# Boundary Control Problem for Heat Convection Equations with Slip Boundary Condition

Authors: Exequiel Mallea-Zepeda; Eber Lenes; Elvis Valero
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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


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## Body

## 1. Introduction

LetΩ⊂R3 be a simply connected bounded domain with Lipschitz boundary Γ. We deal with the existence of weak solutions of a boundary value problem describing the motion of a viscous heat conduction fluid in Ω, with a Navier slip condition on the boundary for the velocity vector. This model is given by the following system of partial differential equations:(1)-νΔu+u·∇u+∇p+βGT=finΩ,(2)-κΔT+u·∇T=ginΩ,(3)divu=0inΩ.Here, u(x) is the velocity field, p(x) represents the hydrostatic pressure, and T(x) is the temperature of the fluid at point x∈Ω; the constant ν>0 denotes the kinematic viscosity; β is the volume extension coefficient; G(x) is the free fall acceleration vector; the field f(x) represents external source of linear momentum; the constant κ>0 is the temperature conduction and g(x) is the volume density of heat sources. Without loss of generality, we assume that density of the fluid is equal to one; thus if the variables in (1)-(2) are nondimensionalized, then the coefficient of viscosity ν is simply the inverse of the Reynolds number Re (see, for example, [1] p. 166 and [2]).We prove the existence of weak solutions for system (1)–(3). Moreover, we study an optimal boundary control problem. To do this, we consider the following boundary conditions:(4)ug1=u0onΓ1,g1onΓ2,u·n=0,Dun+αutang=0onΓ3,T=g2onΓ4,κ∂T∂n+γT=φonΓ5.Here the boundary Γ=Γ1∪Γ2∪Γ3=Γ4∪Γ5, where Γ¯1∩Γ¯2=Γ¯1∩Γ¯3=Γ¯2∩Γ¯3=Γ¯4∩Γ¯5=∅, and n denotes the outward normal vector on Γ. The function u0 is defined on Γ1; γ≥0 and φ are functions givens on Γ5, and the functions g1, g2 describe the Dirichlet boundary control for velocity u on Γ2 and for temperature T on Γ4, respectively. The controls g1, g2 lie in the closed convex sets U1⊂H1/2(Γ2) and U2⊂H1/2(Γ4), respectively. The term [D(u)n+αu]tang≔D(u)n+αu-[(D(u)n+αu)·n]n represents the tangential component of the vector D(u)n+αu, where Du≔1/2(∇u+∇Tu) is the deformation tensor, and the real number α≥0 is the friction coefficient which measures the tendency of the fluid to slip on Γ3. When α=0 and the boundary is flat, the fluid slips along the Γ3 without friction and there is no boundary layers. When α→+∞, the friction is so intense that the fluid is almost at rest near the boundary; the condition [D(u)n+αu]tang=0, u·n=0 on Γ3 converges to the Dirichlet condition. The case when α is a scalar function has been studied in [3], for the 2D nonstationary Navier-Stokes equations. The condition [D(u)n+αu]tang=0, u·n=0 on Γ3 is a Navier friction boundary condition. The Navier boundary condition was proposed by Navier [4], who claimed that the tangential component of the viscous stress at the boundary should be proportional to the tangential velocity. Navier boundary condition was also derived by Maxwell [5] from the kinetic theory of gases and rigorously justified as a homogenization of the no-slip condition on a rough boundary [6].Before we describe in detail our results, let us briefly comment on previous related work. The heat convection equations, or also known as Boussinesq model, have been extensively analyzed, and, among the many articles on these equations, we just mention [7–12]. In these papers are studied the existence of very weak, weak, and strong solutions and uniqueness with Dirichlet boundary conditions on the velocity and temperature or Dirichlet boundary condition on the velocity and mixed boundary conditions on the temperature. The tools used are Galerkin method, linearization, and fixed point results. The results obtained are similar to those known for the classical Navier-Stokes equations.When Navier friction boundary condition is considered, the authors do not know results of existence for the classical Boussinesq model, except work due to Rionero and Mulone [13], where it is considered a slightly more general model on thermodiffusive mixture.In the context of the classic Navier-Stokes equations there is a good amount of work with Navier friction boundary condition, among which we can mention [3, 14–19], and references therein, where the existence, regularity, and uniqueness of weak and strong solutions are studied. In the context of steady-state microstructure fluids, also called micropolar fluids, we can mention works [20, 21]. In [20] a three-dimensional domain is considered where the density of the fluid is constant, while in [21] the study is performed on a two-dimensional domain with variable density.For optimal control problems for the classical Boussinesq stationary model, see [22–27], and the references therein. In such works, standard results have been obtained, such as the existence of an optimal solution and stability as well as first-order necessary optimality conditions, from which the authors derive an optimality system. However, as far as we know, for this model there are no known studies of sufficient second-order conditions for the existence of a local extreme, except for the recent work [28], in which the authors study a problem of control of limits associated with the stationary system of Rayleigh-Bénard-Marangoni (similar to system (1)–(3)), for which they obtain results of existence, uniqueness, and regularity of the model; in addition they provide the existence of optimal solutions, derive optimality conditions, obtain a second-order sufficient condition, and establish a result of uniqueness for the optimal solution.The outline of this paper is as follows. In Section2, we introduce the spaces of functions appropriate for the development of the article. In Section 3, we prove the existence and uniqueness of weak solutions to model (1)–(4), by using the Galerkin method. In Section 4, we establish the optimal control problem and also we prove the existence of optimal solutions, and, using the Lagrange multipliers method, we obtain the first-order optimality conditions, from which we derive an optimality system. In Section 5, we obtain a second-order sufficient optimality condition.


## 2. Preliminaries

Throughout this paper we will use the classical Lebesgue spaceLp(Ω), 1≤p≤∞, with norm denoted by ·p. In particular, the norm and inner product in L2(Ω) will be represented by · and (·,·), respectively. The norm of Lp(Γ) will be denoted by ·Lp(Γ). Also we use the Sobolev space H1(Ω)≔W1,2(Ω)={u∈L2(Ω):Dαu∈L2(Ω)}, for a multi-index α, with α=1. Its norm will be denoted by ·H1. Spaces of R3 valued functions, as well as their elements, are represented by bold face letters; thus, for example, we have H1(Ω), L2(Ω), and so on. We will use the Hilbert space H0={T∈H1(Ω):T=0onΓ4} with the inner product (u,v)H0≔(∇u,∇v) and norm uH0≔∇u. We also consider the following solenoidal Banach spaces Hσ1={u∈H1(Ω):divu=0andu·n=0onΓ3}, endowed with the usual norm of H1(Ω); the space H~σ1={u∈Hσ1:u=0onΓ∖Γ3}, which is a Hilbert space with the inner product (u,v)H~σ1≔(D(u),D(v)) and norm uH~σ1≔D(u).IfX is general Banach space, its topological dual will be denoted by X′ and the duality product by ·,·X′ or simply by ·,· unless this leads to ambiguity. The space H′ denotes the dual of H0, the space Hσ′ denotes the dual of Hσ1, and the space H~σ′ denotes the dual of H~σ1. For simplicity, we consider the spaces X=H~σ1×H0, X′=H~σ′×H′. If Γk is a connected subset of Γ, we consider the trace space by H1/2(Γk)={uΓk:u∈H1(Ω)} (the restriction of the elements of H1(Ω) to Γk) and (5)H001/2Γk=v∈L2Γk:thereexistsv^∈H1/2Γ,v^Γ∖Γk=0,v^Γk=v,H~001/2Γk=v∈H001/2Γk:∫Γkv·ndΓ=0. The space H001/2(Γk) is a closed subspace of H1/2(Γk); moreover, it is continuously embedded in L2(Γk) (see [29, 30]). For the space H1/2(Γk), H-1/2(Γk) denote its dual space and ·,·Γk denote the duality. The letter C will denote diverse positive constants which may change from line to line or even within the same line.In order to establish the weak formulation of problem (1)–(4) we consider the following result.Lemma 1 (see [19]).
Letu∈H2(Ω) and v∈H1(Ω) be divergence-free vector fields tangent to the boundary Γ. Then,(6)-∫ΩΔu·vdx=2∫ΩDu:Dvdx-2∫ΓDuntang·vdΓ.Then, motivated by the formula of integration by parts and Lemma1, we establish the following definition of weak solution of system (1)–(4).Definition 2.
Let(f,g)∈X′, G∈L∞(Ω), u0∈H~001/2(Γ1), g1∈H~001/2(Γ2), g2∈H001/2(Γ4), φ∈L2(Γ5), and γ∈L∞(Γ5). A weak solution of (1)–(4) is a pair (u,T)∈Hσ1×H1(Ω) satisfying(7)2νDu,Dv+2αν∫Γ3u·vdΓ+u·∇u,v+βGT,v=f,vH~σ′,(8)κ∇T,∇z+u·∇T,z+κ∫Γ5γT·zdΓ=g,zH′+φ,zΓ5,(9)u=ug1onΓ∖Γ3,T=g2onΓ4,for all (v,z)∈X.


## 3. Existence of a Weak Solution to Problem (1)–(4)

In order to prove the existence of a solution to system (7)–(9), we reduce the problem to an auxiliary problem with homogeneous conditions for the fields velocity on Γ∖Γ3 and the temperature on Γ4. For this, we introduce the following results.Lemma 3.
Assume thatu0∈H~001/2(Γ1), g1∈H~001/2(Γ2). For any ε>0 there exists uε∈Hσ1 with uε=g1 on Γ2, uε=u0 on Γ1, and uε=0 on Γ3 such that(10)v·∇uε,v≤εvH~σ12∀v∈H~σ1,uεHσ1≤Cug1H1/2Γ∖Γ3,where the constant C>0 depends only on Ω. Moreover, if g2∈H001/2(Γ4), then there exists Tδ∈H1(Ω) such that(11)Tδ=g2onΓ4,∂Tδ∂n+γTδ=0onΓ5,Tδ4≤δ,TδH1≤Cg2H1/2Γ4,where the constant C depends only on Ω.Proof.
The existence ofuε∈Hσ1, satisfying the relations given in (10), follows from [21], Lemma 3 (see, also [20], Lemma 3.2). The existence of Tδ∈H1(Ω) satisfying (11) follows from [31].Lemma 4 (see [32]).
LetX be a finite dimensional Hilbert space with inner product (·,·)X and norm ·X, and let P be a continuous mapping from X into itself such that(12)Pϕ,ϕX>0forϕX=K>0. Then, there exists ϕ∈X, ϕX≤K, such that P(ϕ)=0.Rewriting the unknowns(u,T)∈Hσ1×H1(Ω) in the form u=uε+u^ and T=Tδ+T^, where (u^,T^)∈X are new unknown functions, from (7)–(9) we can obtain the following problem. Find (u^,T^)∈X such that(13)2νDu^,Dv+2αν∫Γ3u^·vdΓ+u^·∇u^,v+u^·∇uε,v+uε·∇u^,v+βGT^,v=f^,vH~σ′∀v∈H~σ1,κ∇T^,∇z+u^·∇T^,z+u^·∇Tδ,z+uε·∇T^,z+κ∫Γ5γT^·zdΓ=g^,zH′∀z∈H0,where(14)f^,vH~σ′≔-2νDuε,Dv-uε·∇uε+βGTδ,v+f,vH~σ′,g^,zH′≔-κ∇Tδ,∇z-uε·∇Tδ,z-κ∫Γ5γTδ·zdΓ+g,zH′+φ,zΓ5.Thus, we have the following result.Theorem 5.
Let(f,g)∈X′, G∈L∞(Ω), u0∈H~001/2(Γ1), g1∈H~001/2(Γ2), g2∈H001/2(Γ4), φ∈L2(Γ5), and γ∈L∞(Γ5). Given these conditions, if ν is large enough and γL∞(Γ5) is small enough such that(15)min⁡2ν,κ-κCγL∞Γ5>CβG∞+δ,where C is a positive constant depending on Ω such that ∇u≤CuH~σ1 and δ is given in (11), then, there exists (u^,T^)∈X, which is a solution of system (13).
Moreover, the following estimate is satisfied:(16)u^,T^X≤1δ~f^,g^X′,where δ~≔min{2ν-ε,κ-κCγL∞(Γ5)}-C(βG∞+δ)>0.Proof.
The existence of a solution will be proved by using the Galerkin method. For this purpose, we consider an approximate solution of (13) and then we pass to the limit. Since H~σ1 and H0 are separable spaces, there exist Hilbert spaces bases {vi}i=1∞ and {zi}i=1∞ of H~σ1 and H0, respectively. For each k∈N, we consider H~k the space spanned by {v1,…,vk} and Hk the space spanned by {z1,…,zk}; let Xk=H~k×Hk⊂X. The scalar product on Xk is the scalar product (·,·)Xk induced by X.
Thus, for eachk∈N, fixed, we define an approximate solution (uk,Tk)∈Xk of system (13), given by (17)uk≔∑i=1kξi,kvix,Tkx≔∑i=1kθi,kTix, where ξi,k,θi,k∈R, and satisfying(18)2νDuk,Dvi+2αν∫Γ3uk·vidΓ+uk·∇uk,vi+uk·∇uε,vi+uε·∇uk,vi+βGTk,vi=f^,viH~σ′,κ∇Tk,∇zi+uk·∇Tk,zi+uk·∇Tδ,zi+uε·∇Tk,zi+κ∫Γ5γTk·zidΓ=g^,ziH′for all (vi,zi)∈Xk.
In order to prove the existence of a solution for system (18) we apply the result of Lemma 4. For this purpose, we consider the continuous mapping P:Xk→Xk, which for wk=(uk,Tk)∈Xk is given by(19)Pwk,wXk=2νDuk,Dv+2αν∫Γ3uk·vdΓ+uk·∇uk,v+uk·∇uε,v+uε·∇uk,v+βGTk,v-f^,vH~σ′+κ∇Tk,∇z+uk·∇Tk,z+uk·∇Tδ,z+uε·∇Tk,z+κ∫Γ5γTk·zdΓ-g^,zH′,for all w=(v,z)∈Xk. By replacing w by (uk,Tk) in (19) and using Lemma 3, we obtain(20)Pwk,wkXk≥2νukH~σ12-εukH~σ12+κTkH02+βGTk,uk+uk·∇Tδ,Tk+κ∫Γ5γTk·TkdΓ-f^,ukH~σ′-g^,TkH′.Now, we will bind the right-hand side of (20). Since ukH~σ12+TkH02=wkXk2, then by using the Hölder and Young inequalities we have(21)f^,ukH~σ′+g^,TkH′≤fHσ′ukH~σ1+g^H′TkH′≤f^,g^X′wkXk.Applying the Hölder and Young inequalities, and taking into account the inequalities u4≤C∇u, ∇u≤CD(u), and Tδ4≤δ, we obtain(22)βGTk,uk≤βG∞Tkuk≤βCG∞TkH0ukH~σ1≤βCG∞ukH~σ12+TkH02,κ∫Γ5γTk·TkdΓ≤κγL∞Γ5TkL2Γ52≤κCγL∞Γ5TkH02,uk·∇Tδ,Tk=uk·∇Tk,Tδ≤uk4∇TkTδ4≤δCukH~σTkH0≤δCukH~σ2+TkH02. By replacing (21) and (22) in (20) and taking into account (11), we have(23)Pwk,wkXk≥2ν-εukH~σ12+κ-κCγL∞Γ5TkH02-βCG∞ukH~σ12+TkH02-δCukH~σ2+TkH02-f^,g^X′wkXk≥min⁡2ν-ε,κ-κCγL∞Γ5wkXk2-βCG∞+δCwkXk2-f^,g^X′wkXk≥wkXkδ~wkXk-f^,g^X′,where, by hypothesis, (24)δ~≔min⁡2ν-ε,κ-κCγL∞Γ5-βCG∞+δC>0, for ε small enough. Then, for wkXk=K with K>1/δ~(f^,g^)X′, from (23) it follows that (25)Pwk,wkXk≥Kδ~K-f^,g^X′>0. Thus, by Lemma 4, there exists (uk,Tk)∈Xk such that (uk,Tk)Xk≥K and(26)Puk,Tk,v,zXk=0∀v,z∈Xk.Moreover, since (vi,0), (0,zi)∈Xk, from (19) and (26), we conclude that (uk,Tk) is a solution to (18).
We now pass to the limit. Sincewk=(uk,Tk)∈Xk, then (26) implies (P(wk),wk)Xk=0 and, from (23), we conclude that(27)uk,TkXk≤1δ~f^,g^X′.Thus, from (27), we conclude that the sequence {(uk,Tk)}k≥1 is bounded in X; then there exists a subsequence, still denoted by {(uk,Tk)}k≥1, and some element (u^,T^)∈X, such that when k→∞,(28)uk⟶u^weaklyinH~σ1andstronglyinL2Ω,Tk⟶T^weaklyinH0andstronglyinL2Ω.The convergences (28) allow us to pass to the limit in (18) and to prove that (u^,T^) is a solution to system (13). Moreover, from (27), we have (29)u^,T^X≤liminfk→∞uk,TkXk≤1δ~f^,g^X′, which implies (16).Theorem 6 (existence of weak solutions).
Under the conditions of Theorem5, there exists a solution (u,T)∈Hσ1×H1(Ω) for system (7)–(9). Moreover, the following estimate holds:(30)uHσ1+TH1≤C~1Θ,where C~1 is a positive constant which depends only on Ω, ν, κ, β, G∞, and γL∞(Γ5) and Θ=ug1H1/2(Γ∖Γ3)+g2H1/2(Γ4)+ug1H1/2(Γ∖Γ3)2+g2H1/2(Γ4)2+φL2(Γ5)+(f,g)X′.Proof.
As a consequence of Theorem5, we have the fact that there exists a solution (u,T)∈Hσ1×H1(Ω) for system (7)–(9), where u=u^+uε and T=T^+Tδ, with (u^,T^)∈X being a solution of (13) given in Theorem 5. Thus, from (16), we have(31)uHσ1+TH1≤u^H~σ1+T^H0+uεHσ1+TδH1≤2u^,T^X+2uε,TδHσ1×H1≤2δf^,g^X′+2uε,TδHσ1×H1. From (14) we can obtain (32)f^,g^X′≤C1uε,TδHσ1×H1+uε,TδHσ1×H12+φL2Γ5+f,gX′, where C1 is a positive constant which depends on β,G∞,γL∞(Γ5),ν, and κ. Thus, from (31), we have(33)uHσ1+TH1≤C1Cδuε,TδHσ1×H1+uε,TδHσ1×H12+C1CδφL2Γ5+f,gX′.Since (uε,Tδ)Hσ1×H1≤uεHσ1+TδH1, then, by Lemma 3, we have (34)uε,TδHσ1×H1≤Cug1H1/2Γ∖Γ3+g2H1/2Γ4,uε,TδHσ1×H12≤Cug1H1/2Γ∖Γ32+g2H1/2Γ42.Therefore, the above inequalities together with (33) imply (30).Remark 7.
SinceRe=1/ν, then condition (15) shows us that the existence of solution for the system (7)–(9) is guaranteed when the Reynolds number is small (the flow is laminar).Theorem 8 (uniqueness of solutions).
Under conditions of Theorem6, if ν, κ are large enough and γL∞(Γ5) is small enough such that(35)min⁡2ν-βCG∞,κ-βCG∞-κCγL∞Γ5>CC~1Θ,where C is a positive constant that depends on the domain Ω and C~1,Θ are given in (30), then the solution (u,T)∈Hσ1×H1(Ω) of system (7)–(9) provided by Theorem 6 is unique.Proof.
Let(u1,T2),(u2,T2)∈Hσ1×H1(Ω) be two solutions of system (7)–(9). Then, subtracting the corresponding equations and denoting u=u1-u2 and T=T1-T2, we obtain (u,T)∈X and the following system:(36)2νDu,Dv+2αν∫Γ3u·vdΓ+u1·∇u,v+u·∇u2,v+βGT,v=0∀v∈H~σ1,(37)κ∇T,∇z+u1·∇T,z+u·∇T2,z+κ∫Γ5γT·zdΓ=0∀z∈H0.Now, by replacing v=u in (36) and z=T in (37) and adding the results, we have(38)2νuH~σ12+κTH02≤u·∇u2,u+βGT,u+u·∇T2,T+κ∫Γ5γT·TdΓ.By using the Hölder and Young inequalities, we obtain(39)u·∇u2,u≤u3∇u2u6≤CuH~σ12u2Hσ1,βGT,u≤βG∞Tu≤βCG∞TH0uH~σ1≤βCG∞uH~σ12+TH02,u·∇T2,T≤u3∇T2T6≤CuH~σ1T2H1TH0≤CT2H1uH~σ12+TH02,κ∫Γ5γT·TdΓ≤κγL∞Γ5TL2Γ52≤κCγL∞Γ5TH02.By replacing (39) in (38), we can obtain(40)2ν-βCG∞uH~σ12+κ-βCG∞-κCγL∞Γ5TH02≤Cu2Hσ1+T2H1uH~σ12+CT2H1TH02.From (30) we have u2Hσ1+T2H1≤C~1Θ; then (40) implies (41)2ν-βCG∞uH~σ12+κ-βCG∞-κCγL∞Γ5TH02≤CC~1ΘuH~σ12+TH02, and thus(42)2ν-βCG∞-CC~1ΘuH~σ12+κ-βCG∞-κCγL∞Γ5-CC~1ΘTH02≤0.Therefore, if min{2ν-βCG∞,κ-βCG∞-κCγL∞(Γ5)}>CC~1Θ, from (42), we have uH~σ1=TH0=0, that is, u1=u2 and T1=T2, and the theorem is proved.Remark 9.
The condition (35) assures us of the uniqueness of solution of system (7)–(9) when the viscosity ν and the temperature conduction coefficient κ are large enough, which implies that the Rayleigh number is small. That is, the uniqueness of weak solution is obtained when the heat transfer is produced primarily by conduction.


## 4. The Optimal Control Problem

In this section the statement of the boundary control problem to study is established. We suppose thatU1⊂H~001/2(Γ2) and U2⊂H001/2(Γ4) are nonempty sets. We consider data (f,g)∈X′, G∈L∞(Ω), u0∈H~001/2(Γ1), φ∈L2(Γ5), γ∈L∞(Γ5), and the functions g1∈U1, g2∈U2, describing the Dirichlet boundary controls for u on Γ2 and T on Γ4, respectively. For simplicity, here and hereinbelow, we will use the product space H=Hσ1×H1(Ω), and we consider the following objective functional J:H×U1×U2→R defined by(43)Ju,T,g1,g2=J^u,T+b12g1H1/2Γ22+b22g2H1/2Γ42,where J^:Hσ1×H1(Ω)→R is a weakly lower semicontinuous functional, and the nonnegative constants b1,b2 measure the cost of the controls.The following are examples of the weakly lower semicontinuous functional in (43) most interesting from a physical viewpoint:(44)J1u=12∫Ωu-ud2dΩ,J2T=12∫ΩT-Td2dΩ,J3u=ν2∫Ω∇u+∇Tu2dΩ,where ud∈L2(Ω) and Td∈L2(Ω) are the desired states. The functionals J1 and J2 describe, respectively, the deviation of the velocity of flow from a given velocity and the deviation of the temperature from a given temperature. The functional J3 describes the resistance in a fluid due to viscous friction (see [22]).For simplicity we will do the study withJ^=J1+J2; also the study can be done considering J^ as the sum of two or more functionals given in (44). Thus, we define the following constrained minimization problem related to system (7)–(9):(45)Findu,T,g1,g2∈HsuchthatJu,T,g1,g2=12u-ud2+12T-Td2+b12g1H1/2Γ22+b22g2H1/2Γ42reaches  its  minimum  over  the  weak  solutions  of  system (1)–(4).The set of admissible solutions of optimal control problem (45) is defined by (46)Sad=s=u,T,g1,g2∈H×UadsuchthatJs<∞andssatisfies  (7)–(9),where Uad≔U1×U2. A similar control problem has been studied in [20, 21] for micropolar fluids. In [20] a three-dimensional domain was considered with density constant; while in [21] the problem was studied in a two-dimensional domain, where the fluid density was considered as an extra variable in the system.


### 4.1. Existence of Optimal Solution

In this subsection we will prove the existence of an optimal solution for problem (45). For this purpose we introduce the following result. The proof follows a technique that is standard for optimal control problems. We will repeat it for convenience of the reader.Theorem 10.
Under the hypotheses of Theorem6, and also assuming that the conditions are satisfied:(47)ibi≥0,fori=1,2,U1⊂H~001/2Γ2,U2⊂H001/2Γ4arebounded,closedandconvexset;iibi>0,fori=1,2,U1⊂H~001/2Γ2,U2⊂H001/2Γ4areclosedandconvexset;then problem (45) has at least one optimal solution; that is, there exist s~=(u~,T~,g~1,g~2)∈Sad such that(48)Ju~,T~,g~1,g~2=mins∈SadJu,T,g1,g2.Proof.
From Theorem6 we have that the set of admissible solutions Sad≠∅, and since the functional J is bounded below, there exists a minimizing sequence {sm=(um,Tm,g1m,g2m)}m≥1∈Sad,m∈N, such that limm→∞J(sm)=infs∈SadJ(s). Also, by definition of set Sad, we have the fact that sm satisfies system (7)–(9); that is,(49)2νDum,Dv+2αν∫Γ3um·vdΓ+um·∇um,v+βGTm,v=f,vH~σ′,κ∇Tm,∇z+um·∇Tm,z+κ∫Γ5γTm·zdΓ=g,zH′+φ,zΓ5,um=ug1monΓ∖Γ3,Tm=g2monΓ4,for all (v,z)∈X. Now, if one of the conditions (i) or (ii) in (47) is satisfied, then there exists a constant C, independent of m, such that g1mH1/2(Γ2)2+g2mH1/2(Γ4)2≤C; thus, from (30), we conclude that umHσ1+TmH1≤C. Therefore, since Uad is a closed convex subset of H~001/2(Γ2)×H001/2(Γ4), there exists s~=(u~,T~,g~1,g~2)∈H×Uad such that, for some subsequence of {sm}m≥1, still denoted by {sm}m≥1, when m→∞, we have (50)um⟶u~weaklyinHσ1,Tm⟶T~weaklyinH1Ω,g1m⟶g~1weaklyinU1,g2m⟶g~2weaklyinU2;by standard Sobolev embeddings, we obtain that(51)um⟶u~stronglyinL2Ω,Tm⟶T~stronglyinL2Ω,g1m⟶g~1stronglyinL2Γ2,g2m⟶g~2stronglyinL2Γ4.
Moreover, sinceum=u0 on Γ1, um=g1m on Γ2, and Tm=g2m on Γ4, then from (51) it follows that u~=ug~1 on Γ∖Γ3 and T~=g~2 on Γ4; thus, s~ satisfies the boundary conditions given in (9). A standard procedure permits passing the limit in (51), as m goes to ∞, and proving that s~ is solution of system (7)–(9). Consequently we have s~∈Sad and(52)limm→∞Jsm=infs∈SadJs≤Js~.Also, since the functional J is weakly lower semicontinuous on Sad, we have that J(s~)≤limm→∞infJ(sm). Therefore, from (52) and last inequality we conclude J(u~,T~,g~1,g~2)=J(s~)=mins∈SadJ(s), which implies the existence of an optimal solution to the control problem (45).


### 4.2. First-Order Necessary Optimality Conditions and an Optimality System

In order to obtain first-order optimality conditions, and thus derive an optimality system, we start by considering the following Banach space:(53)Y≔X′×H~001/2Γ∖Γ3×H001/2Γ4 and the vector operator F=(F1,F2,F3,F4):H×Uad→Y, where F1:H×Uad→H~σ′,F2:H×Uad→H′,F3:H×Uad→H~001/2(Γ∖Γ3), and F4:H×Uad→H001/2(Γ4), at each point s=(u,T,g1,g2)∈H×Uad, are defined by(54)F1s,vH~σ′≔2νDu,Dv+2αν∫Γ3u·vdΓ+u·∇u,v+βGT,v-f,vH~σ′,F2s,zH′≔κ∇T,∇z+u·∇T,z+κ∫Γ5γT·zdΓ-g,zH′+φ,zΓ5,F3s≔uΓ∖Γ3-ug1,F4s≔TΓ4-g2,for all (v,z)∈X. Thus, the optimal control problem (45) is rewritten as follows:(55)Finds=u,T,g1,g2∈H×UadsuchthatthefunctionalJu,T,g1,g2=12u-ud2+12T-Td2+b12g1H1/2Γ22+b22g2H1/2Γ42isminimizedsubjecttoFs=0;thatis,F1s,vH~σ′=0,F2s,zH′=0,F3s=0,F4s=0,for all (v,z)∈X.Recall that problem (45) is a problem with constraint; one method to obtain necessary conditions of optimality is Lagrange method, which consists of transforming the problem (45) into an unrestricted one, using an auxiliary function (Lagrange functional or Lagrangian) and additional variables, called Lagrange multipliers. The Lagrangian associated with optimal control problem (45) is given by(56)Ls,ϕ=Js-F1s,λH~σ′-F2s,ηH′-χ,F3sΓ∖Γ3-ϑ,F4sΓ4,where ϕ=((λ,η),χ,ϑ)∈X×H~00-1/2(Γ∖Γ3)×H00-1/2(Γ4).Concerning differentiation of the objective functionalJ and the constraint operator F, we have the following results.Lemma 11.
The functionalJ is Fréchet differentiable with respect to the point s=((u,T),(g1,g2))∈H×Uad. Moreover, at an arbitrary point s~=((u~,T~),(g~1,g~2))∈H×Uad, the Fréchet derivative of J with respect to s is a linear and bounded functional Js(s~):H×Uad→R such that, at each point t=((ω,ψ),(ϱ,τ))∈H×Uad, it is defined by(57)Jss~t=u~-ud,ω+T~-Td,ψ+b1∫Γ2g~1·ϱdΓ+b2∫Γ4g~2·τdΓ.Lemma 12.
The operatorF is Fréchet differentiable with respect to the point s=((u,T),(g1,g2))∈H×Uad. Moreover, at an arbitrary point s~=((u~,T~),(g~1,g~2))∈H×Uad, the Fréchet derivative of F with respect to s is a linear and bounded operator Fs(s~):H×Uad→R such that, at each point t=((ω,ψ),(ϱ,τ))∈H×Uad, it is defined by(58)F1ss~t,vH~σ′=2νDω,Dv+2αν∫Γ3ω·vdΓ+u~·∇ω,v+ω·∇u~,v+βGψ,v,F2ss~t,zH′=κ∇ψ,∇z+κ∫Γ5γψ·zdΓ+u~·∇ψ,z+ω·∇T~,z,F3ss~t=ωΓ∖Γ3-Bϱ,F4ss~t=ψΓ4-τ,for all (v,z)∈X. The operator B∈L(H~001/2(Γ2),H~001/2(Γ∖Γ3)) is defined by(59)Bϱ≔0onΓ1,ϱonΓ2.The following is an adaptation to the definition of regular point, given by [33, p. 50] in abstract, for optimal control problem (45).Definition 13.
Lets~=((u~,T~),(g~1,g~2))∈H×Uad be an admissible point for the control problem (55). We say that s~ is a regular point of the operator F, defined by (54), if(60)Fss~H×Cg~1×Cg~2=Y,where(61)Cg~1×Cg~2=θ1g1-g~1,θ2g2-g~2:θ1≥0,θ2≥0,g1,g2∈Uad.In the following lemma, we give a condition to assure thats~∈Sad is a regular point for optimal control problem (45). Thereafter the existence of Lagrange multipliers is shown.Lemma 14.
Lets~=(u~,T~,g~1,g~2)∈Sad be a feasible solution for the optimal problem (45). If ν, κ are sufficiently large and γL∞(Γ5) is small enough, such that(62)min⁡2ν-βCG∞-Cu~,T~H,κ-βCG∞-κCγL∞Γ5-Cu~,T~H>0,where the constant C>0 depends on the domain Ω, then s~ is a regular point.Proof.
Given(a,b,c,d)∈Y, it is sufficient to show the existence of the point t=((ω,ψ),(ϱ,τ))∈H×Uad, such that(63)2νDω,Dv+2αν∫Γ3ω·vdΓ+u~·∇ω,v+ω·∇u~,v+βGψ,v=a,vH~σ′∀v∈H~σ1,κ∇ψ,∇z+κ∫Γ5γψ·zdΓ+u~·∇ψ,z+ω·∇T~,z=b,zH′∀z∈H0,ωΓ∖Γ3=c+Bϱ-g~1,ψΓ4=d+τ-g~2.Setting ϱ=g~1 and τ=g~2, we have ωΓ∖Γ3=c and ψΓ=d; thus, from Lemma 3, there exists (ωε,ψε)∈H such that ωΓ∖Γ3ε=c,ωΓ3ε=0, ψΓ4=d, and ψΓ5=0. Then, rewriting the unknown (ω,ψ)∈H in the form ω=ωε+ω^ and ψ=ψε+ψ^, with (ω^,ψ^)∈X new unknowns functions, from system (63), we obtain the following system: Find (ω^,ψ^)∈X such that(64)2νDω^,Dv+2αν∫Γ3ω^·vdΓ+u~·∇ω^,v+ω^·∇u~,v+βGψ^,v=a^,vH~σ′∀v∈H~σ1,κ∇ψ^,∇z+κ∫Γ5γψ^·zdΓ+u~·∇ψ^,z+ω^·∇T~,z=b^,zH′∀z∈H0,where (65)a^,vH~σ′≔-2νDωε,Dv-u~·∇ωε,v-ωε·∇u~,v-βGψε,v+a,vH~σ′,b^,zH′≔-κ∇ψε,∇z-u~·∇ψε,z-ωε·∇T~,z+b,zH′. In order to prove the existence of a solution for system (64) we define the bilinear form a:X×X→R by(66)aω^,ψ^,v,z≔2νDω^,Dv+2αν∫Γ3ω^·vdΓ+u~·∇ω^,v+ω^·∇u~,v+βGψ^,v+κ∇ψ^,∇z+u~·∇ψ^,z+ω^·∇T~,z+κ∫Γ5γψ^·zdΓ,and the linear and continuous functional F:X→R by(67)Fv,z≔a^,vH~σ′+b^,zH′.Thus, from (66)-(67), system (64) is equivalent to the following. Find (ω^,ψ^)∈X such that(68)aω^,ψ^,v,z=Fv,z∀v,z∈X.We notice that the form a(·,·) is continuous. Now, by taking (v,z)=(ω^,ψ^) in (66), we have(69)aω^,ψ^,ω^,ψ^≥2νω^H~σ12+ω^·∇u~,ω^+βGψ^,ω^+κψH02+ω^·∇T~,ψ^+κ∫Γ5γψ^·ψ^dΓ.By applying the Hölder and Young inequalities to the terms in the right-hand side of (69), we have (70)ω^·∇u~,ω^+ω^·∇T~,ψ^≤Cu~Hσ1ω^H~σ12+CT~H1ω^H~σ12+ψ^H02,βGψ^,ω^≤βG∞ψ^ω^≤βCG∞ω^H~σ12+ψ^H02,κ∫Γ5γψ^·ψ^dΓ≤κγL∞Γ5ψ^L2Γ52≤κCγL∞Γ5ψ^H02. Then, by using the last inequalities in (69), we have(71)aω^,ψ^,ω^,ψ^≥2ν-Cu~Hσ1-CT~H1-βCG∞ω^H~σ12+κ-CT~H1-βCG∞-κCγL∞Γ5ψ^H02≥δ¯ω^,ψ^X2,where, by hypotheses, δ¯≔min2ν-βCG∞-Cu~,T~H,κ-βCG∞-κCγL∞Γ5-Cu~,T~H>0. Thus, we obtain that the form a(·,·) is X-coercive; therefore, from (68), (71), and the Lax-Milgram theorem we conclude the existence of a unique (ω^,ψ^)∈X solution of system (64), and, consequently, we obtain that (ω,ψ)∈H is solution of (63).In the following result we prove the existence of Lagrange multipliers provided that a local optimal solutions~=(u~,T~,g~1,g~2)∈Sad verifies the regular point condition provided by Lemma 14.Theorem 15.
Lets~=(u~,T~,g~1,g~2)∈Sad be a local optimal solution for the optimal control problem (45) satisfying the regular point condition given in (62). Then, there exist Lagrange multipliers ((λ,η),χ,ϑ)∈X×H~00-1/2(Γ∖Γ3)×H00-1/2(Γ4) such that for all ((ω,ψ),ϱ,τ)∈H×C(g~1)×C(g~2) it holds that(72)u~-ud,ω+T~-Td,ψ+b1∫Γ2g~1·ϱdΓ+b2∫Γ4g~2·τdΓ-2νDω,Dλ-2αν∫Γ3ω·λdΓ-u~·∇ω,λ-ω·∇u~,λ-βGψ,λ-κ∇ψ,∇η-u~·∇ψ,η-ω·∇T~,η-κ∫Γ5γψ·ηdΓ-χ,ω-BϱΓ∖Γ3-ϑ,ψ-τΓ4≥0,where C(g~1)×C(g~2) is defined in (61).Proof.
Sinces~=(u~,T~,g~1,g~2)∈Sad satisfies the regular point condition (62), then there exist Lagrange multipliers ((λ,η),χ,ϑ)∈X×H~00-1/2(Γ∖Γ3)×H00-1/2(Γ4) such that (73)Jss~t-F1ss~t,λH~σ′-F2ss~t,ηH′-χ,F3ss~tΓ∖Γ3-ϑ,F4ss~tΓ4≥0, for all t=((ω,ψ),ϱ,τ)∈H×C(g~1)×C(g~2). Thus, the proof follows from (57), (58), and (59).


### 4.3. Optimality System

In this subsection we derive the equations that are satisfied by the Lagrange multipliers((λ,η),χ,ϑ)∈X×H~00-1/2(Γ∖Γ3)×H00-1/2(Γ4) provided by Theorem 15.Theorem 16.
Under the conditions of Theorem15, the Lagrange multipliers (λ,η,χ,ϑ) satisfy the following adjoint equations (74)-νΔλ-u~·∇λ+∇Tλ·u~+∇Tη·T~=u~-udinH~σ′,(75)-κΔη-u~·∇η+βG·λ=T~-TdinH′,(76)divλ=0inΩ,(77)ν∇λn=0onΓ∖Γ3,ν∇λn+αλ=0onΓ3,(78)∇ηn=0onΓ4,∇ηn+κγη=0onΓ5,(79)χ=0onΓ∖Γ3,ϑ=0onΓ4,and the optimality conditions(80)b1g~1+χ,g1-g~1Γ2≥0,b2g~2+ϑ,g2-g~2Γ4≥0.Proof.
In fact, from (72), taking (ψ,ϱ,τ)=(0,0,0), we have (81)u~-ud,ω-2νDω,Dλ-2αν∫Γ3ω·λdΓ-u~·∇ω,λ-ω·∇u~,λ-ω·∇T~,η-χ,ωΓ∖Γ3=0∀ω∈Hσ1, and integrating by parts, for all ω∈Hσ1, we obtain(82)u~-ud,ω+νΔλ,ω-2ν∫Γ∇λn·ωdΓ-2αν∫Γ3λ·ωdΓ+u~·∇λ,ω-∇Tλ·u,ω-∇Tη·T~,ω-χ,ωΓ∖Γ3=0.Now, taking (ω,ϱ,τ)=(0,0,0) in (72), we have (83)T~-Td,ψ-βGψ,λ-κ∇ψ,∇η-u~·∇ψ,η-κ∫Γ5γψ·ηdΓ-ϑ,ψΓ4=0, for all ψ∈H1(Ω), and integrating by parts, for all ψ∈H1(Ω), we obtain(84)T~-Td,ψ+κΔη,ψ-κ∫Γ∇ηn·ψdΓ+u~·∇η,ψ-βG·λ,ψ-κ∫Γ5γη·ψdΓ-ϑ,ψΓ4=0.Since (ω,ψ)∈H is arbitrary, then from (82) and (84), we obtain system (74)–(79).
Now, from (72), when (ω,ψ)=(0,0), we obtain (85)b1∫Γ2g~1·ϱdΓ+b2∫Γ4g~2·τdΓ+χ,BϱΓ∖Γ3+ϑ,τΓ4≥0.Taking ϱ=g1-g~1, τ=g2-g~2, and taking into account the definition of operator B given in (59), the last inequality implies(86)b1g~1+χ,g1-g~1Γ2+b2g~2+ϑ,g2-g~2Γ4≥0; thus the optimality conditions (80) follow.Summarizing, the state equations described in (1)–(4), the adjoint equations given in (75)–(79), and the optimality conditions obtained in (80) form the optimality system of the optimal control problem (45).Remark 17.
Since the set of controlsUad is convex, then from (80) we deduce(87)g~1=ProjU1-χb1onΓ2,g~2=ProjU2-ϑb2onΓ4.


## 4.1. Existence of Optimal Solution

In this subsection we will prove the existence of an optimal solution for problem (45). For this purpose we introduce the following result. The proof follows a technique that is standard for optimal control problems. We will repeat it for convenience of the reader.Theorem 10.
Under the hypotheses of Theorem6, and also assuming that the conditions are satisfied:(47)ibi≥0,fori=1,2,U1⊂H~001/2Γ2,U2⊂H001/2Γ4arebounded,closedandconvexset;iibi>0,fori=1,2,U1⊂H~001/2Γ2,U2⊂H001/2Γ4areclosedandconvexset;then problem (45) has at least one optimal solution; that is, there exist s~=(u~,T~,g~1,g~2)∈Sad such that(48)Ju~,T~,g~1,g~2=mins∈SadJu,T,g1,g2.Proof.
From Theorem6 we have that the set of admissible solutions Sad≠∅, and since the functional J is bounded below, there exists a minimizing sequence {sm=(um,Tm,g1m,g2m)}m≥1∈Sad,m∈N, such that limm→∞J(sm)=infs∈SadJ(s). Also, by definition of set Sad, we have the fact that sm satisfies system (7)–(9); that is,(49)2νDum,Dv+2αν∫Γ3um·vdΓ+um·∇um,v+βGTm,v=f,vH~σ′,κ∇Tm,∇z+um·∇Tm,z+κ∫Γ5γTm·zdΓ=g,zH′+φ,zΓ5,um=ug1monΓ∖Γ3,Tm=g2monΓ4,for all (v,z)∈X. Now, if one of the conditions (i) or (ii) in (47) is satisfied, then there exists a constant C, independent of m, such that g1mH1/2(Γ2)2+g2mH1/2(Γ4)2≤C; thus, from (30), we conclude that umHσ1+TmH1≤C. Therefore, since Uad is a closed convex subset of H~001/2(Γ2)×H001/2(Γ4), there exists s~=(u~,T~,g~1,g~2)∈H×Uad such that, for some subsequence of {sm}m≥1, still denoted by {sm}m≥1, when m→∞, we have (50)um⟶u~weaklyinHσ1,Tm⟶T~weaklyinH1Ω,g1m⟶g~1weaklyinU1,g2m⟶g~2weaklyinU2;by standard Sobolev embeddings, we obtain that(51)um⟶u~stronglyinL2Ω,Tm⟶T~stronglyinL2Ω,g1m⟶g~1stronglyinL2Γ2,g2m⟶g~2stronglyinL2Γ4.
Moreover, sinceum=u0 on Γ1, um=g1m on Γ2, and Tm=g2m on Γ4, then from (51) it follows that u~=ug~1 on Γ∖Γ3 and T~=g~2 on Γ4; thus, s~ satisfies the boundary conditions given in (9). A standard procedure permits passing the limit in (51), as m goes to ∞, and proving that s~ is solution of system (7)–(9). Consequently we have s~∈Sad and(52)limm→∞Jsm=infs∈SadJs≤Js~.Also, since the functional J is weakly lower semicontinuous on Sad, we have that J(s~)≤limm→∞infJ(sm). Therefore, from (52) and last inequality we conclude J(u~,T~,g~1,g~2)=J(s~)=mins∈SadJ(s), which implies the existence of an optimal solution to the control problem (45).


## 4.2. First-Order Necessary Optimality Conditions and an Optimality System

In order to obtain first-order optimality conditions, and thus derive an optimality system, we start by considering the following Banach space:(53)Y≔X′×H~001/2Γ∖Γ3×H001/2Γ4 and the vector operator F=(F1,F2,F3,F4):H×Uad→Y, where F1:H×Uad→H~σ′,F2:H×Uad→H′,F3:H×Uad→H~001/2(Γ∖Γ3), and F4:H×Uad→H001/2(Γ4), at each point s=(u,T,g1,g2)∈H×Uad, are defined by(54)F1s,vH~σ′≔2νDu,Dv+2αν∫Γ3u·vdΓ+u·∇u,v+βGT,v-f,vH~σ′,F2s,zH′≔κ∇T,∇z+u·∇T,z+κ∫Γ5γT·zdΓ-g,zH′+φ,zΓ5,F3s≔uΓ∖Γ3-ug1,F4s≔TΓ4-g2,for all (v,z)∈X. Thus, the optimal control problem (45) is rewritten as follows:(55)Finds=u,T,g1,g2∈H×UadsuchthatthefunctionalJu,T,g1,g2=12u-ud2+12T-Td2+b12g1H1/2Γ22+b22g2H1/2Γ42isminimizedsubjecttoFs=0;thatis,F1s,vH~σ′=0,F2s,zH′=0,F3s=0,F4s=0,for all (v,z)∈X.Recall that problem (45) is a problem with constraint; one method to obtain necessary conditions of optimality is Lagrange method, which consists of transforming the problem (45) into an unrestricted one, using an auxiliary function (Lagrange functional or Lagrangian) and additional variables, called Lagrange multipliers. The Lagrangian associated with optimal control problem (45) is given by(56)Ls,ϕ=Js-F1s,λH~σ′-F2s,ηH′-χ,F3sΓ∖Γ3-ϑ,F4sΓ4,where ϕ=((λ,η),χ,ϑ)∈X×H~00-1/2(Γ∖Γ3)×H00-1/2(Γ4).Concerning differentiation of the objective functionalJ and the constraint operator F, we have the following results.Lemma 11.
The functionalJ is Fréchet differentiable with respect to the point s=((u,T),(g1,g2))∈H×Uad. Moreover, at an arbitrary point s~=((u~,T~),(g~1,g~2))∈H×Uad, the Fréchet derivative of J with respect to s is a linear and bounded functional Js(s~):H×Uad→R such that, at each point t=((ω,ψ),(ϱ,τ))∈H×Uad, it is defined by(57)Jss~t=u~-ud,ω+T~-Td,ψ+b1∫Γ2g~1·ϱdΓ+b2∫Γ4g~2·τdΓ.Lemma 12.
The operatorF is Fréchet differentiable with respect to the point s=((u,T),(g1,g2))∈H×Uad. Moreover, at an arbitrary point s~=((u~,T~),(g~1,g~2))∈H×Uad, the Fréchet derivative of F with respect to s is a linear and bounded operator Fs(s~):H×Uad→R such that, at each point t=((ω,ψ),(ϱ,τ))∈H×Uad, it is defined by(58)F1ss~t,vH~σ′=2νDω,Dv+2αν∫Γ3ω·vdΓ+u~·∇ω,v+ω·∇u~,v+βGψ,v,F2ss~t,zH′=κ∇ψ,∇z+κ∫Γ5γψ·zdΓ+u~·∇ψ,z+ω·∇T~,z,F3ss~t=ωΓ∖Γ3-Bϱ,F4ss~t=ψΓ4-τ,for all (v,z)∈X. The operator B∈L(H~001/2(Γ2),H~001/2(Γ∖Γ3)) is defined by(59)Bϱ≔0onΓ1,ϱonΓ2.The following is an adaptation to the definition of regular point, given by [33, p. 50] in abstract, for optimal control problem (45).Definition 13.
Lets~=((u~,T~),(g~1,g~2))∈H×Uad be an admissible point for the control problem (55). We say that s~ is a regular point of the operator F, defined by (54), if(60)Fss~H×Cg~1×Cg~2=Y,where(61)Cg~1×Cg~2=θ1g1-g~1,θ2g2-g~2:θ1≥0,θ2≥0,g1,g2∈Uad.In the following lemma, we give a condition to assure thats~∈Sad is a regular point for optimal control problem (45). Thereafter the existence of Lagrange multipliers is shown.Lemma 14.
Lets~=(u~,T~,g~1,g~2)∈Sad be a feasible solution for the optimal problem (45). If ν, κ are sufficiently large and γL∞(Γ5) is small enough, such that(62)min⁡2ν-βCG∞-Cu~,T~H,κ-βCG∞-κCγL∞Γ5-Cu~,T~H>0,where the constant C>0 depends on the domain Ω, then s~ is a regular point.Proof.
Given(a,b,c,d)∈Y, it is sufficient to show the existence of the point t=((ω,ψ),(ϱ,τ))∈H×Uad, such that(63)2νDω,Dv+2αν∫Γ3ω·vdΓ+u~·∇ω,v+ω·∇u~,v+βGψ,v=a,vH~σ′∀v∈H~σ1,κ∇ψ,∇z+κ∫Γ5γψ·zdΓ+u~·∇ψ,z+ω·∇T~,z=b,zH′∀z∈H0,ωΓ∖Γ3=c+Bϱ-g~1,ψΓ4=d+τ-g~2.Setting ϱ=g~1 and τ=g~2, we have ωΓ∖Γ3=c and ψΓ=d; thus, from Lemma 3, there exists (ωε,ψε)∈H such that ωΓ∖Γ3ε=c,ωΓ3ε=0, ψΓ4=d, and ψΓ5=0. Then, rewriting the unknown (ω,ψ)∈H in the form ω=ωε+ω^ and ψ=ψε+ψ^, with (ω^,ψ^)∈X new unknowns functions, from system (63), we obtain the following system: Find (ω^,ψ^)∈X such that(64)2νDω^,Dv+2αν∫Γ3ω^·vdΓ+u~·∇ω^,v+ω^·∇u~,v+βGψ^,v=a^,vH~σ′∀v∈H~σ1,κ∇ψ^,∇z+κ∫Γ5γψ^·zdΓ+u~·∇ψ^,z+ω^·∇T~,z=b^,zH′∀z∈H0,where (65)a^,vH~σ′≔-2νDωε,Dv-u~·∇ωε,v-ωε·∇u~,v-βGψε,v+a,vH~σ′,b^,zH′≔-κ∇ψε,∇z-u~·∇ψε,z-ωε·∇T~,z+b,zH′. In order to prove the existence of a solution for system (64) we define the bilinear form a:X×X→R by(66)aω^,ψ^,v,z≔2νDω^,Dv+2αν∫Γ3ω^·vdΓ+u~·∇ω^,v+ω^·∇u~,v+βGψ^,v+κ∇ψ^,∇z+u~·∇ψ^,z+ω^·∇T~,z+κ∫Γ5γψ^·zdΓ,and the linear and continuous functional F:X→R by(67)Fv,z≔a^,vH~σ′+b^,zH′.Thus, from (66)-(67), system (64) is equivalent to the following. Find (ω^,ψ^)∈X such that(68)aω^,ψ^,v,z=Fv,z∀v,z∈X.We notice that the form a(·,·) is continuous. Now, by taking (v,z)=(ω^,ψ^) in (66), we have(69)aω^,ψ^,ω^,ψ^≥2νω^H~σ12+ω^·∇u~,ω^+βGψ^,ω^+κψH02+ω^·∇T~,ψ^+κ∫Γ5γψ^·ψ^dΓ.By applying the Hölder and Young inequalities to the terms in the right-hand side of (69), we have (70)ω^·∇u~,ω^+ω^·∇T~,ψ^≤Cu~Hσ1ω^H~σ12+CT~H1ω^H~σ12+ψ^H02,βGψ^,ω^≤βG∞ψ^ω^≤βCG∞ω^H~σ12+ψ^H02,κ∫Γ5γψ^·ψ^dΓ≤κγL∞Γ5ψ^L2Γ52≤κCγL∞Γ5ψ^H02. Then, by using the last inequalities in (69), we have(71)aω^,ψ^,ω^,ψ^≥2ν-Cu~Hσ1-CT~H1-βCG∞ω^H~σ12+κ-CT~H1-βCG∞-κCγL∞Γ5ψ^H02≥δ¯ω^,ψ^X2,where, by hypotheses, δ¯≔min2ν-βCG∞-Cu~,T~H,κ-βCG∞-κCγL∞Γ5-Cu~,T~H>0. Thus, we obtain that the form a(·,·) is X-coercive; therefore, from (68), (71), and the Lax-Milgram theorem we conclude the existence of a unique (ω^,ψ^)∈X solution of system (64), and, consequently, we obtain that (ω,ψ)∈H is solution of (63).In the following result we prove the existence of Lagrange multipliers provided that a local optimal solutions~=(u~,T~,g~1,g~2)∈Sad verifies the regular point condition provided by Lemma 14.Theorem 15.
Lets~=(u~,T~,g~1,g~2)∈Sad be a local optimal solution for the optimal control problem (45) satisfying the regular point condition given in (62). Then, there exist Lagrange multipliers ((λ,η),χ,ϑ)∈X×H~00-1/2(Γ∖Γ3)×H00-1/2(Γ4) such that for all ((ω,ψ),ϱ,τ)∈H×C(g~1)×C(g~2) it holds that(72)u~-ud,ω+T~-Td,ψ+b1∫Γ2g~1·ϱdΓ+b2∫Γ4g~2·τdΓ-2νDω,Dλ-2αν∫Γ3ω·λdΓ-u~·∇ω,λ-ω·∇u~,λ-βGψ,λ-κ∇ψ,∇η-u~·∇ψ,η-ω·∇T~,η-κ∫Γ5γψ·ηdΓ-χ,ω-BϱΓ∖Γ3-ϑ,ψ-τΓ4≥0,where C(g~1)×C(g~2) is defined in (61).Proof.
Sinces~=(u~,T~,g~1,g~2)∈Sad satisfies the regular point condition (62), then there exist Lagrange multipliers ((λ,η),χ,ϑ)∈X×H~00-1/2(Γ∖Γ3)×H00-1/2(Γ4) such that (73)Jss~t-F1ss~t,λH~σ′-F2ss~t,ηH′-χ,F3ss~tΓ∖Γ3-ϑ,F4ss~tΓ4≥0, for all t=((ω,ψ),ϱ,τ)∈H×C(g~1)×C(g~2). Thus, the proof follows from (57), (58), and (59).


## 4.3. Optimality System

In this subsection we derive the equations that are satisfied by the Lagrange multipliers((λ,η),χ,ϑ)∈X×H~00-1/2(Γ∖Γ3)×H00-1/2(Γ4) provided by Theorem 15.Theorem 16.
Under the conditions of Theorem15, the Lagrange multipliers (λ,η,χ,ϑ) satisfy the following adjoint equations (74)-νΔλ-u~·∇λ+∇Tλ·u~+∇Tη·T~=u~-udinH~σ′,(75)-κΔη-u~·∇η+βG·λ=T~-TdinH′,(76)divλ=0inΩ,(77)ν∇λn=0onΓ∖Γ3,ν∇λn+αλ=0onΓ3,(78)∇ηn=0onΓ4,∇ηn+κγη=0onΓ5,(79)χ=0onΓ∖Γ3,ϑ=0onΓ4,and the optimality conditions(80)b1g~1+χ,g1-g~1Γ2≥0,b2g~2+ϑ,g2-g~2Γ4≥0.Proof.
In fact, from (72), taking (ψ,ϱ,τ)=(0,0,0), we have (81)u~-ud,ω-2νDω,Dλ-2αν∫Γ3ω·λdΓ-u~·∇ω,λ-ω·∇u~,λ-ω·∇T~,η-χ,ωΓ∖Γ3=0∀ω∈Hσ1, and integrating by parts, for all ω∈Hσ1, we obtain(82)u~-ud,ω+νΔλ,ω-2ν∫Γ∇λn·ωdΓ-2αν∫Γ3λ·ωdΓ+u~·∇λ,ω-∇Tλ·u,ω-∇Tη·T~,ω-χ,ωΓ∖Γ3=0.Now, taking (ω,ϱ,τ)=(0,0,0) in (72), we have (83)T~-Td,ψ-βGψ,λ-κ∇ψ,∇η-u~·∇ψ,η-κ∫Γ5γψ·ηdΓ-ϑ,ψΓ4=0, for all ψ∈H1(Ω), and integrating by parts, for all ψ∈H1(Ω), we obtain(84)T~-Td,ψ+κΔη,ψ-κ∫Γ∇ηn·ψdΓ+u~·∇η,ψ-βG·λ,ψ-κ∫Γ5γη·ψdΓ-ϑ,ψΓ4=0.Since (ω,ψ)∈H is arbitrary, then from (82) and (84), we obtain system (74)–(79).
Now, from (72), when (ω,ψ)=(0,0), we obtain (85)b1∫Γ2g~1·ϱdΓ+b2∫Γ4g~2·τdΓ+χ,BϱΓ∖Γ3+ϑ,τΓ4≥0.Taking ϱ=g1-g~1, τ=g2-g~2, and taking into account the definition of operator B given in (59), the last inequality implies(86)b1g~1+χ,g1-g~1Γ2+b2g~2+ϑ,g2-g~2Γ4≥0; thus the optimality conditions (80) follow.Summarizing, the state equations described in (1)–(4), the adjoint equations given in (75)–(79), and the optimality conditions obtained in (80) form the optimality system of the optimal control problem (45).Remark 17.
Since the set of controlsUad is convex, then from (80) we deduce(87)g~1=ProjU1-χb1onΓ2,g~2=ProjU2-ϑb2onΓ4.


## 5. Second-Order Sufficient Condition

In this section, we will discuss sufficient conditions fors~=(u~,T~,g~1,g~2)∈Sad being a local optimal solution for control problem (45). We will establish a H×Uad-coercivity condition on the second derivative of the Lagrange functional L given in (56) associated with problem (45) in order to assure that an admissible point s~ is a local optimal solution. Due to Lemmas 11 and 12, the functional L is Fréchet differentiable with respect to the point s=((u,T),(g1,g2))∈H×Uad. Then, we can obtain the fact that that the Lagrange multiplier ϕ~=((λ,η),χ,ϑ)∈X×H~00-1/2(Γ∖Γ3)×H00-1/2(Γ4), given by Theorem 15, for x=(u,T)∈H, satisfies Lx(s~,ϕ~)(ω,ψ)=0; that is,(88)u~-ud,ω+T~-Td,ψ-2νDω,Dλ-2αν∫Γ3ω·λdΓ-u~·∇ω,λ-ω·∇u~,λ-βGψ,λ-κ∇ψ,∇η-u~·∇ψ,η-ω·∇T~,η-κ∫Γ5γψ·ηdΓ-χ,ωΓ∖Γ3-ϑ,ψΓ4=0,for all (ω,ψ)∈H; this equation is called the Lagrange-Euler equation. In this section we will follow the ideas of [20, 30] (see also [28]).Lemma 18.
Lets~=(u~,T~,g~1,g~2)∈Sad be an admissible point for the optimal control problem (45). If ν, κ are sufficiently large and γL∞(Γ5) is small enough, such that (89)δ^≔min⁡ν-βCG∞-Cu~,T~H,κ2-βCG∞-κCγL∞Γ5-Cu~,T~H>0, then the Lagrange multiplier (λ,η)∈X satisfies(90)λ,ηX2≤1δ^C~Nu~,T~,where N≔u~-u22+T~-Td2 and C~≔Cmax⁡1/4ν,1/2κ, with C being a positive constant depending only on Ω.Proof.
Setting(ω,ψ)=(λ,τ)∈X in (88), we have(91)2νλH~σ12+κηH02≤u~-ud,λ+T~-Td,η-λ·∇u~,λ-βGη,λ-λ·∇T~,η-κ∫Γ5γη·ηdΓ.Applying the Hölder and Young inequalities in terms of the right side of (91), we obtain (92)u~-ud,λ≤u~-udλ≤C4νu~-ud2+νλH~σ12,T~-Td,η≤T~-Tdη≤C2κT~-Td2+κ2ηH02,βGη,λ≤βG∞ηλ≤βCG∞λH~σ12+ηH02,λ·∇u~,λ+λ·∇T~,η≤Cu~Hσ1λH~σ12+CT~H1λH~σ12+ηH02≤Cu~,T~HλH~σ12+Cu~,T~HηH02,κ∫Γ5γη·ηdΓ≤κγL∞Γ5ηL2Γ52≤κCγL∞Γ5ηH02.Then, by replacing the last inequalities in (91), we have (93)ν-βCG∞-Cu~,T~HλH~σ12+κ2-βCG∞-κCγL∞Γ5-Cu~,T~HηH02≤C4νu~-ud2+C2κT~-Td2; since, by hypothesis, (94)δ^≔min⁡ν-βCG∞-Cu~,T~H,κ2-βCG∞-κCγL∞Γ5-Cu~,T~H>0, then, from the last inequality, we conclude the proof.Theorem 19.
Lets~=(u~,T~,g~1,g~2)∈Sad be an admissible point for the optimal control problem (45), ν, κ be sufficiently large, and γL∞(Γ5) be small enough, such that (95)δ^=min⁡ν-βCG∞-Cu~,T~H,κ2-βCG∞-κCγL∞Γ5-Cu~,T~H>0. If C~2C2/δ^Π2N(u~,T~)<1, where Π≔minCb1δ^2/(1+δ^)2,Cb2δ^2/(1+δ^)2 and N(u~,T~), C~ are given in Lemma 18, then there exists a constant K>0 such that(96)Lsss~,ϕ~t,t≥KtH×Uad2,for all t∈ker(Fs(s)), with ϕ~=((λ,η),χ,ϑ)∈X×H~00-1/2(Γ∖Γ3)×H00-1/2(Γ4) given by Theorem 15. Consequently, the point s~ is a local optimal solution for problem (45).Proof.
Lett=((ω,ψ),(ϱ,τ))∈H×Uad. Then, the second derivative of Lagrangian given in (56), with respect to s at the point s~ in (t,t), is(97)Lsss~,ϕ~t,t=ω2+ψ2+b1ϱH1/2Γ22+b2τH1/2Γ42-2ω·∇ω,λ-2ω·∇ψ,η.By applying the Hölder and Young inequalities, we can obtain(98)Lsss~,ϕ~t,t≥ω2+ψ2+b1ϱH1/2Γ22+b2τH1/2Γ42-CλH~σ1+ηH0ω,ψH2.On the other hand, if t∈ker(Fs(s~)), then, from (58)-(59), we obtain(99)2νDω,Dv+2αν∫Γ3ω·vdΓ+u~·∇ω,v+ω·∇u~,v+βGψ,v=0,κ∇ψ,∇z+u~·∇ψ,z+ω·∇T~,z+κ∫Γ5γψ·zdΓ=0,ωΓ∖Γ3=Bϱ,ψΓ4=τ, for all (v,z)∈X. Thus, by using Lemma 3, there exists (ωε,ψε)∈H such that ωΓ∖Γ3ε=Bϱ, ωΓ3=0, ψΓ4=τ, and ψΓ5=0. Then, rewriting (ω,ψ)∈H in the form ω=ωε+ω^ and ψ=ψε+ψ^ with (ω^,ψ^)∈X new unknown, and similarly, as in the proof of Lemma 14, we obtain(100)aω^,ψ^,v,z=F~v,z∀v,z∈X,where the form a(·,·) is defined in (66) and F~∈X′ is defined by F~(v,z)≔a~,vH~σ1+b~,zH′, with a~,vH~σ′≔-2ν(D(ωε),D(v))-(u~·∇ωε,v)-(ωε·∇u~,v)-β(Gψε,v) and b~,zH′≔-κ(∇ψε,∇z)-(u~·∇ψε,z)-(ωε·∇T~,z). Also, from (100), we can obtain the fact that there exists a positive constant C depending only on ν,κ and (u~,T~)H, such that (ω^,ψ^)X≤C/δ~(ωε,ψε)H, where δ~ is given as in (71). Thus, we have(101)ω,ψH≤ω^,ψ^X+ωε,ψεH≤C1+δ~δ~ϱH1/2Γ2+τH1/2Γ4.Using (101) in (98) and taking into account the fact that Π≔min{Cb1δ^2/(1+δ^)2,Cb2δ^2/(1+δ^)2}, we obtain(102)Lsss~,ϕ~t,t≥ω2+ψ2+b12ϱH1/2Γ22+b22τH1/2Γ42-CλH~σ1+ηH0ω,ψH2+Πω,ψH2≥Π-CλH~σ1+ηH0ω,ψH2+b12ϱH1/2Γ22+b22τH1/2Γ42.Notice that, from Lemma 18, we can obtain λH~σ1+ηH0≤C~/δ^N(u~,T~)1/2; then, from (102), we have (103)Lsss~,ϕ~t,t≥Π-CC~δ^Nu~,T~1/2ω,ψH2+b12ϱH1/2Γ22+b22τH1/2Γ42.Since CC~/δ^N(u~,T~)1/2<Π, then we have Π~≔Π-CC~/δ^N(u~,T~)1/2>0; therefore (104)Lsss~,ϕ~t,t≥min⁡Π~,b12,b22tH×Uad2. Thus, taking in particular t=((ω,ψ),(ϱ,τ))∈ker(Fs(s~)) with (ϱ,τ)∈C(g~1)×C(g~2)≔{(θ1(g1-g~1),θ2(g2-g~2)):θ1≥0,θ2≥0,(g1,g2)∈Uad}, we obtain that the point s~∈Sad is a local optimal solution of control problem (45) (see [34]).


## 6. Conclusions

In this paper, the existence and uniqueness of weak solutions of the stationary heat convection equations with mixed boundary conditions, including the Navier slip condition, were proved. Also, an optimal boundary control problem was analyzed. Boundary controls for the velocity vector and temperature and the existence of optimal solutions were proved. The optimal control problem analyzed includes the minimization of theL2-distance between the velocities and some desired fields. By using the theorem of Lagrange multipliers, an optimality system was derived. A second-order sufficient condition was also given.



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