Existence of Optimal Control for a Nonlinear-Viscous Fluid Model

Abstract

We consider the optimal control problem for a mathematical model describing steady flows of a nonlinear-viscous incompressible fluid in a bounded three-dimensional (or a two-dimensional) domain with impermeable solid walls. The control parameter is the surface force at a given part of the flow domain boundary. For a given bounded set of admissible controls, we construct generalized (weak) solutions that minimize a given cost functional.

1. Introduction

The control and optimization problems in hydrodynamics have been the focus of attention of the control theory specialists for a long time. Flow boundary control problems have attracted increasing interest in recent years (see, e.g., [1–7]). Such problems are of interest from a theoretical perspective and are beneficial to applications as boundary control is easy to implement in practice.

In this paper, we study the optimal boundary control problem for a mathematical model describing steady flows of a nonlinear-viscous incompressible fluid in a bounded domain of space , , with impermeable solid walls. A distinguishing feature of the problem under consideration is that the surface force at the flow domain boundary is used as a control parameter instead of the nonhomogeneous Dirichlet boundary condition for the velocity field. Such an approach makes it possible to consider the case of flow control in a domain with impermeable solid walls without using external body forces as control parameters.

It should be mentioned at this point that a lot of studies have been conducted towards mathematical models of nonlinear-viscous fluids (see monograph [8] and [9–12]). Nevertheless, there are very few results on the existence and properties of solutions of control problems for nonlinear-viscous fluid flows. To the best of our knowledge, some results have only been obtained for the two-dimensional case (see [13, 14]).

Also, we would mention that there are many mathematical results concerning optimal control problems for the classical Navier-Stokes equations (see [15–17] and the references therein).

The aim of this paper is to prove the solvability of the optimal control problem, which is discussed above. More precisely, for a given bounded set of admissible boundary controls, we will construct generalized (weak) solutions that minimize a given lower weakly semicontinuous cost functional.

2. Problem Formulation and Main Result

Let be a bounded domain in ( or 3) with boundary . Consider the following optimal boundary control problem:where is the velocity field, is the pressure function, is the extra-stress tensor, is the body force, the symbol denotes the gradient with respect to the spatial variables , the divergence is the vector with coordinates is the rate of deformation tensor, is the second invariant of , is a given function, is the unit vector of the outer normal to , is the control, is the scalar product of the vectors and in space , the symbol denotes the tangential component of a vector, that is, is a part of from which the control is realized, is the set of admissible controls, and is a given cost functional.

From here on, the following notations will be used. denotes the space of symmetric -matrices with the norm

We use the standard notations and for the Lebesgue and Sobolev spaces of vector functions defined on with values in a finite-dimensional space (for details, see [18]). The scalar product in the space is denoted by .

By definition, put Moreover, we introduce the spacewith the following norm: In the right-hand side of (15), the restriction of a vector function to is defined by the formula where is the trace operator.

It follows from Korn’s inequality (see [8]) that the norm is equivalent to the norm induced from . Furthermore, we have the following estimates:where and are positive constants.

Suppose the following:(i)the function is measurable and there exist constants and such that (ii)for any , we have(iii)the set is bounded and sequentially weakly closed in ,(iv)the functional is lower weakly semicontinuous; that is, for any sequence such that weakly in , weakly in , and weakly in , we have

Example 1. Let us consider the following cost functionals:where is a favorable velocity field; is an unfavorable velocity field, that is, a velocity field whose appearance is undesirable; is a favorable extra-stress tensor; is a favorable surface force at ; and , , and are positive cost parameters. It is obvious that condition (iv) holds for the functionals , .

Remark 2. We do not assume that the set of admissible controls is convex. As is known, the convexity condition is widely used in studying of optimal control problems (see, e.g., [17]). However, this condition does not always hold in applications. Obviously, condition (iii) is weaker than the convexity condition. For example, (iii) is satisfied if the set can be represented as the union of finite number of convex closed sets in the space .

Now we introduce the concept of admissible triplets of (1)–(8) by analogy with the definition of generalized (weak) solutions to hydrodynamic models with slip boundary conditions (see, e.g., [8, 19, 20]).

Let .

Definition 3. One says that a triplet is an admissible triplet of control system (1)–(8) if the equalityholds for any and if conditions (3) and (7) hold.

Remark 4. Equation (23) appears for the following reasons. Let us assume that is a classical solution of (1)–(7). We take the -scalar product of (1) with . By integrating by parts, we obtain Combining this with (3) and (6), we get (23).
On the other hand, it is not difficult to prove that if an admissible triplet is sufficiently smooth, then there exists a function such that is a classical solution to (1)–(7).

Let be the set of admissible triplets to problem (1)–(8).

Definition 5. A triplet is called a solution of optimization problem (1)–(8) if the equality holds.

Our main result provides existence of solutions to (1)–(8).

Theorem 6. If conditions (i), (ii), (iii), and (iv) hold, then optimization problem (1)–(8) has at least one solution.

3. Proof of Theorem 6

The proof of Theorem 6 is based on the Galerkin method and monotonicity methods [21], as well as the following lemma.

Lemma 7. Let be a closed ball. Suppose the continuous mapping satisfies the following conditions: (a) for any ,(b) for any , where is an isomorphism;then for any the equation has at least one solution .

Lemma 7 can be proved by methods of topological degree theory (see, e.g., [22]).

Proof of Theorem 6. First we show that the set of admissible triplets is nonempty. Let us fix an element . Suppose is an orthonormal basis of the space .
For an arbitrary fixed number , we consider the following auxiliary problem.
Find a vector such thatwhere is a parameter, .
First we prove some a priori estimates of solutions to problem (26) and (27). Let be a solution of system (26) and (27) with a fixed parameter . We multiply (26) by and add the corresponding equalities for . Taking into account the equality we obtainUsing (18) and (19), from (29) we obtain the estimate This yields thatApplying Lemma 7 to system (26) and (27), we see that problem (26) and (27) is solvable for any and .
Let be a sequence of vector functions that satisfy (26) and (27) with . It is clear thatNote that estimate (31) is independent of . This shows the existence of a vector function and a subsequence such that weakly in . For the sake of simplicity, we assume thatMoreover, by the Sobolev embedding theorems, we haveUsing (34), we get Therefore we can pass to the limit in equality (32) and obtainfor any . Since is a basis of the space , it follows that equality (36) remains valid if we replace by an arbitrary vector function :Now we multiply (32) by and add the corresponding equalities for . The result is Hence we find in the limitTaking into account (20), (33), (37), and (39), we obtain the estimate for any number . Multiplying the obtained inequality by , we getfor any and .
Using Krasnoselskii’s theorem [22] on continuity of Nemytskii operators, we can pass to the limit in (41):Since is a basis of the space , it follows that inequality (42) remains valid if we replace by an arbitrary vector function . Furthermore, since is an arbitrary vector function from the space , we have This implies that the triplet is an admissible triplet of problem (1)–(7) and thus .
We will show that is bounded in the space . Suppose is an arbitrary triplet from and . It follows from (23) that This yields thatMoreover, taking into account (19), we obtainRecall that the set is bounded in . Therefore from estimates (45) and (46) it follows that the set is bounded in the space .
Now we will show that the set is sequentially weakly closed. Take a sequence such that weakly in , weakly in , and weakly in as . Let us check that .
By definition, we havefor any . Arguing as above, we conclude that From condition (iii), we get . Thus, it remains to show thatSince weakly in , we see thatNote also thatfor any .
Using the equality , we rewrite (47) as follows: Passing to the limit in this equality, we obtainSubstituting for in (53), we getFurther, substituting for in (47), we find Combining this with (54), we obtainBy [21, Chapter III, Lemma ] and (50), (51), and (56), we get (49).
Applying the generalized Weierstrass theorem (see [23]), we conclude that there exists an element such that This proves Theorem 6.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

The work of the first author was partially supported by Grant 16-31-00182 of the Russian Foundation of Basic Research.