International Journal of Differential Equations

Volume 2011, Article ID 535736, 28 pages

http://dx.doi.org/10.1155/2011/535736

## 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 [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 Here is a bounded domain in the space , with a boundary consisting of two parts and ; , , and denote the velocity and temperature fields, respectively; , where is the pressure and is the density of the medium; is the kinematic viscosity coefficient, is the gravitational acceleration vector, is the volumetric thermal expansion coefficient, is the thermal conductivity coefficient, is a given vector-function on , is a given function on a part of , is a function given on another part of , 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) is a bounded domain in , , with Lipschitz boundary , consisting of coupled components , ; and meas .

Below we shall use the Sobolev spaces and , where , or and for the vector functions where denotes , its subset , or a part of the boundary . In particularly we need the function spaces , , , , and their subspaces

The inner products and norms in , , or are denoted by , , , , or , . The inner products, norms and seminorms in and are denoted by , , and or , and if . The norms in or are denoted by or ; the norm in the dual space is denoted by . Set . Let in addition to condition (i) the following conditions hold:(ii), , .

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

Lemma 2.1. *Under conditions (i) there exist constants , , , , and such that
**
Bilinear form satisfies the inf-sup condition
**
Besides the following identities take place:
*

Let , , , in addition to (i), (ii). We multiply the equations in (2.1), (2.2) by test functions and and integrate the results over 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

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 , , , a weak solution that satisfies the estimates
**
Here , and are nondecreasing continuous functions of the norms , , , , , , . If, additionally, are small in the sense that
**
where , , , and 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: Here is a subdomain of . The functionals , , and where functions (or ) and 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 , , and , which play the role of controls and the group of fixed data comprising the invariable functions **,** and . As to the function 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 in the norm of .

Let , . Denote by a weakly lower semicontinuous functional. We assume that the controls , , and vary in some sets , , . Setting , , , we introduce the functional by the formula Here 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 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), , are nonempty closed convex sets;(iv), or , and is a bounded set, .

Considering the functional 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: Here is the operator acting by formulas The mathematical statement of the optimal control problem is as follows: to seek a pair , where and such that

Let and be the duals of the spaces and . Let denotes the Fréchet derivative of with respect to at the point . By we denote the adjoint operator of which is determined by the relation According to the general theory of extremum problems (see [21]) we introduce an element which is referred to as the adjoint state and define the Lagrangian , where , by Here and below , and is an auxiliary dimensional parameter. Its dimension is chosen so that dimensions of at the adjoint state coincide with those at the basic state, that is, Here denote the SI dimensions of the length, time, mass, and temperature units expressed in meters, seconds, kilograms, and degrees Kelvin, respectively. As a result , and 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 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 , .*

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 be continuously differentiable with respect to at the point . Then there exists a nonzero Lagrange multiplier such that the Euler-Lagrange equation
**
for the adjoint state is satisfied and the minimum principle holds which is equivalent to the inequality
*

Theorem 3.3. *Let the assumptions of Theorem 3.2 be satisfied and condition (2.17) holds for all . Then any nontrivial Lagrange multiplier satisfying (3.9) is regular, that is, has the form and is uniquely determined.*

We note that the functional and Lagrangian given by (3.7) are continuously differentiable functions of controls and its derivatives with respect to , and are given by Here for example is the Gateaux derivative with respect to at the point . Since are convex sets, at the minimum point of the functional the following conditions are satisfied (see [22]):

We also note that the Euler-Lagrange equation (3.9) is equivalent to identities 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 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 entering into (2.3) can vary in a certain set . Let be an arbitrary solution to problem (3.5) for a given function . By we denote a solution to problem It is obtained by replacing the functional in (3.5) by a close functional depending on and by replacing a function by a close function .

By Theorem 3.1 the following estimates hold for triples : Here where , , and are introduced in Theorem 3.1. We introduce “model” Reynolds number , Raley number , and Prandtl number by 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 , , and (where is an arbitrary scalar) are defined as Here is a dimensional factor of dimension whose value is equal to 1.

Assume that the following condition takes place:

Let us denote by , where , , Lagrange multipliers corresponding to solutions . By Theorems 3.2 and 3.3 and (3.12) they satisfy relations We renamed , in (3.20). Set , , , , , , and Let us subtract (2.13)–(2.15), written for from (2.13)–(2.15) for , , , , . We obtain We set , , in the inequality (3.23) under and , , in the same inequality under and add. We obtain

Subtract the identities (3.20)–(3.22), written for from the corresponding identities for ,, set , and add. Using (3.27) we obtain Set further in (3.25), in (3.26), and subtract obtained relations from (3.29). Using inequality (3.28) and arguing as in [18], we obtain Thus we have proved the following result.

Theorem 3.4. *Let under conditions of Theorem 3.2 for functionals and and condition (3.19) quadruples and be solutions to problem (3.5) under and problem (3.14) under , respectively, , be corresponding Lagrange multipliers. Then the inequality (3.30) holds for differences , defined in (3.24), where , .*

Below we shall need the estimates of differences , , entering into (3.25)–(3.27) by differences , , , and . Denote by a vector such that in , , . Here is a constant depending on . The existence of follows from [20, page 24]. We present the difference as , where is a new unknown function. Set , in (3.25). Taking into account (2.9) we obtain Using estimates (2.5), (2.6), (2.7), and (3.15), we deduce from (3.31) that

It follows from (3.19) that Rewriting the inequality (3.32) by (3.33) as we obtain that Taking into account the relation , we come to the following estimate via and :

Denote by a function such that and the estimate holds with a certain constant , which does not depend on . Let us present the difference as , where is a new unknown function. Set , in (3.26). We obtain Using estimates (2.5)–(2.8) and (3.15) we deduce that or Taking into account the relation , we obtain from this estimate that

Using further the estimate (3.36) for , we deduce from (3.40) that From this inequality and (3.17), (3.19) we come to the following estimate:

Using (3.42), we deduce from (3.36) that Taking into account (3.17) we come to the following estimate for :

An analogous estimate holds and for the pressure difference . In order to establish this estimate we make use of inf-sup condition (2.10). By (2.10) for the function and any (small) number there exists a function , , such that where . Set in the identity for in (3.25) and make of this estimate and estimates (2.6), (2.7), (3.15). We shall have Dividing to , we deduce that Using (3.42) and (3.44), we come to the following final estimate for :

*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 is a unique control. This problem can be considered as particular case of the general control problem (3.5), for which the set consists of one element and the set consists of one element *. *For this case the conditions *, * take place, and the estimates (3.42)–(3.47) and inequality (3.30) take the form

#### 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 and the heat flux on the part of is a unique control; that is, we consider one-parametric control problem 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 and consists of one element.

Let be a solution to problem (4.1), that corresponds to given functions and , and let 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 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

Using identities (4.3), (4.4), (3.22) we estimate parameters , , and . Firstly we deduce estimates for norms and . To this end we set , in (4.3), (3.22). Taking into account (2.11), (2.12), and condition , which follows from (4.4), we obtain Using estimates (2.5)–(2.8) and (3.15) we have where

By virtue of (4.8)–(4.10) and (4.12), we deduce from (4.7) and (4.6) that Taking into account (4.14), we obtain from (4.15) that Using (3.33) we deduce successfully from (4.16), (4.14) that

Let us estimate further the norms and from (4.3). In order to estimate we make use of inf-sup condition (2.10). By (2.10) for a function and any small number there exists a function , , such that the inequality holds. Setting in (4.3) and using this estimate together with estimates (2.6), (3.15), (4.11), we have From this inequality we deduce by (4.17) that Taking into account (4.17), we come from (4.20) to the estimate

It remains to estimate . 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 As we obtain from this inequality that

Taking into account (2.6), (3.48), (3.49), and estimates (4.17) for , , we have It follows from (4.24) that Here constants and are given by

Let the data for problem (4.1) and parameters , be such that with a certain constant the following condition takes place: Under condition (4.27) we deduce from (4.25) that Taking into account (4.28) and the estimate which follows from (4.23), we come from (4.5) to the inequality It follows from this inequality that Excluding nonpositive term from the right-hand side of (4.30), we deduce from (4.30) that

Equation (4.31) is a quadratic inequality for . Solving it we come to the following estimate for : As , , , the estimate (4.32) is equivalent to the following estimate for the velocity difference : This estimate under has the sense of the stability estimate in of the component of the solution to problem (4.1) relative to small perturbations of functions and in the norms of and , respectively. In particular case where 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 in , if . This yields together with (4.30), (3.48), (3.50) that , , . The latter means the uniqueness of the solution to problem (4.1) when 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 ; that is,