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.

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., [

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

It should be mentioned at this point that a lot of studies have been conducted towards mathematical models of nonlinear-viscous fluids (see monograph [

Also, we would mention that there are many mathematical results concerning optimal control problems for the classical Navier-Stokes equations (see [

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.

Let

From here on, the following notations will be used.

We use the standard notations

By definition, put

It follows from Korn’s inequality (see [

Suppose the following:

the function

for any

the set

the functional

Let us consider the following cost functionals:

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., [

Now we introduce the concept of admissible triplets of (

Let

One says that a triplet

Equation (

On the other hand, it is not difficult to prove that if an admissible triplet

Let

A triplet

Our main result provides existence of solutions to (

If conditions (i), (ii), (iii), and (iv) hold, then optimization problem (

The proof of Theorem

Let

where

Lemma

First we show that the set of admissible triplets is nonempty. Let us fix an element

For an arbitrary fixed number

Find a vector

First we prove some a priori estimates of solutions to problem (

Applying Lemma

Let

Note that estimate (

Using (

Now we multiply (

Using Krasnoselskii’s theorem [

We will show that

Now we will show that the set

By definition, we have

Since

Using the equality

Applying the generalized Weierstrass theorem (see [

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

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