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 equations

(2.1)

(2.2)

(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,

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 subspaces

(2.4)

The inner products and norms in L²(Ω), L²(Q), or L²(Γ_N) are denoted by (·, ·), ∥·∥, (·, ·) _Q, ∥·∥_Q, or

. 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

; the norm in the dual space

is denoted by ∥·∥_−1/2,Γ. Set

. Let in addition to condition (i) the following conditions hold:

(ii)
f ∈ H⁻¹(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

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

Bilinear form −(div · , ·) satisfies the inf-sup condition

(2.10)

Besides the following identities take place:

(2.11)

(2.12)

Let

, 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

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

satisfying the relations

(2.13)

(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 , χ ∈ L²(Γ_N), f ∈ L²(Ω), ψ ∈ H^1/2(Γ_D) a weak solution (u, p, T) that satisfies the estimates

(2.16)

Here M_u, M_p and M_T are nondecreasing continuous functions of the norms ∥f∥₋₁, ∥b∥, ∥g∥_1/2,Γ,

, ∥f∥,

. If, additionally, f, g, χ, f, ψ, α are small in the sense that

(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:

(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

. Denote by

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 formula

(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:

(3.3)

Here

is the operator acting by formulas

(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 that

(3.5)

Let

and

be the duals of the spaces X and Y. Let

denotes the Fréchet derivative of F with respect to x at the point

. By

we denote the adjoint operator of

which is determined by the relation

(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

, where ℝ⁺ = {x ∈ ℝ : x ≥ 0}, by

(3.7)

Here and below

and κ is an auxiliary dimensional parameter. Its dimension [κ] is chosen so that dimensions of ξ, σ, θ at the adjoint state coincide with those at the basic state, that is,

(3.8)

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

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 . Then there exists at least one solution to problem (3.5) for I = I_k, k = 1,2, 3.

Theorem 3.2. Let under conditions of Theorem 3.1 a pair be a local minimizer in problem (3.5) and let the cost functional I be continuously differentiable with respect to u at the point . Then there exists a nonzero Lagrange multiplier (λ₀, y^*) = (λ₀, ξ, σ, ζ, θ, ζ^t) ∈ ℝ⁺ × Y^* such that the Euler-Lagrange equation

(3.9)

for the adjoint state y^* is satisfied and the minimum principle holds which is equivalent to the inequality

(3.10)

Theorem 3.3. Let the assumptions of Theorem 3.2 be satisfied and condition (2.17) holds for all u ≡ (χ, ψ, f) ∈ K. Then any nontrivial Lagrange multiplier satisfying (3.9) is regular, that is, has the form (1, y^*) and is uniquely determined.

We note that the functional J and Lagrangian ℒ given by (3.7) are continuously differentiable functions of controls χ, ψ, f and its derivatives with respect to χ, ψ, and f are given by

(3.11)

Here for example

is the Gateaux derivative with respect to χ at the point

. Since K₁, K₂, K₃ are convex sets, at the minimum point

of the functional

the following conditions are satisfied (see [22]):

(3.12)

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

(3.13)

Relations (3.13), the minimum principle which is equivalent to the inequalities (3.10) or (3.12), and the operator constraint (3.3) which is equivalent to (2.13)–(2.15) constitute the optimality system for control problem (3.5).

Theorems 3.1 and 3.2 above are valid without any smallness conditions in relation to the data of Problem 1. The natural smallness condition (2.17) arises only when proving the uniqueness of solution to boundary problem (2.1)–(2.3) and Lagrange multiplier regularity. However, condition (2.17) does not provide the uniqueness of problem (3.5) solution. 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

. Let (x₁, u₁)≡(u₁, p₁, T₁, χ₁, ψ₁, f₁) ∈ X × K be an arbitrary solution to problem (3.5) for a given function g = g₁ ∈ G. By (x₂, u₂)≡(u₂, p₂, T₂, χ₂, ψ₂, f₂) ∈ X × K we denote a solution to problem

(3.14)

It is obtained by replacing the functional I in (3.5) by a close functional

depending on u and by replacing a function g ∈ G by a close function

By Theorem 3.1 the following estimates hold for triples (u_i, p_i, T_i):

(3.15)

Here

(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 by

(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

(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:

(3.19)

Let us denote by

, where

, i = 1,2, Lagrange multipliers corresponding to solutions (x_i, u_i). By Theorems 3.2 and 3.3 and (3.12) they satisfy relations

(3.20)

(3.21)

(3.22)

(3.23)

We renamed I¹ ≡ I,

in (3.20). Set ξ = ξ₁ − ξ₂, σ = σ₁ − σ₂, ζ = ζ₁ − ζ₂, θ = θ₁ − θ₂,

, g = g₁ − g₂, and

(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 obtain

(3.25)

(3.26)

(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

(3.28)

Subtract the identities (3.20)–(3.22), written for

from the corresponding identities for

,g₁), set w = u, τ = T and add. Using (3.27) we obtain

(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

(3.30)

Thus we have proved the following result.

Theorem 3.4. Let under conditions of Theorem 3.2 for functionals I and 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, , 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 div u₀ = 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

, where

is a new unknown function. Set

in (3.25). Taking into account (2.9) we obtain

(3.31)

Using estimates (2.5), (2.6), (2.7), and (3.15), we deduce from (3.31) that

(3.32)

It follows from (3.19) that

(3.33)

Rewriting the inequality (3.32) by (3.33) as

(3.34)

we obtain that

(3.35)

Taking into account the relation

, we come to the following estimate ∥u∥₁ via ∥g∥_1/2,Γ and ∥T∥₁:

(3.36)

Denote by T₀ ∈ H¹(Ω) a function such that

and the estimate

holds with a certain constant C₁, which does not depend on ψ. Let us present the difference T = T₁ − T₂ as

, where

is a new unknown function. Set

in (3.26). We obtain

(3.37)

Using estimates (2.5)–(2.8) and (3.15) we deduce that

(3.38)

(3.39)

Taking into account the relation

, we obtain from this estimate that

(3.40)

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

(3.41)

From this inequality and (3.17), (3.19) we come to the following estimate:

(3.42)

Using (3.42), we deduce from (3.36) that

(3.43)

Taking into account (3.17) we come to the following estimate for ∥u∥₁:

(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 ≠ 0, such that −(div v₀, p) ≥ β₀∥v₀∥₁∥p∥ where β₀ = (β − δ) > 0. Set v = v₀ in the identity for u in (3.25) and make of this estimate and estimates (2.6), (2.7), (3.15). We shall have

(3.45)

Dividing to ∥v₀∥₁ ≠ 0, we deduce that

(3.46)

Using (3.42) and (3.44), we come to the following final estimate for ∥p∥:

(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 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

(3.48)

(3.49)

(3.50)

(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 problem

(4.1)

In accordance to Remark 3.5 we can consider problem (4.1) as a particular case of the general control problem (3.5), which corresponds to the situation when every of sets 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

and

, and let (x₂, u₂)≡(u₂, p₂, T₂, χ₂) be a solution to problem (4.1), that corresponds to perturbed functions

and

. Setting

in addition to (3.24) we note that under conditions of problem (4.1) we have

(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 form

(4.3)

(4.4)

(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 obtain

(4.6)

(4.7)

Using estimates (2.5)–(2.8) and (3.15) we have

(4.8)

(4.9)

(4.10)

(4.11)

(4.12)

where

(4.13)

By virtue of (4.8)–(4.10) and (4.12), we deduce from (4.7) and (4.6) that

(4.14)

(4.15)

Taking into account (4.14), we obtain from (4.15) that

(4.16)

Using (3.33) we deduce successfully from (4.16), (4.14) that

(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

and any small number δ > 0 there exists a function

, v_i ≠ 0, such that the inequality

(4.18)

holds. Setting in (4.3) w = v_i and using this estimate together with estimates (2.6), (3.15), (4.11), we have

(4.19)

From this inequality we deduce by (4.17) that

(4.20)

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

(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

(4.22)

As ζ = ζ₁ − ζ₂ we obtain from this inequality that

(4.23)

Taking into account (2.6), (3.48), (3.49), and estimates (4.17) for ξ_i, θ_i, we have

(4.24)

It follows from (4.24) that

(4.25)

Here constants b and c are given by

(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:

(4.27)

Under condition (4.27) we deduce from (4.25) that

(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 inequality

(4.29)

It follows from this inequality that

(4.30)

Excluding nonpositive term

from the right-hand side of (4.30), we deduce from (4.30) that

(4.31)

Equation (4.31) is a quadratic inequality for ∥u∥_Q. Solving it we come to the following estimate for ∥u∥_Q:

(4.32)

As u = u₁ − u₂,

, g = g₁ − g₂, the estimate (4.32) is equivalent to the following estimate for the velocity difference u₁ − u₂:

(4.33)

This estimate under Q = Ω has the sense of the stability estimate in L²(Ω) of the component

of the solution

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

. It was obtained when studying control problems for Navier-Stokes and in [18] when studying control problems for heat convection equations. If besides

it follows from (4.33) that u₁ = u₂ in Ω, if Q = Ω. This yields together with (4.30), (3.48), (3.50) that χ₁ = χ₂, T₁ = T₂, p₁ = p₂. The latter means the uniqueness of the solution to problem (4.1) when Q = Ω and condition (4.27) holds.

It is important to note that the uniqueness and stability of the solution to problem (4.1) under condition (4.27) take place and in the case where Q ⊂ Ω; that is, Q is only a part of domain Ω. In order to prove this fact let us consider the inequality (4.30). Using (4.32) we deduce from (4.30) that

(4.34)

From (4.34) and (3.48)–(3.50) we come to the following stability estimates:

(4.35)

(4.36)

(4.37)

(4.38)

where

(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 and , i = 1,2, where Q ⊂ Ω is an arbitrary open subset, and let the parameters a and b, c are defined in (4.23) and (4.26) in which parameters γ and ℛe⁰ are given by (4.13). Suppose that condition (4.27) is satisfied. Then stability estimates (4.33) and (4.35)–(4.38) hold true where Δ is defined in (4.39).

Now we consider three-parametric control problem

(4.40)

corresponding to the cost functional

. Let (x₁, u₁)≡(u₁, p₁, T₁, χ₁, ψ₁, f₁) be a solution to problem (4.40) corresponding to given functions

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

and

. Setting

in addition to (3.24), we note that under conditions of problem (4.40) identities (3.20) and (3.21) transform to identities (4.3), (4.4), identity (3.22) does not change, while inequality (3.30) takes by (4.2) a form

(4.41)

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

(4.42)

Here parameters γ and ℛe⁰ are given by (4.13). From (4.42) we obtain that

(4.43)

Here constants b, c₁, c₂, and c₃ are given by relations

(4.44)

Let the data for problem (4.40) and parameters μ₀, μ₁, μ₂, and μ₃ be such that

(4.45)

Under condition (4.45) we deduce from (4.43) that

(4.46)

Taking into account (4.46) and (4.23), we come from (4.41) to the inequality

(4.47)

It follows from this inequality that

(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

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) as

(4.49)

From this inequality and from (3.42)–(3.47) we come to the following stability estimates:

(4.50)

(4.51)

(4.52)

(4.53)

Here a constant d depending on μ₁, μ₂, and μ₃ is given by

(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 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

(4.55)

corresponding to the cost functional

. Let us denote by (x₁, u₁)≡(u₁, p₁, T₁, χ₁) a solution to problem (4.55) which corresponds to given functions

and g = g₁ ∈ G; by (x₂, u₂)≡(u₂, p₂, T₂, χ₂) we denote a solution to problem (4.1) which corresponds to perturbed functions

and

. Setting

in addition to (3.24) we note that under conditions of problem (4.55) we have

(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 relations

(4.57)

(4.58)

Using identities (4.57), (4.4), and (3.22) we estimate parameters ξ_i, θ_i, σ_i and ζ_i. To this end we set w = ξ_i, τ = θ_i in (4.57), (3.22). Taking into account (2.11), (2.12) and condition ξ_i ∈ V which follows from (4.4) we obtain (4.7) and relation

(4.59)

Using estimates (3.15) we deduce in addition to (4.8)–(4.10) that

(4.60)

where

(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 inequality

(4.62)

It follows from this inequality that

(4.63)

Excluding nonpositive term

, we deduce from (4.63) that

(4.64)

Equation (4.31) is a quadratic inequality relative to ∥u∥_1,Q. By solving it we come to the estimate

(4.65)

which is equivalent to the following estimate for u₁ − u₂:

(4.66)

The estimate (4.66) under Q = Ω has the sense of the stability estimate in the norm H¹(Ω) of the component

of the solution

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

and g₁ = g₂ it follows from (4.66) that u₁ = u₂ in Ω, if Q = Ω. This yields together with (4.63), (3.48), (3.50) that χ₁ = χ₂, T₁ = T₂, p₁ = p₂. The latter means the uniqueness of the solution to problem (4.55) when Q = Ω and (4.27) holds.

We note again that using (4.63), (4.65) we can deduce rougher stability estimates of the solution to problem (4.55) which take place even in the case where Q ≠ Ω. In fact we deduce from (4.63) (4.65) that

(4.67)

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

(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 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 problem

(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 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 problem

(5.1)

which corresponds to the cost functional

. Let (x₁, u₁)≡(u₁, p₁, T₁, χ₁) be a solution to problem (5.1) corresponding to given functions

and g = g₁ ∈ G, and let (x₂, u₂)≡(u₂, p₂, T₂, χ₂) be a solution to problem (4.1) corresponding to perturbed functions

and

. Setting

in addition to (3.24), we have under conditions of problem (4.1)

(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 relations

(5.3)

(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 relation

(5.5)

Using (2.9), (3.15) we deduce in addition to (4.8)–(4.10) that

(5.6)

where

(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 inequality

(5.8)

It follows from this inequality that

(5.9)

Excluding nonpositive term

, we deduce from (5.9) that

(5.10)

Equation (5.10) is a quadratic inequality relative to ∥rot u∥_Q. Solving it we come to the estimate

(5.11)

which is equivalent to the following estimate for the difference rot u₁ − rot u₂:

(5.12)

The estimate (5.12) under Q = Ω has the sense of the stability estimate in the norm L²(Ω) of the vorticity of the component of the solution to problem (5.1) relative to small perturbations of functions ζ_d ∈ L²(Ω) and in the norms of L²(Ω) and H^1/2(Γ), respectively. In particular case where and g₁ = g₂ it follows from (5.11) that rot u₁ = rot u₂ 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) that

(5.13)

From (5.13) and (3.48)–(3.50) we come to the estimates (4.35)–(4.38) where

(5.14)

Thus the following theorem is proved.

Theorem 5.1. Let, under conditions (i), (ii), (iii) for 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 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 problem

(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 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 in the expression for minimized functional in (4.1) or the terms , 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).

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Stability of Optimal Controls for the Stationary Boussinesq Equations

Abstract

1. Introduction

2. Statement of Boundary Problem

3. Statement of Control Problems

4. Control Problems for Velocity Tracking-Type Cost Functionals

5. Control Problem for Vorticity Tracking-Type Cost Functional

6. Conclusion

Acknowledgments

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Stability of Optimal Controls for the Stationary Boussinesq Equations

Abstract

1. Introduction

2. Statement of Boundary Problem

3. Statement of Control Problems

4. Control Problems for Velocity Tracking-Type Cost Functionals

5. Control Problem for Vorticity Tracking-Type Cost Functional

6. Conclusion

Acknowledgments

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