝜁𝑖,𝐰Γ||𝜈+2𝛾0𝑀0𝐮𝜉𝑖1+𝛾1𝑀0𝑇𝜅𝜃𝑖1+𝐶𝑑𝜎𝑖+𝜇0𝛿0𝜈𝛾𝑒+𝑒0𝐰1𝜇0𝛿0𝜈𝛾1+𝐶𝑑𝛽01𝑒+𝑒0(2+2𝑎+1)𝐰1𝐇𝐰1(Ω).(4.22) As 𝜁=𝜁1𝜁2 we obtain from this inequality that𝜁1/2,Γ𝜇0𝑎,𝑎=2𝛿0𝜈𝛾1+𝐶𝑑𝛽01𝑒+𝑒0(2+2𝑎+1).(4.23)

Taking into account (2.6), (3.48), (3.49), and estimates (4.17) for 𝜉𝑖, 𝜃𝑖, we have ||(𝐮)𝐮,𝜉1+𝜉2||𝛾0𝐮21𝜉11+𝜉214𝜇0𝛾0𝛾𝑒+𝑒02𝛽1𝛾2𝜒Γ𝑁𝛿0𝜈𝛿1+𝐶𝜆(12𝑎)0(2+1)𝐠1/2,Γ12𝑎2,𝜅||𝐮𝑇,𝜃1+𝜃2||4𝜇0𝛾1𝛾𝛽1𝑒+𝑒0𝛿1𝜆2𝛽1𝛾2𝜒Γ𝑁𝛿0𝜈𝛿1+𝐶𝜆(12𝑎)0(2+1)𝐠1/2,Γ×𝛾12𝑎2𝜒Γ𝑁𝛿1𝜆+𝐶(12𝑎)0(2+1)𝛾12𝑎1𝑀0𝑇𝛿1𝜆𝐠1/2,Γ.(4.24) It follows from (4.24) that||(𝐮)𝐮,𝜉1+𝜉2+𝜅𝐮𝑇,𝜃1+𝜃2||𝜇0𝑏𝐠21/2,Γ+𝑐𝜒2Γ𝑁.(4.25) Here constants 𝑏 and 𝑐 are given by𝑏=4𝛾𝛾0𝐶20(2+1)2𝑒+𝑒0(12𝑎)2𝛾3+1𝛾02𝒫2𝑎2,𝑐=4𝛾𝛾0𝛽1𝛿0𝜈𝛾2𝛿1𝜆2𝑒+𝑒0(12𝑎)2𝛾12+1𝛾02𝒫2.(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:(1𝜀)𝜇1𝜇0𝑐,𝜀=const>0.(4.27) Under condition (4.27) we deduce from (4.25) that||(𝐮)𝐮,𝜉1+𝜉2+𝜅𝐮𝑇,𝜃1+𝜃2||𝜇0𝑏𝐠21/2,Γ+(1𝜀)𝜇1𝜒2Γ𝑁.(4.28) Taking into account (4.28) and the estimate |𝜁,𝐠Γ|𝜁1/2,Γ𝐠1/2,Γ𝜇0𝑎𝐠1/2,Γ which follows from (4.23), we come from (4.5) to the inequality𝜇0𝐮2𝑄𝐮,𝐮𝑑𝑄(𝐮)𝐮,𝜉1+𝜉2𝜅𝐮𝑇,𝜃1+𝜃2𝜁,𝐠Γ𝜇1𝜒2Γ𝑁𝜀𝜇1𝜒2Γ𝑁+𝜇0𝑎𝐠1/2,Γ+𝜇0𝑏𝐠21/2,Γ.(4.29) It follows from this inequality that𝜇0𝐮2𝑄𝜇0𝐮,𝐮𝑑𝑄𝜀𝜇1𝜒2Γ𝑁+𝜇0𝑎𝐠1/2,Γ+𝜇0𝑏𝐠21/2,Γ.(4.30) Excluding nonpositive term 𝜀𝜇1𝜒2Γ𝑁 from the right-hand side of (4.30), we deduce from (4.30) that𝐮2𝑄𝐮𝑑𝑄𝐮𝑄+𝑎𝐠1/2,Γ+𝑏𝐠21/2,Γ.(4.31)

Equation (4.31) is a quadratic inequality for 𝐮𝑄. Solving it we come to the following estimate for 𝐮𝑄:𝐮𝑄𝐮𝑑𝑄+𝑎𝐠1/2,Γ+𝑏𝐠21/2,Γ1/2.(4.32) As 𝐮=𝐮1𝐮2, 𝐮𝑑=𝐮𝑑(1)𝐮𝑑(2), 𝐠=𝐠1𝐠2, the estimate (4.32) is equivalent to the following estimate for the velocity difference 𝐮1𝐮2:𝐮1𝐮2𝑄𝐮𝑑(1)𝐮𝑑(2)𝑄+𝑎𝐠1𝐠21/2,Γ𝐠+𝑏1𝐠221/2,Γ1/2.(4.33) This estimate under 𝑄=Ω has the sense of the stability estimate in 𝐋2(Ω) of the component ̂𝐮 of the solution (̂𝐮,̂𝑝,𝑇,𝜒) to problem (4.1) relative to small perturbations of functions 𝐯𝑑𝐋2(Ω) and 𝐠𝐺 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 𝐮1𝐮2𝑄𝐮𝑑(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 𝑄Ω; that is, 𝑄