Optimal Dirichlet Boundary Control for the Corotational Oldroyd Model

Abstract

In this article, we investigate an optimal control problem for the coupled system of partial differential equations describing the steady-state flow of a corotational-type Oldroyd fluid through a bounded 3D (or 2D) domain. The control function is included in Dirichlet boundary conditions for the velocity field; in other words, we consider a model of inflow–outflow control. The main result is a theorem that states sufficient conditions for the solvability of the corresponding optimization problem in the set of admissible weak solutions. Namely, we establish the existence of a weak solution that minimizes the cost functional under given constraints on controls and states.

1. Introduction

The Oldroyd model is one of the basic macroscopic models for viscoelastic fluid dynamics. This model was originally introduced by Oldroyd [1]. Following the work [2], we write the system of governing equations for the isothermal flow of an incompressible Oldroyd fluid with diffusive stress in the dimensionless form:

$$\left\{\begin{array}{c}{\displaystyle Re\left(\frac{\partial \overrightarrow{v}}{\partial t}+\mathrm{div}(\overrightarrow{v}\otimes \overrightarrow{v})\right)-(1-r)\Delta \overrightarrow{v}+\nabla \pi -\mathrm{div}\mathbb{E}=\overrightarrow{f},}\hfill \\ \nabla ·\overrightarrow{v}=0,\hfill \\ {\displaystyle We\frac{{\mathcal{D}}_a\mathbb{E}}{\mathcal{D}t}+\mathbb{E}-\zeta \Delta \mathbb{E}=2r\mathbb{D}\left(\overrightarrow{v}\right),}\hfill \end{array}\right.$$

(1)

where

• $Re$ denotes the Reynolds number, $Re\ge 0$ (this number characterizes the inertial effects in the fluid flow);
• $We$ denotes the Weissenberg number, $We\ge 0$ (this number characterizes elasticity properties of the fluid);
• $\overrightarrow{v}$ is the velocity vector;
• the symbol ⊗ denotes the tensor product of vectors, $\overrightarrow{v}\otimes \overrightarrow{w}\stackrel{\mathrm{def}}{=}{\left(v_i{w}_j\right)}_{i,j=1}^d$ for any vectors $\overrightarrow{v},\overrightarrow{w}\in {\mathbb{R}}^d$, $d=3$ (or 2);
• the operator ∇ denotes the gradient with respect to the spatial variables $x_1,\dots ,x_d$, that is, $\nabla \stackrel{\mathrm{def}}{=}\left(\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_d}\right)$;
• the operator $\Delta$ denotes the Laplacian with respect to the spatial variables $x_1,\dots ,x_d$, that is, $\Delta \stackrel{\mathrm{def}}{=}\nabla ·\nabla ={\displaystyle \sum _{i=1}^d}\frac{{\partial }^2}{\partial x_i^2}$;
• the operator div is defined by the formula

  $$\mathrm{div}\mathbb{S}\stackrel{\mathrm{def}}{=}\left({\displaystyle \sum _{i=1}^d}\frac{\partial {\mathrm{S}}_{i1}}{\partial x_i},\dots ,{\displaystyle \sum _{i=1}^d}\frac{\partial {\mathrm{S}}_{id}}{\partial x_i}\right),$$

  for any matrix-valued function $\mathbb{S}={\left({\mathrm{S}}_{ij}\right)}_{i,j=1}^d$;
• r is an interpolation parameter, $0\le r\le 1$ (in the literature, the case $0<r<1$ is often referred to as the Jeffreys model, whereas the case $r=1$ is referred to as the Maxwell model);
• $\pi$ is the pressure;
• $\mathbb{E}$ is the non-Newtonian (elastic) part of the stress tensor;
• $\zeta$ is the diffusive parameter of the elastic stress, $\zeta \ge 0$;
• $\mathbb{D}\left(\overrightarrow{v}\right)$ is the rate-of-strain tensor, $\mathbb{D}\left(\overrightarrow{v}\right)=(\nabla \overrightarrow{v}+{(\nabla \overrightarrow{v})}^{\top })/2;$
• $\mathbb{W}\left(\overrightarrow{v}\right)$ is the vorticity tensor, $\mathbb{W}\left(\overrightarrow{v}\right)=(\nabla \overrightarrow{v}-{(\nabla \overrightarrow{v})}^{\top })/2;$
• $\overrightarrow{f}$ is the external force applied to the fluid;
• ${\mathcal{D}}_a/\mathcal{D}t$ denotes Oldroyd’s objective derivative defined by the formula

  $$\frac{{\mathcal{D}}_a\mathbb{S}}{\mathcal{D}t}\stackrel{\mathrm{def}}{=}\frac{\partial \mathbb{S}}{\partial t}+(\overrightarrow{v}·\nabla )\mathbb{S}+\mathbb{S}\mathbb{W}\left(\overrightarrow{v}\right)-\mathbb{W}\left(\overrightarrow{v}\right)\mathbb{S}-a(\mathbb{S}\mathbb{D}\left(\overrightarrow{v}\right)+\mathbb{D}\left(\overrightarrow{v}\right)\mathbb{S}),$$

  in which the parameter a interpolates between the upper-convected model ($a=1$) and the lower-convected model ($a=-1$), while the median case $a=0$ corresponds to the corotational model.

As can be seen from (1), Oldroyd’s model describes the behaviour of a viscoelastic fluid, using an approach that is based on a consideration of the extra stress interpolating between the purely viscous contribution ($r=0$) and the purely elastic contribution ($r=1$).

The Oldroyd model has been carefully scrutinized by mathematicians. We refer readers to Saut [3] and Renardy & Thomases [4] for the excellent reviews on mathematical issues (such as the existence and uniqueness of solutions to initial value problems and steady-state flows, flow stability, numerical simulations, etc.) and some open problems for this model and its modifications.

In this paper, we deal with an optimal boundary control problem for the stationary version of system (1) with the corotational (Jaumann) derivative ${\mathcal{D}}_0/\mathcal{D}t$. It is assumed that the flow region is a bounded domain $\Omega \subset {\mathbb{R}}^d$, $d=2,3$, with locally Lipschitz boundary $\Gamma$. The control parameter is included in Dirichlet boundary conditions for the velocity field $\overrightarrow{v}$ on $\Gamma$. Main our aim is to obtain sufficient conditions for the solvability of the optimization problem in the class of $H^1$-weak solutions.

The present work continues the investigation by Artemov [5], in which a flow control problem is considered for a simplified version of system (1) with the material derivative $d/dt$ instead of the objective derivative ${\mathcal{D}}_a/\mathcal{D}t$ under the assumption that $\zeta =0$. Note that Doubova & Fernandez-Cara [6] obtained the approximate controllability results for the linearized motion equations (in which all nonlinear terms are neglected) of the Oldroyd fluid in a bounded flow domain with boundary of class $C^2$. Later, Artemov [7] established the unique solvability of the optimal starting control problem for the linear partial differential equations describing the unsteady creeping flow of a viscoelastic fluid of the Oldroyd kind. We also mention here that there are many mathematical results related to various control problems for the classical Euler equations (inviscid fluid) and the Navier–Stokes equations (Newtonian fluid) [8,9,10,11,12,13] as well as for magnetohydrodynamic equations [14,15,16,17,18]. In general, control and optimization problems for hydrodynamic models demand careful investigation because these are not merely of academic interest but have important applications in engineering sciences and industry, for example, in the formation of fluid flows with required properties through channels and pipeline networks.

Our article is organized as follows. In Section 2, we introduce necessary notations and functional spaces as well as give some results on the solvability of a class of finite-dimensional operator equations with a numerical parameter. Section 3 is devoted mainly to studying the solvability and energy estimates of solutions to the Dirichlet–Neumann problem for the corotational Oldroyd model in the weak formulation (see Theorem 1). In Section 4, we formulate an optimal boundary control problem for Oldroyd fluid flows and prove the main result of the work—Theorem 2 about the existence of weak solutions that minimize the cost functional under given constraints on controls and states.

2. Preliminaries

2.1. Notations and Function Spaces

For $\overrightarrow{x},\overrightarrow{y}\in {\mathbb{R}}^d$, $\mathbb{A},\mathbb{B}\in {\mathbb{R}}^{d\times d}$, $\widehat{\Phi },\widehat{\Psi }\in {\mathbb{R}}^{d\times d\times d}$, by $\overrightarrow{x}·\overrightarrow{y}$, $\mathbb{A}:\mathbb{B}$, $\widehat{\Phi }⋮\widehat{\Psi }$ denote the scalar products in the spaces ${\mathbb{R}}^d$, ${\mathbb{R}}^{d\times d}$, ${\mathbb{R}}^{d\times d\times d}$, respectively, that is,

$$\overrightarrow{x}·\overrightarrow{y}\stackrel{\mathrm{def}}{=}{\displaystyle \sum _{i=1}^d}{x}_i{y}_i,\mathbb{A}:\mathbb{B}\stackrel{\mathrm{def}}{=}{\displaystyle \sum _{i,j=1}^d}{A}_{ij}{B}_{ij},\widehat{\Phi }⋮\widehat{\Psi }\stackrel{\mathrm{def}}{=}{\displaystyle \sum _{i,j,k=1}^d}{\widehat{\Phi }}_{ijk}{\widehat{\Psi }}_{ijk}.$$

By ${\mathbb{M}}_{\mathrm{sym}}^{d\times d}$ we denote the set of all symmetric $d\times d$-matrices.

Suppose $\Omega$ is a bounded locally Lipschitz domain in the space ${\mathbb{R}}^d$, where $d=2,3$, with boundary $\Gamma$, and the symbol X denotes one of the finite-dimensional spaces $\mathbb{R},$${\mathbb{R}}^d$, ${\mathbb{R}}^{d\times d}$, ${\mathbb{R}}^{d\times d\times d}$, and ${\mathbb{M}}_{\mathrm{sym}}^{d\times d}$.

Let $p\ge 1$ and $k\in \mathbb{N}$. We use the notation $L^p(\Omega ,X)$ ($H^k(\Omega ,X)$, resp.) for Lebesgue (Sobolev, resp.) space of functions defined on $\Omega$ with values in X. Definitions of these spaces and description of their properties can be found in [19,20,21].

By $C_0^{\infty }(\Omega ,X)$ denote the space of $C^{\infty }$-smooth functions $\omega :\Omega \to X$ such that

$$\mathrm{supp}\left(\omega \right)\stackrel{\mathrm{def}}{=}\mathrm{the}\mathrm{closure}\mathrm{of}\mathrm{the}\mathrm{set}\{\overrightarrow{x}\in \Omega :\omega \left(\overrightarrow{x}\right)\ne 0\}\subset \Omega .$$

Let

$$\begin{array}{cc}\hfill C_{0,\sigma }^{\infty }(\Omega ,{\mathbb{R}}^d)& \stackrel{\mathrm{def}}{=}\{\overrightarrow{v}\in C_0^{\infty }(\Omega ,{\mathbb{R}}^d):\nabla ·\overrightarrow{v}=0in\Omega \},\hfill \\ \hfill H_{\sigma }^k(\Omega ,{\mathbb{R}}^d)& \stackrel{\mathrm{def}}{=}\{\overrightarrow{v}\in H^k(\Omega ,{\mathbb{R}}^d):\nabla ·\overrightarrow{v}=0in\Omega \},\hfill \\ \hfill H_0^k(\Omega ,{\mathbb{R}}^d)& \stackrel{\mathrm{def}}{=}\mathrm{the}\mathrm{closure}\mathrm{of}\mathrm{the}\mathrm{set}{C}_0^{\infty }(\Omega ,{\mathbb{R}}^d)\mathrm{in}\mathrm{the}\mathrm{Sobolev}\mathrm{space}{H}^k(\Omega ,{\mathbb{R}}^d),\hfill \\ \hfill H_{0,\sigma }^k(\Omega ,{\mathbb{R}}^d)& \stackrel{\mathrm{def}}{=}\mathrm{the}\mathrm{closure}\mathrm{of}\mathrm{the}\mathrm{set}{C}_{0,\sigma }^{\infty }(\Omega ,{\mathbb{R}}^d)\mathrm{in}\mathrm{the}\mathrm{Sobolev}\mathrm{space}{H}^k(\Omega ,{\mathbb{R}}^d).\hfill \end{array}$$

Let us introduce the scalar products and the associated norms in the spaces $H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)$ and $H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)$ as follows:

$$\begin{array}{cc}\hfill {(\overrightarrow{v},\overrightarrow{w})}_{H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)}& \stackrel{\mathrm{def}}{=}{(\nabla \overrightarrow{v},\nabla \overrightarrow{w})}_{L^2(\Omega ,{\mathbb{R}}^{d\times d})}+{(\overrightarrow{v},\overrightarrow{w})}_{L^2(\Gamma ,{\mathbb{R}}^d)},\hfill \\ \hfill \parallel \overrightarrow{v}{\parallel }_{H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)}& \stackrel{\mathrm{def}}{=}{(\overrightarrow{v},\overrightarrow{v})}_{H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)}^{1/2},\hfill \\ \hfill {(\overrightarrow{v},\overrightarrow{w})}_{H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)}& \stackrel{\mathrm{def}}{=}{(\nabla \overrightarrow{v},\nabla \overrightarrow{w})}_{L^2(\Omega ,{\mathbb{R}}^{d\times d})},\hfill \\ \hfill \parallel \overrightarrow{v}{\parallel }_{H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)}& \stackrel{\mathrm{def}}{=}{(\overrightarrow{v},\overrightarrow{v})}_{H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)}^{1/2}.\hfill \end{array}$$

Here the restriction of a vector function $\overrightarrow{v}$ to $\Gamma$ is defined by the formula

$$\overrightarrow{v}{|}_{\Gamma }\stackrel{\mathrm{def}}{=}{\gamma }_{\Gamma }\overrightarrow{v},$$

where ${\gamma }_{\Gamma }:H^1(\Omega ,{\mathbb{R}}^d)\to H^{1/2}(\Gamma ,{\mathbb{R}}^d)$ is the trace operator (see [22], Section 2.5).

By $L_n^2(\Gamma ,{\mathbb{R}}^d)$ ($H_n^{1/2}(\Gamma ,{\mathbb{R}}^d)$, resp.) denote the space of vector functions $\overrightarrow{\psi }\in L^2(\Gamma ,{\mathbb{R}}^d)$ ($\overrightarrow{\psi }\in H^{1/2}(\Gamma ,{\mathbb{R}}^d)$, resp.) that satisfy the condition of zero total flux on $\Gamma$:

$${\int }_{\Gamma }\overrightarrow{\psi }·\overrightarrow{n}d\Gamma =0,$$

where $\overrightarrow{n}=\overrightarrow{n}\left(\overrightarrow{x}\right)$ is the unit exterior (with respect to $\Omega$) normal vector to $\Gamma$ at a point $\overrightarrow{x}$.

Lemma 1. 

The trace operator ${\gamma }_{\Gamma }:H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)\to L_n^2(\Gamma ,{\mathbb{R}}^d)$ is compact.

The proof of this lemma is given in the monograph [22], Section 2.6.

Let F be a Banach space. By $F^{\ast }$ denote the dual space of F. As usual, ${\langle ·,·\rangle }_{F^{\ast }\times F}$ denotes the duality bracket between spaces $F^{\ast }$ and F.

The weak (strong) convergence in Banach and Hilbert spaces is denoted by ⇀ (→).

2.2. Solvability Results for a Class of Finite-Dimensional Operator Equations

Lemma 2. 

Let $\mathfrak{B}$ be a bounded domain in ${\mathbb{R}}^N$ such that

• the zero vector $\overrightarrow{0}$ belongs to $\mathfrak{B};$
• the domain $\mathfrak{B}$ has the central symmetry property: if $\overrightarrow{h}\in \mathfrak{B}$, then $-\overrightarrow{h}\in \mathfrak{B}$.

Suppose that $\overrightarrow{\varphi }:\overline{\mathfrak{B}}\times [0,1]\to {\mathbb{R}}^N$ is a continuous mapping that satisfies the following conditions:

• $\overrightarrow{\varphi }(\overrightarrow{h},\lambda )\ne \overrightarrow{0}$, for all $(\overrightarrow{h},\lambda )\in \partial \mathfrak{B}\times [0,1];$
• $\overrightarrow{\varphi }(·,0):\overline{\mathfrak{B}}\to {\mathbb{R}}^N$ is an odd mapping, i. e., $\overrightarrow{\varphi }(-\overrightarrow{h},0)=-\overrightarrow{\varphi }(\overrightarrow{h},0)$, for all $\overrightarrow{h}\in \overline{\mathfrak{B}}$.

Then the operator equation $\overrightarrow{\varphi }(\overrightarrow{h},\lambda )=\overrightarrow{0}$ has at least one solution ${\overrightarrow{h}}_{\lambda }\in \mathfrak{B}$, for any $\lambda \in [0,1]$.

The proof of this statement is based on applying methods of Brouwer’s degree theory (for details, see [23], Section 5).

3. Dirichlet–Neumann Problem for the Corotational Oldroyd Model

Let us consider the Dirichlet–Neumann boundary value problem describing the steady-state flow of an incompressible Oldroyd fluid through the domain $\Omega$:

$$\left\{\begin{array}{cc}Re\mathrm{div}(\overrightarrow{v}\otimes \overrightarrow{v})-(1-r)\Delta \overrightarrow{v}+\nabla \pi -\mathrm{div}\mathbb{E}=\overrightarrow{f}\hfill & \mathrm{in}\Omega ,\hfill \\ \nabla ·\overrightarrow{v}=0\hfill & \mathrm{in}\Omega ,\hfill \\ {\displaystyle We\frac{{\mathcal{D}}_0\mathbb{E}}{\mathcal{D}t}+\mathbb{E}-\zeta \Delta \mathbb{E}=2r\mathbb{D}\left(\overrightarrow{v}\right)}\hfill & \mathrm{in}\Omega ,\hfill \\ \overrightarrow{v}=\overrightarrow{u}\hfill & \mathrm{on}\Gamma ,\hfill \\ {\displaystyle \frac{\partial \mathbb{E}}{\partial \overrightarrow{n}}=\mathbb{O}}\hfill & \mathrm{on}\Gamma ,\hfill \end{array}\right.$$

(2)

where

• $\overrightarrow{u}$ is a given vector-valued function;
• $\mathbb{O}$ is a given matrix-valued function;

For the sake of simplicity of further discussion, we will assume that $\mathbb{O}\left(\overrightarrow{x}\right)$ is equal to the zero matrix for all $\overrightarrow{x}\in \Gamma$. Moreover, assume that the following two inclusions hold:

$$\overrightarrow{f}\in {\left[H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\right]}^{\ast },\overrightarrow{u}\in H_n^{1/2}(\Gamma ,{\mathbb{R}}^d).$$

Definition 1 

(Weak solution). A pair $(\overrightarrow{v},\mathbb{E})$ is said to be a weak solution of boundary value problem (2) if

$$(\overrightarrow{v},\mathbb{E})\in H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)\times H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d}),{\gamma }_{\Gamma }\overrightarrow{v}=\overrightarrow{u}$$

and the equalities

$$\begin{array}{cc}\hfill & -Re{\displaystyle \sum _{i=1}^d}{\int }_{\Omega }{v}_i\overrightarrow{v}·\frac{\partial \overrightarrow{w}}{\partial x_i}d\overrightarrow{x}+(1-r){\int }_{\Omega }\nabla \overrightarrow{v}:\nabla \overrightarrow{w}d\overrightarrow{x}\hfill \\ \hfill & +{\int }_{\Omega }\mathbb{E}:\mathbb{D}\left(\overrightarrow{w}\right)d\overrightarrow{x}={\langle \overrightarrow{f},\overrightarrow{w}\rangle }_{{\left[H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\right]}^{\ast }\times H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & -We{\displaystyle \sum _{i=1}^d}{\int }_{\Omega }{v}_i\mathbb{E}:\frac{\partial \mathbb{F}}{\partial x_i}d\overrightarrow{x}+We{\int }_{\Omega }(\mathbb{E}\mathbb{W}\left(\overrightarrow{v}\right)-\mathbb{W}\left(\overrightarrow{v}\right)\mathbb{E}):\mathbb{F}d\overrightarrow{x}\hfill \\ \hfill & +{\int }_{\Omega }\mathbb{E}:\mathbb{F}d\overrightarrow{x}+\zeta {\int }_{\Omega }\nabla \mathbb{E}⋮\nabla \mathbb{F}d\overrightarrow{x}=2r{\int }_{\Omega }\mathbb{D}\left(\overrightarrow{v}\right):\mathbb{F}d\overrightarrow{x},\hfill \end{array}$$

(4)

hold for any $\overrightarrow{w}\in H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)$ and $\mathbb{F}\in H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})$.

Remark 1. 

If a triplet $(\overrightarrow{v},\mathbb{E},\pi )$ is a classical solution of problem (2), then the pair $(\overrightarrow{v},\mathbb{E})$ is a weak solution of this problem. On the other hand, having a weak solution in hand, we can obtain the pressure $\pi$ by using the De Rham theory (see, for example, [24], Chapter I, Section 1).

Theorem 1 

(Solvability and energy estimates). Suppose that

$$0\le r<1,\zeta >0,\overrightarrow{u}=\overrightarrow{0}.$$

Then boundary value problem (2) has at least one weak solution $({\overrightarrow{v}}_0,{\mathbb{E}}_0)$ satisfying the following energy estimates:

$$\parallel {\overrightarrow{v}}_0{\parallel }_{H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)}\le \frac{1}{1-r}{\parallel \overrightarrow{f}\parallel }_{{\left[H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\right]}^{\ast }},$$

(5)

$$\begin{array}{cc}\hfill & 2r(1-r)\parallel {\overrightarrow{v}}_0{\parallel }_{H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)}^2+{\parallel {\mathbb{E}}_0\parallel }_{L^2(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})}^2\hfill \\ \hfill & +\zeta \parallel \nabla {\mathbb{E}}_0{\parallel }_{L^2(\Omega ,{\mathbb{R}}^{d\times d\times d})}^2\le \frac{2r}{1-r}{\parallel \overrightarrow{f}\parallel }_{{\left[H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\right]}^{\ast }}^2.\hfill \end{array}$$

(6)

Proof. 

Using the Galerkin scheme and Lemma 2, we perform the proof in five steps.

Step 1: Galerkin’s approximation. Consider two sequences ${\left\{{\overrightarrow{w}}_j\right\}}_{j=1}^{\infty }$ and ${\left\{{\mathbb{F}}_j\right\}}_{j=1}^{\infty }$ such that

• ${\left\{{\overrightarrow{w}}_j\right\}}_{j=1}^{\infty }$ is an orthonormal basis in $H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)$;
• ${\left\{{\mathbb{F}}_j\right\}}_{j=1}^{\infty }$ is an orthonormal basis in $H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})$.

Let us fix $m\in \mathbb{N}$ and consider the auxiliary finite-dimensional problem:

Find a pair $\left({\overrightarrow{v}}_m,{\mathbb{E}}_m\right)$ such that

$$\begin{array}{cc}\hfill & -\lambda Re{\displaystyle \sum _{i=1}^d}{\int }_{\Omega }{v}_{mi}{\overrightarrow{v}}_m·\frac{\partial {\overrightarrow{w}}_j}{\partial x_i}d\overrightarrow{x}+(1-r){\int }_{\Omega }\nabla {\overrightarrow{v}}_m:\nabla {\overrightarrow{w}}_{j}d\overrightarrow{x}\hfill \\ \hfill & +\lambda {\int }_{\Omega }{\mathbb{E}}_m:\mathbb{D}\left({\overrightarrow{w}}_j\right)d\overrightarrow{x}=\lambda {\langle \overrightarrow{f},{\overrightarrow{w}}_j\rangle }_{{\left[H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\right]}^{\ast }\times H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)},j=1,\dots ,m,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & -\lambda We{\displaystyle \sum _{i=1}^d}{\int }_{\Omega }{v}_{mi}{\mathbb{E}}_m:\frac{\partial {\mathbb{F}}_j}{\partial x_i}d\overrightarrow{x}+\lambda We{\int }_{\Omega }({\mathbb{E}}_m\mathbb{W}\left({\overrightarrow{v}}_m\right)-\mathbb{W}\left({\overrightarrow{v}}_m\right){\mathbb{E}}_m):{\mathbb{F}}_{j}d\overrightarrow{x}\hfill \\ \hfill & +{\int }_{\Omega }{\mathbb{E}}_m:{\mathbb{F}}_{j}d\overrightarrow{x}+\zeta {\int }_{\Omega }\nabla {\mathbb{E}}_m⋮\nabla {\mathbb{F}}_{j}d\overrightarrow{x}=2\lambda r{\int }_{\Omega }\mathbb{D}\left({\overrightarrow{v}}_m\right):{\mathbb{F}}_{j}d\overrightarrow{x},j=1,\dots ,m,\hfill \end{array}$$

(8)

$${\overrightarrow{v}}_m={\displaystyle \sum _{j=1}^m}{a}_{mj}{\overrightarrow{w}}_j,{\mathbb{E}}_m={\displaystyle \sum _{j=1}^m}{b}_{mj}{\mathbb{F}}_j,$$

(9)

where

• $a_{m1},\dots ,a_{mm}$, $b_{m1},\dots ,b_{mm}$ are unknown real numbers;
• $\lambda$ is the homotopy parameter, $\lambda \in [0,1]$.

Step 2: A priori estimates for Galerkin’s solutions. Suppose a pair $({\overrightarrow{v}}_m,{\mathbb{E}}_m)$ satisfies relations (7)–(9). We multiply (7) by $a_{mj}$ and sum up the corresponding equalities over j from 1 to m. Since

$$\begin{array}{cc}\hfill {\displaystyle \sum _{i=1}^d}{\int }_{\Omega }{v}_{mi}{\overrightarrow{v}}_m·\frac{\partial {\overrightarrow{v}}_m}{\partial x_i}d\overrightarrow{x}=& \frac{1}{2}{\displaystyle \sum _{i=1}^d}{\int }_{\Omega }{v}_{mi}\frac{\partial |{\overrightarrow{v}}_m{|}^2}{\partial x_i}d\overrightarrow{x}\hfill \\ \hfill =& \frac{1}{2}{\int }_{\Gamma }\underset{=0}{\underbrace{({\overrightarrow{v}}_m·\overrightarrow{n}){\left|{\overrightarrow{v}}_m\right|}^2}}d\overrightarrow{x}-\frac{1}{2}{\int }_{\Omega }\underset{=0}{\underbrace{(\nabla ·{\overrightarrow{v}}_m)}}{\left|{\overrightarrow{v}}_m\right|}^{2}d\overrightarrow{x}\hfill \\ \hfill =& 0,\hfill \end{array}$$

we arrive at the equality

$$(1-r)\parallel {\overrightarrow{v}}_m{\parallel }_{H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)}^2+\lambda {\int }_{\Omega }{\mathbb{E}}_m:\mathbb{D}\left({\overrightarrow{v}}_m\right)d\overrightarrow{x}=\lambda {\langle \overrightarrow{f},{\overrightarrow{v}}_m\rangle }_{{\left[H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\right]}^{\ast }\times H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)}.$$

(10)

Next, we multiply (8) by $b_{mj}$ and sum up the corresponding equalities over j from 1 to m. Taking into account the following relationships

$$\begin{array}{cc}\hfill {\displaystyle \sum _{i=1}^d}{\int }_{\Omega }{v}_{mi}{\mathbb{E}}_m:\frac{\partial {\mathbb{E}}_m}{\partial x_i}d\overrightarrow{x}& =\frac{1}{2}{\displaystyle \sum _{i=1}^d}{\int }_{\Omega }{v}_{mi}\frac{\partial |{\mathbb{E}}_m{|}^2}{\partial x_i}d\overrightarrow{x}\hfill \\ \hfill & =\frac{1}{2}{\int }_{\Gamma }\underset{=0}{\underbrace{({\overrightarrow{v}}_m·\overrightarrow{n}){\left|{\mathbb{E}}_m\right|}^2}}d\overrightarrow{x}-\frac{1}{2}{\int }_{\Omega }\underset{=0}{\underbrace{(\nabla ·{\overrightarrow{v}}_m)}}{\left|{\mathbb{E}}_m\right|}^{2}d\overrightarrow{x}\hfill \\ \hfill & =0,\hfill \end{array}$$

$$({\mathbb{E}}_m\mathbb{W}\left({\overrightarrow{v}}_m\right)-\mathbb{W}\left({\overrightarrow{v}}_m\right){\mathbb{E}}_m):{\mathbb{E}}_m=0,$$

we obtain

$$\parallel {\mathbb{E}}_m{\parallel }_{L^2(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})}^2+\zeta {\parallel \nabla {\mathbb{E}}_m\parallel }_{L^2(\Omega ,{\mathbb{R}}^{d\times d\times d})}^2=2\lambda r{\int }_{\Omega }\mathbb{D}\left({\overrightarrow{v}}_m\right):{\mathbb{E}}_{m}d\overrightarrow{x}.$$

(11)

Let us multiply (10) by $2r$ and sum with (11). This yields that

$$\begin{array}{cc}\hfill & 2r(1-r)\parallel {\overrightarrow{v}}_m{\parallel }_{H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)}^2+{\parallel {\mathbb{E}}_m\parallel }_{L^2(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})}^2\hfill \\ \hfill & +\zeta \parallel \nabla {\mathbb{E}}_m{\parallel }_{L^2(\Omega ,{\mathbb{R}}^{d\times d\times d})}^2=2r\lambda {\langle \overrightarrow{f},{\overrightarrow{v}}_m\rangle }_{{\left[H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\right]}^{\ast }\times H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)},\hfill \end{array}$$

whence

$$\begin{array}{cc}\hfill & 2r(1-r)\parallel {\overrightarrow{v}}_m{\parallel }_{H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)}^2+{\parallel {\mathbb{E}}_m\parallel }_{L^2(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})}^2\hfill \\ \hfill & +\zeta \parallel \nabla {\mathbb{E}}_m{\parallel }_{L^2(\Omega ,{\mathbb{R}}^{d\times d\times d})}^2\le 2r\parallel \overrightarrow{f}{\parallel }_{{\left[H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\right]}^{\ast }}{\parallel {\overrightarrow{v}}_m\parallel }_{H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)}.\hfill \end{array}$$

(12)

Multiplying both sides of the last inequality by $1/(2r(1-r)\parallel {\overrightarrow{v}}_m{\parallel }_{H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)})$, we arrive at

$$\parallel {\overrightarrow{v}}_m{\parallel }_{H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)}\le \frac{1}{1-r}{\parallel \overrightarrow{f}\parallel }_{{\left[H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\right]}^{\ast }}.$$

(13)

Finally, it follows from (12) and (13) that

$$\begin{array}{cc}\hfill & 2r(1-r)\parallel {\overrightarrow{v}}_m{\parallel }_{H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)}^2+{\parallel {\mathbb{E}}_m\parallel }_{L^2(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})}^2\hfill \\ \hfill & +\zeta \parallel \nabla {\mathbb{E}}_m{\parallel }_{L^2(\Omega ,{\mathbb{R}}^{d\times d\times d})}^2\le \frac{2r}{1-r}{\parallel \overrightarrow{f}\parallel }_{{\left[H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\right]}^{\ast }}^2.\hfill \end{array}$$

(14)

Step 3: Solvability of the Galerkin approximation. Note that the right-hand side of estimate (14) is independent of $\lambda$ and m. Applying Lemma 2, we establish the solvability of system (7)–(9) for any $\lambda \in [0,1]$ and $m\in \mathbb{N}$.

Step 4: Passage to the limit as $m\to \infty$. Fix $\lambda =1$. Let ${\left\{({\overrightarrow{v}}_m,{\mathbb{E}}_m)\right\}}_{m=1}^{\infty }$ be a sequence of solutions of system (7)–(9). It follows from (14) that the norms $\parallel {\overrightarrow{v}}_m{\parallel }_{H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)}$ and $\parallel {\mathbb{E}}_m{\parallel }_{H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})}$ are majorized by a positive constant that does not depend on m. Hence, without loss of generality it can be assumed that

$$\begin{array}{cc}\hfill & {\overrightarrow{v}}_m\rightharpoonup {\overrightarrow{v}}_0\mathrm{weakly}\mathrm{in}\mathrm{the}\mathrm{space}{H}_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\mathrm{as}m\to \infty ,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & {\mathbb{E}}_m\rightharpoonup {\mathbb{E}}_0\mathrm{weakly}\mathrm{in}\mathrm{the}\mathrm{space}{H}^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})\mathrm{as}m\to \infty .\hfill \end{array}$$

(16)

Since the following embeddings

$$H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\hookrightarrow L^4(\Omega ,{\mathbb{R}}^d),H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})\hookrightarrow L^4(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})$$

are compact (see [22], Section 2.6), we see that

$$\begin{array}{cc}\hfill & {\overrightarrow{v}}_m\to {\overrightarrow{v}}_0\mathrm{strongly}\mathrm{in}\mathrm{the}\mathrm{space}{L}^4(\Omega ,{\mathbb{R}}^d)\mathrm{as}m\to \infty ,\hfill \\ \hfill & {\mathbb{E}}_m\to {\mathbb{E}}_0\mathrm{strongly}\mathrm{in}\mathrm{the}\mathrm{space}{L}^4(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})\mathrm{as}m\to \infty .\hfill \end{array}$$

Therefore, one can pass to the limit as $m\to \infty$ in equalities (7) and (8) with $\lambda =1$; this gives

$$\begin{array}{c}\hfill -Re{\displaystyle \sum _{i=1}^d}{\int }_{\Omega }{v}_{0i}{\overrightarrow{v}}_0·\frac{\partial {\overrightarrow{w}}_j}{\partial x_i}d\overrightarrow{x}+(1-r){\int }_{\Omega }\nabla {\overrightarrow{v}}_0:\nabla {\overrightarrow{w}}_{j}d\overrightarrow{x}\\ \hfill +{\int }_{\Omega }{\mathbb{E}}_0:\mathbb{D}\left({\overrightarrow{w}}_j\right)d\overrightarrow{x}={\langle \overrightarrow{f},{\overrightarrow{w}}_j\rangle }_{{\left[H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\right]}^{\ast }\times H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)},\end{array}$$

(17)

$$\begin{array}{cc}\hfill & -We{\displaystyle \sum _{i=1}^d}{\int }_{\Omega }{v}_{0i}{\mathbb{E}}_0:\frac{\partial {\mathbb{F}}_j}{\partial x_i}d\overrightarrow{x}+We{\int }_{\Omega }({\mathbb{E}}_0\mathbb{W}\left({\overrightarrow{v}}_0\right)-\mathbb{W}\left({\overrightarrow{v}}_0\right){\mathbb{E}}_0):{\mathbb{F}}_{j}d\overrightarrow{x}\hfill \\ \hfill & +{\int }_{\Omega }{\mathbb{E}}_0:{\mathbb{F}}_{j}d\overrightarrow{x}+\zeta {\int }_{\Omega }\nabla {\mathbb{E}}_0⋮\nabla {\mathbb{F}}_{j}d\overrightarrow{x}=2r{\int }_{\Omega }\mathbb{D}\left({\overrightarrow{v}}_0\right):{\mathbb{F}}_{j}d\overrightarrow{x},\hfill \end{array}$$

(18)

for any $j\in \mathbb{N}$.

Since the system ${\left\{{\overrightarrow{w}}_j\right\}}_{j=1}^{\infty }$ is a basis in the subspace $H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)$ and the system ${\left\{{\mathbb{F}}_j\right\}}_{j=1}^{\infty }$ is a basis in the space $H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})$, relations (17) and (18) remain valid if ${\overrightarrow{w}}_j$ and ${\mathbb{F}}_j$ are replaced by arbitrary vector functions $\overrightarrow{w}\in H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)$ and $\mathbb{F}\in H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})$, respectively. Thus, the pair $({\overrightarrow{v}}_0,{\mathbb{E}}_0)$ is a weak solution to problem (2).

Step 5: Energy estimates. From (13)–(16) it follows that energy estimates (5) and (6) hold. This completes the proof of Theorem 1. □

Analyzing the above proof, one can arrive at the following proposition.

Proposition 1 

(Passage to the limit as $\zeta \to 0$). If ${\left\{({\overrightarrow{v}}_{{\zeta }_k},{\mathbb{E}}_{{\zeta }_k})\right\}}_{k=1}^{\infty }$ is a sequence such that $({\overrightarrow{v}}_{{\zeta }_k},{\mathbb{E}}_{{\zeta }_k})$ is a weak solution of problem (2) with $\zeta ={\zeta }_k>0$, $0\le r<1$, and $\overrightarrow{u}=\overrightarrow{0}$. Suppose

$$\begin{array}{cc}\hfill & {\overrightarrow{v}}_{{\zeta }_k}\rightharpoonup \overrightarrow{v}weakly in H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)ask\to \infty ,\hfill \\ \hfill & {\mathbb{E}}_{{\zeta }_k}\rightharpoonup \mathbb{E}weakly in L^2(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})ask\to \infty ,\hfill \\ \hfill & \underset{k\to \infty }{lim}{({\mathbb{E}}_{{\zeta }_k},\mathbb{D}\left({\overrightarrow{v}}_{{\zeta }_k}\right))}_{L^2(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})}={(\mathbb{E},\mathbb{D}\left(\overrightarrow{v}\right))}_{L^2(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})},\hfill \\ \hfill & \underset{k\to \infty }{lim}{\zeta }_k=0;\hfill \end{array}$$

then the pair $(\overrightarrow{v},\mathbb{E})$ is a weak solution of the Dirichlet boundary value problem for the stationary corotational Oldroyd model without diffusive stress:

$$\left\{\begin{array}{cc}Re\mathrm{div}(\overrightarrow{v}\otimes \overrightarrow{v})-(1-r)\Delta \overrightarrow{v}+\nabla \pi -\mathrm{div}\mathbb{E}=\overrightarrow{f}\hfill & \mathrm{in}\Omega ,\hfill \\ \nabla ·\overrightarrow{v}=0\hfill & \mathrm{in}\Omega ,\hfill \\ {\displaystyle We\frac{{\mathcal{D}}_0\mathbb{E}}{\mathcal{D}t}+\mathbb{E}=2r\mathbb{D}\left(\overrightarrow{v}\right)}\hfill & \mathrm{in}\Omega ,\hfill \\ \overrightarrow{v}=\overrightarrow{0}\hfill & \mathrm{on}\Gamma .\hfill \end{array}\right.$$

Remark 2 

(Uniqueness and regularity). Chupin & Martin [2] showed that problem (2) with $\overrightarrow{u}=\overrightarrow{0}$ admits at most one weak solution if one of the following conditions is satisfied:

• the norm $\parallel \overrightarrow{f}{\parallel }_{{\left[H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\right]}^{\ast }}$ is small enough;
• the numbers $Re$ and $We$ are small enough.

Moreover, if the domain Ω is of class $C^{\infty }$ and each component of the vector function $\overrightarrow{f}$ belongs to the space $C^{\infty }\left(\overline{\Omega }\right)$, then each component of any weak solution of (2) belongs to the space $C^{\infty }\left(\overline{\Omega }\right)$.

Remark 3. 

Lions & Masmoudi [25] showed the global-in-time existence of weak solutions to evolution system (1) with $\zeta =0$ in dimensions 2 and 3 for general initial conditions, but the global solvability in the class of smooth solutions is an open problem for this model. Recently, Ye [26] obtained some results related to global regularity of solutions of high-dimensional Oldroyd’s model (in the corotational case) with the fractional dissipation term ${(-\Delta )}^{\alpha }\overrightarrow{v}$. Various theorems on the well-posedness of the Oldroyd model with slip boundary conditions are proved in [27,28,29]. The first problem of Stokes for an Oldroyd-B fluid is investigated in [30]. Analytical solutions corresponding to the starting flow of an Oldroyd-type fluid between rotating cylinders are given in [31]. The paper [32] is devoted to studying the flow of Oldroyd fluids with fractional derivatives over a plate that applies shear stress to the fluid. Semi-analytical solutions for rivulet flows of non-Newtonian fluids are obtained in [33,34].

4. Optimal Boundary Control Problem for the Corotational Oldroyd Model

4.1. Problem Formulation and the Main Result

Consider the following optimal boundary control problem

$$\left\{\begin{array}{cc}Re\mathrm{div}(\overrightarrow{v}\otimes \overrightarrow{v})-(1-r)\Delta \overrightarrow{v}+\nabla \pi -\mathrm{div}\mathbb{E}=\overrightarrow{f}\hfill & \mathrm{in}\Omega ,\hfill \\ \nabla ·\overrightarrow{v}=0\hfill & \mathrm{in}\Omega ,\hfill \\ {\displaystyle We\frac{{\mathcal{D}}_0\mathbb{E}}{\mathcal{D}t}+\mathbb{E}-\zeta \Delta \mathbb{E}=2r\mathbb{D}\left(\overrightarrow{v}\right)}\hfill & \mathrm{in}\Omega ,\hfill \\ \overrightarrow{v}=\overrightarrow{u}\hfill & \mathrm{on}\Gamma ,\hfill \\ {\displaystyle \frac{\partial \mathbb{E}}{\partial \overrightarrow{n}}=\mathbb{O}}\hfill & \mathrm{on}\Gamma ,\hfill \\ (\overrightarrow{v},\overrightarrow{u})\in {\mathcal{V}}_{\mathrm{ad}}\times {\mathcal{U}}_{\mathrm{ad}},\hfill \\ \mathcal{J}(\overrightarrow{v},\mathbb{E},\overrightarrow{u})\to min.\hfill \end{array}\right.$$

(19)

where

• $\overrightarrow{u}$ is the control function;
• ${\mathcal{U}}_{\mathrm{ad}}$ is the admissible controls set;
• ${\mathcal{V}}_{\mathrm{ad}}$ is the admissible states set;
• $\mathcal{J}$ is the cost functional.

For simplicity, we will assume that

$${\mathcal{V}}_{\mathrm{ad}}={\mathcal{V}}_{\mathrm{ad}}\left(R\right)\stackrel{\mathrm{def}}{=}\{\overrightarrow{v}\in H_{\sigma }^1(\Omega ,{\mathbb{R}}^d):\parallel \overrightarrow{v}{\parallel }_{H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)}\le R\},$$

where R is a positive constant.

Further, assume that $\mathcal{J}$ is a real-valued function defined on the Cartesian product $H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)\times H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})\times L_n^2(\Gamma ,{\mathbb{R}}^d)$.

For $K\in {\mathbb{R}}_{+}$, let

$${\mathfrak{N}}_K\stackrel{\mathrm{def}}{=}\{(\overrightarrow{v},\mathbb{E},\overrightarrow{u})\in H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)\times H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})\times L_n^2(\Gamma ,{\mathbb{R}}^d):|\mathcal{J}(\overrightarrow{v},\mathbb{E},\overrightarrow{u})|\le K\}.$$

Definition 2 

(Admissible weak solution). We shall say that a triplet $(\overrightarrow{v},\mathbb{E},\overrightarrow{u})$ is an admissible weak solution of problem (19) if

• $(\overrightarrow{v},\mathbb{E},\overrightarrow{u})\in H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)\times H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})\times L_n^2(\Gamma ,{\mathbb{R}}^d);$
• relations (3) and (4) hold for any $\overrightarrow{w}\in H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)$ and $\mathbb{F}\in H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d});$
• ${\gamma }_{\Gamma }\overrightarrow{v}=\overrightarrow{u};$
• $(\overrightarrow{v},\overrightarrow{u})\in {\mathcal{V}}_{\mathrm{ad}}\left(R\right)\times {\mathcal{U}}_{\mathrm{ad}}$.

Denote the set of all admissible weak solutions of problem (19) by ${\mathfrak{M}}_{\mathrm{ad}}$.

Definition 3 

(Optimal weak solution). A triplet $({\overrightarrow{v}}_{★},{\mathbb{E}}_{★},{\overrightarrow{u}}_{★})$ is said to be an optimal weak solution of problem (19) if

• $({\overrightarrow{v}}_{★},{\mathbb{E}}_{★},{\overrightarrow{u}}_{★})\in {\mathfrak{M}}_{\mathrm{ad}}$;
• $\mathcal{J}({\overrightarrow{v}}_{★},{\mathbb{E}}_{★},{\overrightarrow{u}}_{★})=\underset{(\overrightarrow{v},\mathbb{E},\overrightarrow{u})\in {\mathfrak{M}}_{\mathrm{ad}}}{inf}\mathcal{J}(\overrightarrow{v},\mathbb{E},\overrightarrow{u})$.

Theorem 2 

(Main result). Suppose that

$$0\le r<1,\zeta >0,R\ge \frac{1}{1-r}{\parallel \overrightarrow{f}\parallel }_{{\left[H_{0,\sigma }^1(\Omega ,{\mathbb{R}}^d)\right]}^{\ast }}$$

(20)

and the following five conditions hold:

(i)

the cost functional $\mathcal{J}$ is bounded from below on its domain, that is, there exists a constant ${\mathcal{J}}_{\min }$ such that

$${\mathcal{J}}_{\min }\le \mathcal{J}(\overrightarrow{v},\mathbb{E},\overrightarrow{u})$$

for any $(\overrightarrow{v},\mathbb{E},\overrightarrow{u})\in H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)\times H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})\times L_n^2(\Gamma ,{\mathbb{R}}^d);$

(ii)

the set ${\mathfrak{N}}_K$ is bounded in the Cartesian product $H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)\times H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})\times L_n^2(\Gamma ,{\mathbb{R}}^d)$, for each positive number $K;$

(iii)

for any sequence ${\left\{({\overrightarrow{v}}_s,{\mathbb{E}}_s,{\overrightarrow{u}}_s)\right\}}_{s=1}^{\infty }$ such that

$$\begin{array}{cc}\hfill & {\overrightarrow{v}}_s\rightharpoonup \overrightarrow{v}\mathit{weakly}\mathit{in}{H}_{\sigma }^1(\Omega ,{\mathbb{R}}^d)\mathrm{as}s\to \infty ,\hfill \\ \hfill & {\mathbb{E}}_s\rightharpoonup \mathbb{E}\mathit{weakly}\mathit{in}{H}^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})\mathrm{as}s\to \infty ,\hfill \\ \hfill & {\overrightarrow{u}}_s\to \overrightarrow{u}\mathit{strongly}\mathit{in}{L}_n^2(\Gamma ,{\mathbb{R}}^d)\mathrm{as}s\to \infty ,\hfill \end{array}$$

we have

$$\mathcal{J}(\overrightarrow{v},\mathbb{E},\overrightarrow{u})\le \underset{s\to \infty }{lim\; inf}\mathcal{J}({\overrightarrow{v}}_s,{\mathbb{E}}_s,{\overrightarrow{u}}_s);$$

(iv)

the admissible controls set ${\mathcal{U}}_{\mathrm{ad}}$ is closed in the subspace $L_n^2(\Gamma ,{\mathbb{R}}^d)$;

(v)

the inclusion $\overrightarrow{0}\in {\mathcal{U}}_{\mathrm{ad}}$ holds.

Then problem(19)has at least one optimal weak solution.

Example 1. 

Consider the cost functional $\mathcal{J}:H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)\times H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})\times L_n^2(\Gamma ,{\mathbb{R}}^d)\to \mathbb{R}$ defined as follows:

$$\mathcal{J}(\overrightarrow{v},\mathbb{E},\overrightarrow{u})\stackrel{\mathrm{def}}{=}{\theta }_1\parallel \overrightarrow{v}-{\overrightarrow{v}}_{\#}{\parallel }_{H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)}^2+{\theta }_2\parallel \mathbb{E}-{\mathbb{E}}_{\#}{\parallel }_{H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})}^2+{\theta }_3{\parallel \overrightarrow{u}-{\overrightarrow{u}}_{\#}\parallel }_{L_n^2(\Gamma ,{\mathbb{R}}^d)}^2,$$

where

• ${\theta }_1$, ${\theta }_2$, ${\theta }_3$ are given positive constants (weight coefficients) such that $0<{\theta }_i<1$, $i=1,2,3$, and ${\theta }_1+{\theta }_2+{\theta }_3=1;$
• ${\overrightarrow{v}}_{\#}$, ${\mathbb{E}}_{\#}$, ${\overrightarrow{u}}_{\#}$ are given mappings characterizing the desired velocity field in Ω, the desired elastic extra-stress in Ω and the desired inflow-outflow regime on Γ, respectively.

Clearly, the functional $\mathcal{J}$ satisfies conditions (i)–(iii).

Remark 4. 

Another approach to the statement of boundary control problems for flow models is based on the use of boundary conditions for the dynamic pressure at permeable parts of the boundary of a flow region [35,36,37].

4.2. Proof of the Main Result

The proof of Theorem 2 will be performed in three steps.

Step 1: Existence of admissible weak solutions. First we show that the set ${\mathfrak{M}}_{\mathrm{ad}}$ is not empty. Consider a triplet $({\overrightarrow{v}}_0,{\mathbb{E}}_0,\overrightarrow{0})$, where the pair $({\overrightarrow{v}}_0,{\mathbb{E}}_0)$ is a weak solution of problem (2) with the zero boundary data for the velocity field and the vector function ${\overrightarrow{v}}_0$ satisfies estimate (5) (see Theorem 1). Taking into account the third inequality from (20) and condition (v), we deduce that

$$({\overrightarrow{v}}_0,{\mathbb{E}}_0,\overrightarrow{0})\in {\mathfrak{M}}_{\mathrm{ad}},$$

and hence ${\mathfrak{M}}_{\mathrm{ad}}\ne \varnothing$.

Step 2: Minimizing sequence. Let ${\left\{({\overrightarrow{v}}_{\ell },{\mathbb{E}}_{\ell },{\overrightarrow{u}}_{\ell })\right\}}_{\ell =1}^{\infty }\subset {\mathfrak{M}}_{\mathrm{ad}}$ be a sequence such that

$$\underset{\ell \to \infty }{lim}\mathcal{J}({\overrightarrow{v}}_{\ell },{\mathbb{E}}_{\ell },{\overrightarrow{u}}_{\ell })=\underset{(\overrightarrow{v},\mathbb{E},\overrightarrow{u})\in {\mathfrak{M}}_{\mathrm{ad}}}{inf}\mathcal{J}(\overrightarrow{v},\mathbb{E},\overrightarrow{u}).$$

(21)

From condition (i) it follows that the sequence ${\left\{\mathcal{J}({\overrightarrow{v}}_{\ell },{\mathbb{E}}_{\ell },{\overrightarrow{u}}_{\ell })\right\}}_{\ell =1}^{\infty }$ is bounded. Using condition (ii), we deduce that the sequence ${\left\{({\overrightarrow{v}}_m,{\mathbb{E}}_m,{\overrightarrow{u}}_m)\right\}}_{m=1}^{\infty }$ is bounded in the Cartesian product $H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)\times H^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})\times L_n^2(\Gamma ,{\mathbb{R}}^d)$. Hence, there exist a triplet $({\overrightarrow{v}}_{★},{\mathbb{E}}_{★},{\overrightarrow{u}}_{★})$ and a subsequence ${\left\{{\ell }_s\right\}}_{s=1}^{\infty }$ such that

$$\begin{array}{cc}\hfill & {\overrightarrow{v}}_{{\ell }_s}\rightharpoonup {\overrightarrow{v}}_{★}\mathrm{weakly}\mathrm{in}{H}_{\sigma }^1(\Omega ,{\mathbb{R}}^d)\mathrm{as}s\to \infty ,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & {\mathbb{E}}_{{\ell }_s}\rightharpoonup {\mathbb{E}}_{★}\mathrm{weakly}\mathrm{in}{H}^1(\Omega ,{\mathbb{M}}_{\mathrm{sym}}^{d\times d})\mathrm{as}s\to \infty ,\hfill \\ \hfill & {\overrightarrow{u}}_{{\ell }_s}\rightharpoonup {\overrightarrow{u}}_{★}\mathrm{weakly}\mathrm{in}{L}_n^2(\Gamma ,{\mathbb{R}}^d)\mathrm{as}s\to \infty .\hfill \end{array}$$

(23)

Moreover, since the trace operator ${\gamma }_{\Gamma }:H_{\sigma }^1(\Omega ,{\mathbb{R}}^d)\to L_n^2(\Gamma ,{\mathbb{R}}^d)$ is compact (see Lemma 1) and ${\gamma }_{\Gamma }{\overrightarrow{v}}_{{\ell }_s}={\overrightarrow{u}}_{{\ell }_s}$, we have

$${\overrightarrow{u}}_{{\ell }_s}\to {\overrightarrow{u}}_{★}\mathrm{strongly}\mathrm{in}{L}_n^2(\Gamma ,{\mathbb{R}}^d)\mathrm{as}s\to \infty .$$

(24)

Step 3: Existence of optimal weak solutions. Now we shall show that the triplet $({\overrightarrow{v}}_{★},{\mathbb{E}}_{★},{\overrightarrow{u}}_{★})$ is an optimal weak solution to problem (19).

From condition (iv), ${\left\{{\overrightarrow{u}}_{{\ell }_s}\right\}}_{s=1}^{\infty }\subset {\mathcal{U}}_{\mathrm{ad}}$, and the strong convergence (24) it follows that ${\overrightarrow{u}}_{★}\in {\mathcal{U}}_{\mathrm{ad}}$. Note also that ${\overrightarrow{v}}_{★}\in {\mathcal{V}}_{\mathrm{ad}}\left(R\right)$. Moreover, it is not difficult to prove that the pair $({\overrightarrow{v}}_{★},{\mathbb{E}}_{★})$ is a weak solution to problem (2) with $\overrightarrow{u}={\overrightarrow{u}}_{★}$. This implies that

$$({\overrightarrow{v}}_{★},{\mathbb{E}}_{★},{\overrightarrow{u}}_{★})\in {\mathfrak{M}}_{\mathrm{ad}}.$$

(25)

Taking into account condition (iii) and (21)–(25), we derive

$$\begin{array}{cc}\hfill \mathcal{J}({\overrightarrow{v}}_{★},{\mathbb{E}}_{★},{\overrightarrow{u}}_{★})& \le \underset{s\to \infty }{lim\; inf}\mathcal{J}({\overrightarrow{v}}_{{\ell }_s},{\mathbb{E}}_{{\ell }_s},{\overrightarrow{u}}_{{\ell }_s})\hfill \\ \hfill & =\underset{(\overrightarrow{v},\mathbb{E},\overrightarrow{u})\in {\mathfrak{M}}_{\mathrm{ad}}}{inf}\mathcal{J}(\overrightarrow{v},\mathbb{E},\overrightarrow{u})\hfill \\ \hfill & \le \mathcal{J}({\overrightarrow{v}}_{★},{\mathbb{E}}_{★},{\overrightarrow{u}}_{★}),\hfill \end{array}$$

whence

$$\mathcal{J}({\overrightarrow{v}}_{★},{\mathbb{E}}_{★},{\overrightarrow{u}}_{★})=\underset{(\overrightarrow{v},\mathbb{E},\overrightarrow{u})\in {\mathfrak{M}}_{\mathrm{ad}}}{inf}\mathcal{J}(\overrightarrow{v},\mathbb{E},\overrightarrow{u}).$$

(26)

From (25) and (26) it follows that the triplet $({\overrightarrow{v}}_{★},{\mathbb{E}}_{★},{\overrightarrow{u}}_{★})$ is an optimal weak solution of problem (19). Thus, Theorem 2 is proved.