Bilinear Optimal Control Problem for the Stationary Navier–Stokes Equations with Variable Density and Slip Boundary Condition

Abstract

An optimal control problem for the stationary Navier–Stokes equations with variable density is studied. A bilinear control is applied on the flow domain, while Dirichlet and Navier boundary conditions for the velocity are assumed on the boundary. As a first step, we enunciate a result on the existence of weak solutions of the dynamical equation; this is done by firstly expressing the fluid density in terms of the stream-function. Then, the bilinear optimal control problem is analyzed, and the existence of optimal solutions are proved; their corresponding characterization regarding the first-order optimality conditions are obtained. Such optimality conditions are rigorously derived by using a penalty argument since the weak solutions are not necessarily unique neither isolated, and so standard methods cannot be applied.

Introduction

In this work we consider an optimal control problem restricted by the stationary Navier–Stokes equations for the flow of an incompressible viscous fluid with variable density. More precisely, being \(\mathbf{u}\) the velocity field, p the pressure and \(\rho \) the fluid density, and \(\varOmega \subset \mathbb {R}^2\) the flow domain, which is assumed to be a bounded open and connected set, with boundary \(\varGamma \) of class \(C^1\). We consider the following system of partial differential equations:

$$\begin{aligned} \left\{ \begin{array}{lll} -\mu \varDelta \mathbf{u}+\rho (\mathbf{u}\cdot \nabla )\mathbf{u}+\nabla p&{}=&{}\rho \mathbf{f}\quad \text{ in } \varOmega ,\\ \mathbf{u}\cdot \nabla \rho &{}=&{}0 \text{ in } \varOmega ,\\ \mathrm{div}\,\mathbf{u}&{}=&{}0 \! \text{ in } \varOmega . \end{array}\right. \end{aligned}$$
(1)

Here, the constant \(\mu >0\) denotes the dynamic viscosity, and the field \(\mathbf{f}\) is the specific density of external forces. Also, the function \(\mathbf{f}\), in the optimal control problem that we will study, describe the bilinear control for dynamical equation in \(\varOmega \) and lie in closed convex set \(\mathcal {U}\subset \mathbf{L}^2(\varOmega )\).

To describe the boundary conditions for the velocity, we split the boundary of \(\varOmega \) in three parts (see Fig. 1 below): \(\varGamma = \overline{\varGamma }_1 \cup \overline{\varGamma }_2 \cup \overline{\varGamma }_3\), with \(\varGamma _1\), \(\varGamma _2\) and \(\varGamma _3\) disjoint open sets (relatively to \(\varGamma \)); \(\varGamma _1\) is the outflow/inflow part of the boundary and \(\varGamma _3 := \varGamma {\setminus } (\overline{\varGamma }_1 \cup \overline{\varGamma }_2)\) are the fixed walls of the flow domain. Then, given suitable functions \(\mathbf{u}_0\) and \(\rho _0\) defined on \(\varGamma _1\), and \(\mathbf{g}\) defined on \(\varGamma _2\); we impose the following Dirichlet boundary conditions:

$$\begin{aligned} \mathbf{u}= & {} \mathbf{u}_{\mathbf{g}}= \left\{ \begin{array}{lll} \mathbf{u}_0 &{}\quad \text{ on }&{} \varGamma _1, \\ \mathbf{g}&{}\quad \text{ on }&{} \varGamma _2, \end{array} \right. \end{aligned}$$
(2)
$$\begin{aligned} \rho= & {} \rho _0>0\ \text{ on } \varGamma _1. \end{aligned}$$
(3)

On the part \(\varGamma _3\) of the boundary we impose the called Navier friction boundary condition:

$$\begin{aligned} \mathbf{u}\cdot \mathbf{n}=0\ \text{ and } \ [D(\mathbf{u})\mathbf{n}+\alpha \mathbf{u}]_\mathrm{tang}=0. \end{aligned}$$
(4)

Here \(\mathbf{n}\) denotes the outward normal vector on \(\varGamma \). The term \([D(\mathbf{u})\mathbf{n} + \alpha \mathbf{u}]_\mathrm{tang}:= D(\mathbf{u})\mathbf{n}\,+\,\alpha \mathbf{u}-[(D(\mathbf{u})\mathbf{n}\,+\,\alpha \mathbf{u})\cdot \mathbf{n}]\mathbf{n}\) is the tangential component of the vector \(D(\mathbf{u})\mathbf{n}+\alpha \mathbf{u}\), where \(D(\mathbf{u}):= \frac{1}{2}(\nabla \mathbf{u}+\nabla ^T\mathbf{u})\) is twice the standard symmetric part of the rate of deformation tensor, and the real \(\alpha \ge 0\) is the friction coefficient which measures the tendency of the fluid to slip on \(\varGamma _3\). When \(\alpha =0\) and the boundary is flat, the fluid slips along of \(\varGamma _3\) without friction and there is no boundary layers, and as \(\alpha \rightarrow \infty \), the friction is so intense that the fluid is almost at rest near the boundary.

We additionally assume that

$$\begin{aligned}&\varGamma _1\ \text{ is } \text{ an } \text{ arcwise } \text{ connected } \text{ closed } \text{ set } \text{ on } \varGamma , \text{ meas } (\varGamma _1) >0; \end{aligned}$$
(5)
$$\begin{aligned}&\text{ either }\quad \mathbf{u}_0\cdot \mathbf{n}>0 \; \text{ on } \; \varGamma _1 \, \text{(outflow) } \quad \text{ or } \quad \mathbf{u}_0\cdot \mathbf{n}<0 \; \text{ on } \; \varGamma _1 \; \text{(inflow) }. \end{aligned}$$
(6)

The Dirichlet boundary condition, almost exclusively, has been considered for motions of viscous incompressible fluids in hydrodynamics as well as in mathematics. However, there exist some flow phenomena, modeling of which might require introduction of Navier slip boundary condition in the reality. As examples, we can refer to Fujita (1994, 2002): flow through a drain or canal with its bottom covered be sherbet of mud and pebbles; flow of melted iron coming out from a smelting furnace; avalanche of water and rocks and blood flow in a vein of an arterial sclerosis patient. Other applications can be found in Beirão da Veiga (2004, 2005).

Fig. 1
figure1

Sketch of the domain

We recall that the Navier boundary condition was proposed by Navier (1827), who claimed that the tangential component of the viscous stress at the boundary should be proportional to the tangential velocity. Navier boundary condition was also derived by Maxwell (1879) from the kinetic theory of gases and rigorously justified as a homogenization of the no-slip condition on a rough boundary (Jägger and Mikelić 2001). We also remark that the boundary condition for \(\rho \) is given only on \(\varGamma _1\) (it could also be given only on \(\varGamma _2\)) because the equation for \(\rho \) is a first-order transport equation (see (1)\(_2\)).

We must stress that nonhomogeneous fluids, in particular the ones with variable density, appear in many important situations both in nature as in industrial applications. Such fluids present a much larger class of interesting phenomena than the homogeneous (constant density) fluids, and there exist many different mathematical models correspond to different physical situations. For the particular case of mixtures of incompressible fluids, one can find specific information on their physical modeling in Joseph (2010), for instance. Still in the context of mixtures of incompressible fluids, but now focusing on the rigorous mathematical analysis we can mention the works (Antonsev et al. 1990; Kazhikov 1974; Simon 1989, 1990).

For the stationary Navier–Stokes equations with variable density, there exist some results on the existence of weak solutions (Ammar-Khodja and Santos 2006; Frolov 1993, 1996; Illarionov 2001; Santos 2002). In all these works the authors only consider Dirichlet boundary conditions for the velocities field. Results on the existence of weak and strong solutions for the stationary Navier–Stokes system with Navier friction boundary condition, with constant density, can be consulted in Mulone and Salemi (1983, 1985).

The purpose of this paper is to study a bilinear control problem associated with the weak solutions of system (1)–(4), we will use a stream-function formulation as Frolov (1993) has done, in which the fluid density depends on the stream-function by means of another function determined by the boundary conditions. This allows for dropping some of the equations, most notably the continuity equation (1)\(_2\). This representation has been used by Illarionov (2001) for the Navier–Stokes equations with Dirichlet boundary conditions and in Mallea-Zepeda et al. (2017), Vitoriano (2013) in the context of the micropolar fluids equations.

The exact formulation of this problem is given in Sect. 3. We prove the existence of optimal solutions, and we derive first-order optimality conditions of Pontryagin type (existence of Lagrange multipliers). We stress that standard procedures to obtain optimality conditions cannot be applied in the present context since the weak solutions of our stationary problem are not necessarily unique neither isolated. To obtain the required optimality conditions, we adapt the technique of Lions (1971, Chapter 7), which consists in introducing a family of penalized control problems \((P)_\varepsilon \), \(\varepsilon >0\), whose solutions converge towards a solution to original optimal control problem. We then derive the optimality conditions associated with problems \((P)_\varepsilon \) and pass to the limit to obtain the optimality condition of the original problem (see also Hettich et al. 1997a, b; Lions 1985). This method has been used in Abergel and Casas (1993), Illarionov (2001), Lee and Imanuvilov (2000) in the context of Navier–Stokes and Boussinesq equations; and in Mallea-Zepeda et al. (2017), Rodríguez-Bellido et al. (2017), Stavre (2015) in other situations. The details of this arguments are presented in Sect. 4; see in particular the result stated in Theorem 4.

Also, as far as we know, the literature related to optimal control problems with bilinear control for partial differential equations is few, see (Borzì et al. 2016; Köner and Vexler 2009; Schöberl et al. 2011; Vallejos and Borzì 2011). The main difficulty is that the solution of the state equations depends linearly on the variable control.

The outline of this paper is as follows: In Sect. 2, we fix the notation, introduce the functional spaces to be used, rewrite the problem using the given functional setting, give the precise definition of weak solution to be considered in this work, and we give a result on existence of weak solutions. In Sect. 3, we introduce our bilinear control problem to study, and we prove the existence of optimal solutions. Finally, in Sect. 4, we derive the associated first-order optimality conditions.

Preliminaries

In this section, we introduce the notations, function spaces and the main results that we will use to study the optimal control problem related to weak solutions of system (1)–(4).

Function Spaces

For \(s\ge 0\) we consider the standard Sobolev space \(W^{s,2}(\varOmega ):=H^s(\varOmega )\), and the Lebesgue space \(L^p(\varOmega )\), \(1\le p\le \infty \), with norms denoted by \(\Vert \cdot \Vert _{H^s}\) and \(\Vert \cdot \Vert _{L^p}\), respectively. In particular, the norm and inner product in \(L^2(\varOmega )\) will be represented by \(\Vert \cdot \Vert \) and \((\cdot ,\cdot )_2\), respectively. The norm \(L^p(\varGamma )\) will be denoted by \(\Vert \cdot \Vert _{L^p(\varGamma )}\). Corresponding Sobolev spaces of vector-valued functions will be denoted by \(\mathbf{H}^1(\varOmega )\), \(\mathbf{L}^2(\varOmega )\), and so on. Also, We will use the following solenoidal Banach spaces: the function space \(\mathbf{H}_\sigma =\{\mathbf{u}\in \mathbf{H}^1(\varOmega )\,:\, \mathrm{div}\,\mathbf{u}=0 \text{ and } {} \mathbf{u}\cdot \mathbf{n}=0 \text{ on } \varGamma _3\}\), endowed with the usual norm of \(\mathbf{H}^1(\varOmega )\), and also the function space \(\mathbf{V}_\sigma =\{\mathbf{u}\in \mathbf{H}_\sigma \,:\, \mathbf{u}=\mathbf{0} \text{ on } \varGamma {\setminus }\varGamma _3\}\), which is a Hilbert space with the inner product \((\mathbf{u},\mathbf{v})_{\mathbf{V}_\sigma }:=(D(\mathbf{u}), D(\mathbf{v}))_2\) and norm \(\Vert \mathbf{u}\Vert _{\mathbf{V}_\sigma }:=\Vert D(\mathbf{u})\Vert \). If X is a general Banach space, its topological dual will be denoted by \(X'\) and the duality product by \(\langle \cdot ,\cdot \rangle _{X'}\) or simply by \(\langle \cdot ,\cdot \rangle \) unless this leads to ambiguity. The space \(\mathbf{H}'_\sigma \) denotes the dual of \(\mathbf{H}_\sigma \), and the space \(\mathbf{V}'_\sigma \) denotes the dual of \(\mathbf{V}_\sigma \).

If \(\varGamma _i\) is a connected subset of \(\varGamma ,\) then the restriction of elements of the space \(\mathbf{H}^1(\varOmega )\) to \(\varGamma _i\) we will denote by \(\mathbf{H}^{1/2}(\varGamma _i)\) and we consider the following subspaces

$$\begin{aligned} \mathbf{H}^{1/2}_{0}(\varGamma _i)= & {} \{\mathbf{u}\in \mathbf{L}^2(\varGamma _i):\ \text{ there } \text{ exists }\ \mathbf{v} \in \mathbf{H}^{1}(\varOmega ),\ \mathbf{v}_{|_{\varGamma {\setminus }\varGamma _i}}=\mathbf{0},\ \mathbf{v}_{|_{\varGamma _i}}=\mathbf{u}\},\\ \mathbf{H}^{1/2}_{00}(\varGamma _i)= & {} \left\{ \mathbf{u}\in \mathbf{H}^{1/2}_{0}(\varGamma _i):\int _{\varGamma _i}{} \mathbf{u} \cdot \mathbf{n}=0\right\} . \end{aligned}$$

The space \(\mathbf{H}^{1/2}_0(\varGamma _i)\) is a closed subspace of \(\mathbf{H}^{1/2}(\varGamma _i)\) and is continuously embedded in \(\mathbf{L}^2(\varGamma _i)\) (see Dautray and Lions 2000; De los Reyes and Kunisch 2005). For the space \(\mathbf{H}^{1/2}(\varGamma _i),\)\(\mathbf{H}^{-1/2}(\varGamma _i)\) denotes its dual space, and \(\langle \cdot ,\cdot \rangle _{\varGamma _i}\), their duality product. As it is usual, the letter C will denote diverse positive constants which may change from line to line or even within the same line.

Definition of Weak Solution

In order to present the concept of weak solutions of system (1)–(4); first, following the ideas of Frolov (1993), we will express the fluid density in terms of the stream-function.

We assume that

$$\begin{aligned} \rho _0\in C^0(\varGamma _1),\quad \mathbf{u}_0\in \mathbf{H}^{1/2}_{00}(\varGamma _1). \end{aligned}$$
(7)

For each \(\mathbf{u}\in \mathbf{H}_\sigma \) there exists a stream-function \(\varPsi \in H^2(\varOmega )\) such that

$$\begin{aligned} \mathbf{u}=\mathrm{rot}\,\varPsi =\left( \frac{\partial \varPsi }{\partial x_2},-\frac{\partial \varPsi }{\partial x_1}\right) \text{ in } \varOmega . \end{aligned}$$
(8)

Also, the assumption \(\mathrm{rot}\,\varPsi =\mathbf{u}_0\) on \(\varGamma _1\) implies that

$$\begin{aligned} \frac{\partial \varPsi }{\partial \mathbf{n}}=\mathbf{u}_0\cdot \varvec{\tau },\quad \frac{\partial \varPsi }{\partial \varvec{\tau }}=-\mathbf{u}_0\cdot \mathbf{n}\ \text{ on } \ \varGamma _1, \end{aligned}$$
(9)

where \(\varvec{\tau }\) denotes the outward tangent vector on \(\varGamma \). Thus, the boundary values of \(\varPsi \) can be obtained as follow

$$\begin{aligned} \varPsi (\mathbf{x})=\int _{\varGamma _1(\mathbf{x}_0,\mathbf{x})}{} \mathbf{u}_0\cdot \mathbf{n},\ \mathbf{x}\in \varGamma _1, \end{aligned}$$
(10)

where \(\mathbf{x}_0\) is the initial point of the curve \(\varGamma _1\) and \(\varGamma _1(\mathbf{x}_0,\mathbf{x})\) is the part of \(\varGamma _1\) lying between the point \(\mathbf{x}_0\) and \(\mathbf{x}\) (see Frolov 1993, for details).

Since \(\mathbf{u}_0\in \mathbf{H}^{1/2}_{00}(\varGamma _1),\) we have \(\mathbf{u}_0\cdot \mathbf{n}\in \mathbf{H}^{1/2}(\varGamma _1)\); thus we deduce that \(\varPsi \in H^{3/2}(\varGamma _1)\subset C^0(\varGamma _1)\). Moreover, from (5) and (6) the function \(\varPsi \) is strictly monotone on \(\varGamma _1\); therefore, there exists \(\varPsi ^{-1}:\varPsi (\varGamma _1)\subseteq \mathbb {R}\rightarrow \varGamma _1\), and then from (7)\(_1\) we can define \(\widehat{\eta }(y)=\rho _0(\varPsi ^{-1}(y))\), \(y\in \varPsi (\varGamma _1)\subseteq \mathbb {R}\). Then, using that \(\rho _0(\mathbf{x})>0\), for all \(\mathbf{x}\in \varGamma _1\), and taking into account that \(\varGamma _1\) is an arcwise connected closed set in \(\varGamma \), we extend \(\widehat{\eta }\) to \(\mathbb {R}\) as a strictly positive scalar function \(\eta \) such that

$$\begin{aligned} \eta \in C^0(\mathbb {R}),\quad \eta (z)>0\ \forall z\in \mathbb {R},\quad \eta (z)=\rho _0(\varPsi ^{-1}(z)),\ z\in \varGamma _1. \end{aligned}$$
(11)

Moreover, following (Illarionov 2001; Mallea-Zepeda et al. 2017), we define the linear and continuous operator \(N:\mathbf{H}_\sigma \rightarrow H^2(\varOmega )\) which to each vector field \(\mathbf{u}\in \mathbf{H}_\sigma \) assigns its stream-function satisfying (8); then, from (11) we can define the density \(\rho :\varOmega \rightarrow \mathbb {R}\) as being

$$\begin{aligned} \rho (\mathbf{x})=(\eta \circ \varPsi )(\mathbf{x})=\eta (N\mathbf{u})(\mathbf{x}),\ \mathbf{x}\in \varOmega . \end{aligned}$$
(12)

Remark 1

The density \(\rho \) defined by (12) satisfies the boundary condition (3), that is, \(\eta (N\mathbf{u})=\rho _0\) on \(\varGamma _1\). Moreover, for \(\mathbf{u}\in \mathbf{H}_\sigma \) and \(\eta \in C^1(\mathbb {R})\), the Eq. (1)\(_2\) is satisfied in the weak sense

$$\begin{aligned} \int _\varOmega (\mathbf{u}\cdot \nabla \varphi )\eta (N\mathbf{u})=\int _\varOmega (\mathbf{u}\cdot \nabla \varphi )\varPsi =0,\ \forall \varphi \in H^1(\varOmega ) \text{ with } \varphi =0 \text{ on } \varGamma . \end{aligned}$$
(13)

Regularization and passage to the limit give the result for \(\eta \in \text{ C }^0(\mathbb {R})\).

Now, we consider the following operators

$$\begin{aligned} \begin{array}{cc} \mathcal {B}:\mathbf{H}_\sigma \times \mathbf{H}_\sigma \times \mathbf{H}_\sigma \rightarrow \mathbf{V}'_\sigma ,&\ \mathcal {F}:\mathbf{H}_\sigma \rightarrow \mathbf{V}'_\sigma , \end{array} \end{aligned}$$

defined by

$$\begin{aligned} \begin{array}{cc} \mathcal {B}(\mathbf{u},\mathbf{v},\mathbf{w})=\eta (N\mathbf{u})\mathbf{v}\cdot \nabla \mathbf{w},&\ \mathcal {F}(\mathbf{u})=\eta (N\mathbf{u})\mathbf{f}. \end{array} \end{aligned}$$
(14)

Lemma 1

For any \(\mathbf{u}\in \mathbf{H}_\sigma \) and \(\mathbf{v}\in \mathbf{V}_\sigma \), we have

$$\begin{aligned} \langle \mathcal {B}(\mathbf{u},\mathbf{u},\mathbf{v}),\mathbf{v}\rangle _{\mathbf{V}_\sigma '} = 0. \end{aligned}$$
(15)

Proof

If \(\eta \in C^1(\mathbb {R})\), there holds \(\mathrm{div}(\eta (N\mathbf{u})\mathbf{u})=\mathrm{div}(\rho \mathbf{u})=\mathbf{u}\cdot \nabla \rho =0\); thus, we have (15). When \(\eta \in C^0(\mathbb {R})\), by regularization of \(\eta \) and passage to the limit, the results in (15) remain true. \(\square \)

In order to define the concept of weak solutions of system (1)–(4), we recall the following result.

Lemma 2

(Solonnikov and Scadilov 1973; Verfüth 1987) Let \(\mathbf{u}\in \mathbf{H}^2_\sigma :=\{\mathbf{u}\in \mathbf{H}^2(\varOmega )\,:\, \mathrm{div}\,\mathbf{u}=0\}\) satisfying the Navier boundary condition and \(\mathbf{v}\in \mathbf{H}^1_\sigma :=\{\mathbf{v}\in \mathbf{H}^1(\varOmega )\,:\, \mathrm{div}\,\mathbf{v}=0\}\) tangent to the boundary. Then,

$$\begin{aligned} -\int _\varOmega \varDelta \mathbf{u}\cdot \mathbf{v} = 2 \int _\varOmega D(\mathbf{u}):D(\mathbf{v}) - 2 \int _\varGamma [D(\mathbf{u})\mathbf{n}]_\mathrm{tang}\cdot \mathbf{v}. \end{aligned}$$

Motived by integration by parts and Lemma 2, we introduce the following definition of weak solutions of system (1)–(4).

Definition 1

Let \(\mathbf{f} \in \mathbf{L}^2(\varOmega )\), \(\mathbf{u}_0\in \mathbf{H}^{1/2}_{00}(\varGamma _1)\), \(\mathbf{g}\in \mathbf{H}^{1/2}_{00}(\varGamma _2)\), and \(\eta \in C^0(\mathbb {R})\). A weak solution of (1)–(4) is a pair \((\mathbf{u},\rho )\) such that \(\mathbf{u}\in \mathbf{H}_\sigma \) and \(\rho =\eta (N\mathbf{u})\), and satisfies

$$\begin{aligned}&2\mu (D(\mathbf{u}),D(\mathbf{v}))_2+2\alpha \mu \int _{\varGamma _3}\mathbf{u}\cdot \mathbf{v}+\langle \mathcal {B}(\mathbf{u},\mathbf{u},\mathbf{u}),\mathbf{v}\rangle _{\mathbf{V}_\sigma '} =\langle \mathcal {F}(\mathbf{u}),\mathbf{v}\rangle _{\mathbf{V}_\sigma '}, \end{aligned}$$
(16)
$$\begin{aligned}&\qquad \qquad \qquad \qquad \eta (N\mathbf{u})=\rho _0\ \text{ on } \varGamma _1,\quad \mathbf{u}=\mathbf{u}_{\mathbf{g}}\ \text{ on } \varGamma {\setminus }\varGamma _3, \end{aligned}$$
(17)

for all \(\mathbf{v}\in \mathbf{V}_\sigma .\)

Existence of Weak Solutions

To obtain the existence of a solution of (16)–(17), we make a change of variables to obtain an auxiliary problem with homogeneous conditions for the velocity on \(\varGamma {\setminus }\varGamma _3\). For this, it will be necessary to use the following result; and its proof can be revised in Mallea-Zepeda et al. (2017).

Lemma 3

Assume that \(\mathbf{u}_0\in \mathbf{H}^{1/2}_{00}(\varGamma _1)\), \(\mathbf{g}\in \mathbf{H}^{1/2}_{00}(\varGamma _2)\). For any \(\varepsilon >0\) there exists \(\mathbf{u}^\varepsilon \in \mathbf{H}_\sigma \) with \(\mathbf{u}^\varepsilon =\mathbf{g}\) on \(\varGamma _2\), \(\mathbf{u}^\varepsilon =\mathbf{u}_0\) on \(\varGamma _1\) and \(\mathbf{u}^\varepsilon =\mathbf{0}\) on \(\varGamma _3\) such that

$$\begin{aligned} |\langle \mathbf{v}\cdot \nabla \mathbf{u}^\varepsilon ,\mathbf{v}\rangle _{\mathbf{V}_\sigma '}|\le \varepsilon \Vert \mathbf{v}\Vert ^2_{\mathbf{V}_\sigma }\ \forall \mathbf{v}\in \mathbf{V}_\sigma . \end{aligned}$$
(18)

Following the arguments of (Illarionov 2001, Theorem 2.1) and using Lemma 3, we have the following result on existence of solution to system (16)–(17).

Theorem 1

Let \(\mathbf{f}\in \mathbf{L}^2(\varOmega )\), \(\mathbf{u}_0\in \mathbf{H}^{1/2}_{00}(\varGamma _1)\), \(\mathbf{g}\in \mathbf{H}^{1/2}_{00}(\varGamma _2)\) and \(\eta \in C^0(\mathbb {R})\). There exist \(\mathbf{u}\in \mathbf{H}_\sigma \) and \(\rho =\eta (N\mathbf{u})\) satisfying the system (16)–(17). Moreover, the function \(\mathbf{u}\) satisfies the estimate

$$\begin{aligned} \Vert \mathbf{u}\Vert _{\mathbf{H}_\sigma }\le C\delta (\Vert \eta \Vert _{L^\infty }\Vert \mathbf{f}\Vert +\Vert \mathbf{u}^\varepsilon \Vert _{\mathbf{H}_\sigma }+\Vert \eta \Vert _{L^\infty }\Vert \mathbf{u}^\varepsilon \Vert ^2_{\mathbf{H}_\sigma }), \end{aligned}$$
(19)

where \(\delta :=\frac{1}{2\mu -\varepsilon \Vert \eta \Vert _{L^\infty }}\) and C is a positive constant which depends only on \(\varOmega \).

The Optimal Control Problem

In this section, we establish the statement of the optimal bilinear control problem under study. We suppose that \(\mathcal {U}\subset \mathbf{L}^2(\varOmega )\) is nonempty closed and convex set. We consider data \(\mathbf{u}_0\in \mathbf{H}^{1/2}_{00}(\varGamma _1)\), \(\mathbf{g}\in \mathbf{H}^{1/2}_{00}(\varGamma _2)\) and \(\eta \in C^0(\mathbb {R})\), and the function \(\mathbf{f}\in \mathcal {U}\) describing the bilinear control for the dynamical equation in \(\varOmega \). For simplicity, here and in what follows, we will use the space \(\mathbb {H}:=\mathbf{H}_\sigma \times \mathcal {U},\) and we consider the objective functional \(J:\mathbb {H}\rightarrow \mathbb {R}\) defined by

$$\begin{aligned} J({\mathbf{u}}, \mathbf{f}):= \bar{J}(\mathbf{u})+\frac{\beta }{2}\int _\varOmega |\mathbf{f}|^2, \end{aligned}$$
(20)

where \(\bar{J}:\mathbf{H}_\sigma \rightarrow \mathbb {R}\) is a weakly lower semicontinuous functional, and the nonnegative real \(\beta \) measure the cost of the control. Examples of weakly lower semicontinuous functionals \(\bar{J}\) for which our results apply and are interesting from the physical point of view are the following:

$$\begin{aligned} \left\{ \begin{array}{llllllll} J_1(\mathbf{u})&{}=&{}\displaystyle \frac{1}{2}\int _\varOmega |\mathbf{u}-\mathbf{u}_d|^2, &{} &{}J_2(\mathbf{u})&{}=&{}\displaystyle \frac{1}{2}\int _\varOmega |\eta (N\mathbf{u})-\rho _d|^2,\\ J_3(\mathbf{u})&{}=&{}\displaystyle \frac{1}{2}\int _\varOmega |\mathrm{rot}\,\mathbf{u}|^2,&{} &{} J_4(\mathbf{u})&{}=&{}\displaystyle \frac{\mu }{2}\int _\varOmega |D(\mathbf{u})|^2. \end{array} \right. \end{aligned}$$
(21)

The previous functionals \(J_1\), \(J_2\) and \(J_3\) describe the deviation of the velocity of fluid from a given target velocity \(\mathbf{u}_d\), the deviation of the fluid density from a given density \(\rho _d\), the vorticity of the velocity field, respectively. The functional \(J_4\) is known in the literature as the dissipation function; it represents the rate at which heat energy is conducted into the fluid, or equivalently, the rate at which heat is generated by deformations of the velocity field (Gunzburger et al. 1992). It is known that for an aerodynamic body moving at uniform velocity, the main contribution to retardation is the friction force. Thus, the functional \(J_4\) describes the total resistance in a fluid due to viscous friction (see Alekseev 1998; Illarionov 2001). Obviously, the sum of two or more of the functionals given in (21) could also be considered.

Due to physical importance, mentioned above, we will do our study with \(\bar{J}=J_4\). Thus, we define the following constrained minimization problem related to system (16)–(17):

$$\begin{aligned} \left\{ [Arrayl]\begin{array}{lll} \text{ Find }\ ({\mathbf{u}}, \mathbf{f})\in \mathbb {H} \text{ such } \text{ that } \text{ the } \text{ functional } \\ J({\mathbf{u}}, \mathbf{f})=\dfrac{\mu }{2}\Vert D(\mathbf{u})\Vert ^2+\dfrac{\beta }{2}\Vert \mathbf{f}\Vert ^2_{\mathcal {U}}\\ \text{ is } \text{ minimized, } \text{ subject } \text{ to } (\mathbf{u},\mathbf{f}) \text{ satisfies } (16){-}(17). \end{array} \right. \end{aligned}$$
(22)

The set of admissible solutions of problem (22) is defined by

$$\begin{aligned} \mathcal {S}_{ad}=\{({\mathbf{u}}, \mathbf{f})\in \mathbb {H}\,:\, (\mathbf{u},\mathbf{f})\ \text{ satisfies }\ (16){-}(17)\}. \end{aligned}$$

Existence of Optimal Solution

In this subsection, we will prove the existence of an optimal solution for problem (22). First we introduce the concept of optimal solution for problem (22).

Definition 2

A pair \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\in \mathcal {S}_{ad}\) will be called optimal solution of problem (22) if

$$\begin{aligned} J({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})=\min _{(\mathbf{u},\mathbf{f})\in \mathcal {S}_{ad}} J(\mathbf{u},\mathbf{f}). \end{aligned}$$
(23)

Theorem 2

Under the conditions of Theorem 1. We assume that either \(\beta >0\) or \(\mathcal {U}\) is bounded in \(\mathbf{L}^2(\varOmega )\), then the optimal control problem (22) has at least one optimal solution \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\in \mathcal {S}_{ad}\).

Proof

From Theorem 1 we conclude that the set \(\mathcal {S}_{ad}\ne \emptyset \), and since the functional J is bounded below, there exists a minimizing sequence \(\{(\mathbf{u}^m,\mathbf{f}^m)\}_{m\ge 1}\subset \mathcal {S}_{ad}\), \(m\in \mathbb {N}\), such that

$$\begin{aligned} \lim _{m\rightarrow \infty } J(\mathbf{u}^m,\mathbf{f}^m)=\inf _{(\mathbf{u},\mathbf{f})\in \mathcal {S}_{ad}}J(\mathbf{u},\mathbf{f}). \end{aligned}$$

Also, by definition of admissible set \(\mathcal {S}_{ad}\), for each m, \((\mathbf{u}^m,\mathbf{f}^m)\) satisfies the system (16)–(17), that is

$$\begin{aligned}&2\mu (D(\mathbf{u}^m),D(\mathbf{v}))_2+2\alpha \mu \int _{\varGamma _3}{} \mathbf{u}^m\cdot \mathbf{v}+\langle \mathcal {B}(\mathbf{u}^m,\mathbf{u}^m,\mathbf{u}^m),\mathbf{v}\rangle _{\mathbf{V}_\sigma '}\nonumber \\&\quad =\langle \mathcal {F}^m(\mathbf{u}^m),\mathbf{v}\rangle _{\mathbf{V}_\sigma '}\ \forall \mathbf{v}\in \mathbf{V}_\sigma , \end{aligned}$$
(24)
$$\begin{aligned}&\eta (N\mathbf{u}^m)=\rho _0\ \text{ on } \varGamma _1,\quad \mathbf{u}^m=\mathbf{u}_{\mathbf{g}}\ \text{ on } \varGamma {\setminus }\varGamma _3, \end{aligned}$$
(25)

where \(\mathbf{u}_\mathbf{g}\) is defined in (2).

Now, since we assume that \(\beta >0\) or \(\mathcal {U}\) is bounded in \(\mathbf{L}^2(\varOmega )\), we conclude

$$\begin{aligned} \{\mathbf{f}^m\}_{m\ge 1} \text{ is } \text{ bounded } \text{ in } {} \mathbf{L}^2(\varOmega ) \end{aligned}$$
(26)

Also, from (19) we deduce that there exists a positive constant C, independent of m, such that

$$\begin{aligned} \Vert \mathbf{u}^m\Vert _{\mathbf{H}_\sigma }\le C. \end{aligned}$$
(27)

Therefore, from (26), (27) and taking into account that \(\mathcal {U}\subset \mathbf{L}^2(\varOmega )\) is a closed convex set (then is weakly closed in \(\mathbf{L}^2(\varOmega )\)), we have that there exists \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\in \mathbb {H}\) such that, for some subsequence of \(\{(\mathbf{u}^m,\mathbf{f}^m)\}_{m\ge 1}\), still denoted by \(\{(\mathbf{u}^m,\mathbf{f}^m)\}_{m\ge 1}\), the following convergences hold, as \(m\rightarrow \infty \)

$$\begin{aligned}&\mathbf{u}^m\rightarrow {\hat{\mathbf{u}}} \text{ weakly } \text{ in } \mathbf{H}_\sigma \text{ and } \text{ strongly } \text{ in } {} \mathbf{L}^2(\varOmega ), \end{aligned}$$
(28)
$$\begin{aligned}&\mathbf{f}^m\rightarrow {\hat{\mathbf{f}}} \text{ weakly } \text{ in } \mathbf{L}^2(\varOmega ), \text{ and } {\hat{\mathbf{f}}}\in \mathcal {U}. \end{aligned}$$
(29)

We can pass to the limit in (24)-(25) to determine that \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\) satisfies (16)-(17). Indeed, since \(\mathbf{u}^m=\mathbf{u}_0\) on \(\varGamma _1\), \(\mathbf{u}^m=\mathbf{g}\) on \(\varGamma _2\) and \(\eta (N\mathbf{u}^m)=\rho _0\) on \(\varGamma _1\) we deduce that \(\mathbf{u}^m=\mathbf{u}_\mathbf{g}\) on \(\varGamma {\setminus }\varGamma _3\) and \(\eta (N{\hat{\mathbf{u}}})=\rho _0\) on \(\varGamma _1\); thus \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\) satisfies (17). When one passes to the limit in (24), the only difficulties appear in the nonlinear terms \(\mathcal {B}(\mathbf{u}^m,\mathbf{u}^m,\mathbf{u}^m)\) and \(\mathcal {F}^m(\mathbf{u}^m)\). To control this problem, by using the Hölder inequality, for \(\mathbf{v}\in \mathbf{V}_\sigma \), we have

$$\begin{aligned}&|\langle \mathcal {B}(\mathbf{u}^m,\mathbf{u}^m,\mathbf{u}^m)-\mathcal {B}({\hat{\mathbf{u}}},{\hat{\mathbf{u}}},{\hat{\mathbf{u}}}) ,\mathbf{v} \rangle _{\mathbf{V}_\sigma '}|\nonumber \\&\quad \le |\langle (\eta (N\mathbf{u}^m)-\eta (N{\hat{\mathbf{u}}}))\mathbf{u}^m\cdot \nabla \mathbf{u}^m,\mathbf{v}\rangle _{\mathbf{V}_\sigma '}|+|\langle \eta (N{\hat{\mathbf{u}}})(\mathbf{u}^m-{\hat{\mathbf{u}}})\cdot \nabla \mathbf{u}^m,\mathbf{v}\rangle _{\mathbf{V}_\sigma '}|\nonumber \\&\qquad +|\langle \eta (N{\hat{\mathbf{u}}}){\hat{\mathbf{u}}}\cdot (\nabla \mathbf{u}^m-\nabla {\hat{\mathbf{u}}}),\mathbf{v}\rangle _{\mathbf{V}_\sigma '}|\nonumber \\&\quad \le \Vert \eta (N\mathbf{u}^m)-\eta (N{\hat{\mathbf{u}}})\Vert _{L^\infty }\Vert \mathbf{u}^m\Vert _{L^4}\Vert \nabla \mathbf{u}^m\Vert \Vert \mathbf{v}\Vert _{L^4}\nonumber \\&\qquad +\Vert \eta (N{\hat{\mathbf{u}}})\Vert _{L^\infty }\Vert \mathbf{u}^m-{\hat{\mathbf{u}}}\Vert _{L^4}\Vert \nabla \mathbf{u}^m\Vert \Vert \mathbf{v}\Vert _{L^4}\nonumber \\&\qquad +\left| \sum ^2_{i,j=1}(\partial _i u^m_j-\partial _i\hat{u}_j,\eta (N{\hat{\mathbf{u}}})\hat{u}_iv_j)\right| , \end{aligned}$$
(30)

and

$$\begin{aligned} |\langle \mathcal {F}^m(\mathbf{u}^m)-\hat{\mathcal {F}}({\hat{\mathbf{u}}}),\mathbf{v}\rangle _{\mathbf{V}_\sigma '}|\le & {} |\langle (\eta (N\mathbf{u}^m)-\eta (N{\hat{\mathbf{u}}}))\mathbf{f}^m,\mathbf{v}\rangle _{\mathbf{V}_\sigma '}|\nonumber \\&+|\langle \eta (N{\hat{\mathbf{u}}})(\mathbf{f}^m-{\hat{\mathbf{f}}}),\mathbf{v}\rangle _{\mathbf{V}_\sigma '}|\nonumber \\\le & {} \Vert \eta (N\mathbf{u}^m)-\eta (N{\hat{\mathbf{u}}})\Vert _{L^\infty }|\langle \mathbf{f}^m,\mathbf{v}\rangle |\nonumber \\&+\Vert \eta (N{\hat{\mathbf{u}}})\Vert _{L^\infty }|\langle \mathbf{f}^m-{\hat{\mathbf{f}}},\mathbf{v}\rangle | \end{aligned}$$
(31)

Now, since the operator \(N:\mathbf{V}_\sigma \rightarrow H^2(\varOmega )\) is continuous (cf. Illarionov 2001, Lemma 2.1) and taking into account the compact injection \(H^2(\varOmega )\hookrightarrow C^0(\overline{\varOmega })\), we have

$$\begin{aligned} \eta (N\mathbf{u}^m)\rightarrow \eta (N{\hat{\mathbf{u}}})\ \text{ strongly } \text{ in } \ C^0(\overline{\varOmega }). \end{aligned}$$
(32)

Moreover, since \(\Vert \eta (N{\hat{\mathbf{u}}})\tilde{u}_i{ v}_j\Vert \le \Vert \eta (N{\hat{\mathbf{u}}})\Vert _{L^\infty }\Vert \hat{ u}_i\Vert _4\Vert { v}_j\Vert _4<\infty ,\) for each \(i,j=1,2\), then using standard Sobolev embeddings we obtain

$$\begin{aligned} \partial _i{ u}_j^m\rightarrow \partial _i\hat{ u}_j\ \text{ weakly } \text{ in } L^2(\varOmega ). \end{aligned}$$
(33)

Thus, from (28)–(33) we deduce that

$$\begin{aligned} \langle \mathcal {B}(\mathbf{u}^m,\mathbf{u}^m,\mathbf{u}^m),\mathbf{v}\rangle _{\mathbf{V}_\sigma '}\rightarrow & {} \langle \mathcal {B}({\hat{\mathbf{u}}},{\hat{\mathbf{u}}},{\hat{\mathbf{u}}}),\mathbf{v}\rangle _{\mathbf{V}_\sigma '}\ \forall \mathbf{v}\in \mathbf{V}_\sigma , \end{aligned}$$
(34)
$$\begin{aligned} \langle \mathcal {F}^m(\mathbf{u}^m),\mathbf{v}\rangle _{\mathbf{V}_\sigma '}\rightarrow & {} \langle \hat{\mathcal {F}}({\hat{\mathbf{u}}}),\mathbf{v}\rangle _{\mathbf{V}_\sigma '}\ \forall \mathbf{v}\in \mathbf{V}_\sigma . \end{aligned}$$
(35)

Therefore, from convergences (28), (29), (34) and (35), taking the limit in (24)-(25) as m goes to \(\infty \), we obtain that \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\) is solution of the system (16)-(17). Consequently, we have \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\in \mathcal {S}_{ad}\) and

$$\begin{aligned} \lim _{m\rightarrow \infty }J(\mathbf{u}^m,\mathbf{f}^m)=\inf _{(\mathbf{u},\mathbf{f})\in \mathcal {S}_{ad}}J(\mathbf{u},\mathbf{f})\le J({\hat{\mathbf{u}}},{\hat{\mathbf{f}}}). \end{aligned}$$
(36)

Also, since the functional J is weakly lower semicontinuous on \(\mathcal {S}_{ad}\), we have that \(J({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\le \lim _{m\rightarrow \infty }\inf J(\mathbf{u}^m,\mathbf{f}^m)\), which jointly to (36), implies (23).\(\square \)

Optimality Conditions

This section is devoted to obtaining an optimality system to optimal control problem (22). We shall use the method of Lagrange multipliers to turn the constrained optimization problem (22) into an unconstrained one.

Since a solution of the system (16)–(17) is not necessarily isolated, a characterization of an optimal control cannot be obtained directly. To overcome this difficulty, we use a penalty method proposed by Lions (1971). This method consists in introducing a family of problems \((P)_{\varepsilon }\), \(\varepsilon >0,\) whose solutions converge towards a solution to problem (22); then we derive the optimality conditions for the problems \((P)_\varepsilon \), and finally we pass to the limit in these optimality conditions.

Penalized Problem

For an optimal solution \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\in \mathcal {S}_{ad}\) of the optimal control problem (22) we consider the following family of auxiliary extremal problems depending on \(\varepsilon >0\):

$$\begin{aligned} \min _{(\mathbf{u},\mathbf{f})\in \mathbb {H}} J_{\varepsilon }(\mathbf{u}, \mathbf{f}), \end{aligned}$$
(37)

where the functional \(J_\varepsilon : \mathbb {H} \rightarrow \mathbb {R}\) is defined by

$$\begin{aligned} J_{\varepsilon }(\mathbf{u}, \mathbf{f})= & {} J(\mathbf{u}, \mathbf{f}) +\frac{1}{2}\Vert \mathbf{u}-{\hat{\mathbf{u}}}\Vert ^2_{\mathbf{H}_\sigma } +\frac{1}{2}\Vert \mathbf{f}-{\hat{\mathbf{f}}}\Vert ^2_{\mathcal {U}}\nonumber \\&+\frac{1}{2\varepsilon }\Vert \mu \mathcal {A}\mathbf{u}+\mathcal {B}(\mathbf{u},\mathbf{u},\mathbf{u})-\mathcal {F}(\mathbf{u})\Vert ^2_{\mathbf{V}'_\sigma } +\frac{1}{2\varepsilon }\Vert \mathbf{u}-\mathbf{u}_{\mathbf{g}}\Vert ^2_{\mathbf{H}^{1/2}(\varGamma {\setminus }\varGamma _3)}, \end{aligned}$$
(38)

with the functional \(J(\mathbf{u}, \mathbf{f})\) defined in (20), the operators \(\mathcal {B}\) and \(\mathcal {F}\) are given in (14), and the operator \(\mathcal {A}:\mathbf{V}_\sigma \rightarrow \mathbf{V}'_\sigma \) is defined by

$$\begin{aligned} \langle \mathcal {A}{} \mathbf{u},\mathbf{v}\rangle _{\mathbf{V}_\sigma '}=2(D(\mathbf{u}),D(\mathbf{v}))_2+2\alpha \int _{\varGamma _3}{} \mathbf{u}\cdot \mathbf{v}\quad \forall \mathbf{v}\in \mathbf{V}_\sigma . \end{aligned}$$
(39)

Remark 2

Denoting by \(\tilde{\mathcal {A}}:\mathbf{V}_\sigma \rightarrow \mathbf{V}'_\sigma \) the restriction of the operator \(\mathcal {A}\) to subspace \(\mathbf{V}_\sigma \subset \mathbf{H}_\sigma \), by Lax–Milgram Lemma we deduce that \(\tilde{\mathcal {A}}\) is an invertible operator with continuous inverse \(\tilde{\mathcal {A}}^{-1}:\mathbf{V}'_\sigma \rightarrow \mathbf{V}_\sigma \). Then, we can consider the following inner-product in \(\mathbf{V}'_\sigma \), given by \((\mathbf{u},\mathbf{v}):=\langle \tilde{\mathcal {A}}^{-1}{} \mathbf{u},\mathbf{v}\rangle \) for any \(\mathbf{u}, \mathbf{v}\in \mathbf{V}'_\sigma \). The norm \(\Vert \cdot \Vert _{\mathbf{V}'_\sigma }\) appearing in (38) is the norm associated with such inner-product.

Also, the terms \(\Vert \mathbf{u}-{\hat{\mathbf{u}}}\Vert ^2_{\mathbf{H}_\sigma }\) and \(\Vert \mathbf{f}-{\hat{\mathbf{f}}}\Vert ^2_{\mathcal {U}}\) in (38) are necessary because they help to obtain strong convergence of \((\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\) to \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\), where \((\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\) is a solution of problem (37); thus we can pass to the limit in the system (54) and (55) below.

Since the functional \(J_\varepsilon \) is weakly lower semicontinuous on \(\mathbb {H}\), then following the arguments of the proof of Theorem 2 we deduce the following result.

Lemma 4

Under conditions of Theorem 2, there exists at least an optimal solution of problem (37); that is, for each \(\varepsilon >0\), there exists \((\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\in \mathbb {H}\) such that

$$\begin{aligned} J_\varepsilon (\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )=\min _{(\mathbf{u},\mathbf{f})\in \mathbb {H}} J_\varepsilon (\mathbf{u},\mathbf{f}). \end{aligned}$$

Theorem 3

Assume the hypotheses of Theorem 2; let \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\in \mathcal {S}_{ad}\) be an optimal solution of the optimal control problem (22) and \((\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\in \mathbb {H}\) be as Lemma 4. Then, as \(\varepsilon \rightarrow 0\), there hold the following convergences

$$\begin{aligned}&\mathbf{u}^\varepsilon \rightarrow {\hat{\mathbf{u}}} \text{ strongly } \text{ in } {} \mathbf{H}_\sigma , \end{aligned}$$
(40)
$$\begin{aligned}&\mathbf{f}^\varepsilon \rightarrow {\hat{\mathbf{f}}} \text{ strongly } \text{ in } \mathcal {U},\end{aligned}$$
(41)
$$\begin{aligned}&J_\varepsilon (\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\rightarrow J({\hat{\mathbf{u}}},{\hat{\mathbf{f}}}). \end{aligned}$$
(42)

Proof

Since \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\in \mathcal {S}_{ad}\subset \mathbb {H}\) and the functional \(J_\varepsilon \) attains its minimum in \((\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\in \mathbb {H}\), we have

$$\begin{aligned} J_\varepsilon (\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\le J_\varepsilon ({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})=J({\hat{\mathbf{u}}},{\hat{\mathbf{f}}}). \end{aligned}$$
(43)

Now, from (38) and (43), we obtain

$$\begin{aligned}&\frac{1}{2}\Vert \mathbf{u}^\varepsilon -{\hat{\mathbf{u}}}\Vert ^2_{\mathbf{H}_\sigma } +\frac{1}{2}\Vert \mathbf{f}^\varepsilon -{\hat{\mathbf{f}}}\Vert ^2_{\mathcal {U}} +\frac{1}{2\varepsilon }\Vert \mu \mathcal {A}\mathbf{u}^\varepsilon +\mathcal {B}(\mathbf{u}^\varepsilon ,\mathbf{u}^\varepsilon ,\mathbf{u}^\varepsilon )-\mathcal {F}^\varepsilon (\mathbf{u}^\varepsilon )\Vert ^2_{\mathbf{V}'_\sigma }\nonumber \\&\quad +\frac{1}{2\varepsilon }\Vert \mathbf{u}^\varepsilon -\mathbf{u}_{\mathbf{g}^\varepsilon }\Vert ^2_{\mathbf{H}^{1/2}(\varGamma {\setminus }\varGamma _3)} \le J_\varepsilon (\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\le J({\hat{\mathbf{u}}},{\hat{\mathbf{f}}}). \end{aligned}$$
(44)

Then, from (43) and (44) we have deduce

$$\begin{aligned} \Vert \mathbf{u}^\varepsilon \Vert ^2_{\mathbf{H}_\sigma }+\Vert \mathbf{f}^\varepsilon \Vert ^2_{\mathcal {U}}\le & {} 2(\Vert \mathbf{u}^\varepsilon -{\hat{\mathbf{u}}}\Vert ^2_{\mathbf{H}_\sigma }+\Vert \mathbf{f}^\varepsilon -{\hat{\mathbf{f}}}\Vert ^2_{\mathcal {U}} +\Vert {\hat{\mathbf{u}}}\Vert ^2_{\mathbf{H}_\sigma }+\Vert {\hat{\mathbf{f}}}\Vert ^2_{\mathcal {U}})\nonumber \\\le & {} 4J({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})+2(\Vert {\hat{\mathbf{u}}}\Vert ^2_{\mathbf{H}_\sigma }+\Vert {\hat{\mathbf{f}}}\Vert ^2_{\mathcal {U}})\le C, \end{aligned}$$
(45)

where C is a constant independent of \(\varepsilon \).

Thus, (45) implies that there exists \((\bar{\mathbf{u}},\bar{\mathbf{f}})\in \mathbb {H}\) and a subsequence of \(\{(\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\}_{\varepsilon >0}\), still denoted by \(\{(\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\}_{\varepsilon >0}\), such that, as \(\varepsilon \rightarrow 0\), the following convergence holds

$$\begin{aligned} (\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\rightarrow (\bar{\mathbf{u}},\bar{\mathbf{f}}) \text{ weakly } \text{ in } \mathbb {H}. \end{aligned}$$
(46)

Moreover, from (20), (38) and (43) we have

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0}\frac{1}{2}\left( \Vert \mathbf{u}^\varepsilon -\bar{\mathbf{u}}\Vert ^2_{\mathbf{H}_\sigma }+\Vert \mathbf{f}-\bar{\mathbf{f}}\Vert ^2_{\mathcal {U}}\right)\le & {} \limsup _{\varepsilon \rightarrow 0}(J_\varepsilon (\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )-J(\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon ))\nonumber \\\le & {} \limsup _{\varepsilon \rightarrow 0}J_\varepsilon (\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )-\liminf _{\varepsilon \rightarrow 0}J(\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\nonumber \\\le & {} C. \end{aligned}$$
(47)

Thus, from (47) we have

$$\begin{aligned} (\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\rightarrow (\bar{\mathbf{u}},\bar{\mathbf{f}}) \text{ strongly } \text{ in } \mathbb {H}. \end{aligned}$$
(48)

Also, estimate (44) implies that

$$\begin{aligned} \Vert \mu \mathcal {A}{} \mathbf{u}^\varepsilon +\mathcal {B}(\mathbf{u}^\varepsilon ,\mathbf{u}^\varepsilon ,\mathbf{u}^\varepsilon )-\mathcal {F}^\varepsilon (\mathbf{u}^\varepsilon )\Vert ^2_{\mathbf{V}'_\sigma } \rightarrow 2\varepsilon J({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\rightarrow 0,\ \text{ as } \varepsilon \rightarrow 0. \end{aligned}$$
(49)

Using (46), (49), and following the same arguments as in the proof of Theorem 2 we obtain that \((\bar{\mathbf{u}},\bar{\mathbf{f}})\) satisfies (16) and (17), which implies that \((\bar{\mathbf{u}},\bar{\mathbf{f}})\in \mathcal {S}_{ad}\). Now, we observe that \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\in \mathcal {S}_{ad}\) is the unique minimal point of the functional

$$\begin{aligned} \hat{J}(\mathbf{u},\mathbf{f}):=J(\mathbf{u},\mathbf{f})+\frac{1}{2}\Vert \mathbf{u}-{\hat{\mathbf{u}}}\Vert ^2_{\mathbf{H}_\sigma }+\frac{1}{2}\Vert \mathbf{f}-{\hat{\mathbf{f}}}\Vert ^2_{\mathcal {U}} \end{aligned}$$

in \(\mathcal {S}_{ad}\). Then, from (20), (38) and (43) we obtain \( \hat{J}(\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\le J_\varepsilon (\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\le J({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})=\hat{J}({\hat{\mathbf{u}}},{\hat{\mathbf{f}}}), \) and taking into account that \(\hat{J}\) is weakly lower semicontinuous on \(\mathcal {S}_{ad}\), then from (46), for any \((\mathbf{u},\mathbf{f})\in \mathcal {S}_{ad}\), we have

$$\begin{aligned} \hat{J}(\bar{\mathbf{u}},\bar{\mathbf{f}})\le \liminf _{\varepsilon \rightarrow 0}\hat{J}(\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon ) \le \hat{J}({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\le \hat{J}(\mathbf{u},\mathbf{f}), \end{aligned}$$

which implies that \((\bar{\mathbf{u}},\bar{\mathbf{f}})\) is a minimal point of \(\hat{J}\) in \(\mathcal {S}_{ad}\). Since \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\in \mathcal {S}_{ad}\) is the unique minimal point of \(\hat{J}\) in \(\mathcal {S}_{ad}\), we conclude that \((\bar{\mathbf{u}},\bar{\mathbf{f}})=({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\); which, jointly to (48), implies (40) and (41).

Finally, from (20), (38) and (43) we obtain \(J(\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\le J_\varepsilon (\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\le J({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\). Then, taking into account that J is weakly lower semicontinuous on \(\mathcal {S}_{ad}\), we obtain

$$\begin{aligned} J({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\le \liminf _{\varepsilon \rightarrow 0}J(\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\le \liminf _{\varepsilon \rightarrow 0}J_\varepsilon (\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon ) \le \limsup _{\varepsilon \rightarrow 0}J_\varepsilon (\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\le J({\hat{\mathbf{u}}},{\hat{\mathbf{f}}}), \end{aligned}$$

which implies \(\lim _{\varepsilon \rightarrow 0}J_\varepsilon (\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )=J({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\), and (42) follow.\(\square \)

In the following result we obtain an optimality system associated to problem (22).

Theorem 4

Let \(\mathbf{u}_0\in \mathbf{H}^{1/2}_{00}(\varGamma _1)\), \(\mathbf{g}\in \mathbf{H}^{1/2}_{00}(\varGamma _2)\) and \(\eta \in C^1(\mathbb {R})\). Then, for any optimal solution \(({\hat{\mathbf{u}}},{\hat{\mathbf{f}}})\in \mathcal {S}_{ad}\) of control problem (22) there exist Lagrange multipliers, not all zero, \((\lambda _0,\varvec{\lambda },\varvec{\xi })\in (\mathbb {R}^+\cup \{0\})\times \mathbf{V}_\sigma \times \mathbf{H}^{-1/2}_{00}(\varGamma {\setminus }\varGamma _3)\) satisfying the adjoint system

$$\begin{aligned}&\mu \langle \mathcal {A}\lambda ,\mathbf{v}\rangle _{\mathbf{V}_\sigma '}+\langle \eta '(N{\hat{\mathbf{u}}})(N\mathbf{v}){\hat{\mathbf{u}}} \cdot \nabla {\hat{\mathbf{u}}},\varvec{\lambda }\rangle _{\mathbf{V}_\sigma '} +\langle \eta (N{\hat{\mathbf{u}}})({\hat{\mathbf{u}}}\cdot \nabla \mathbf{v} +\mathbf{v}\cdot \nabla {\hat{\mathbf{u}}}),\varvec{\lambda }\rangle _{\mathbf{V}_\sigma '}\nonumber \\&\quad +\lambda _0\mu (D({\hat{\mathbf{u}}}),D(\mathbf{v}))_2 +\langle \varvec{\xi },\mathbf{v}\rangle _{\varGamma {\setminus }\varGamma _3} =\langle \eta '(N{\hat{\mathbf{u}}})(N\mathbf{v}){\hat{\mathbf{f}}},\varvec{\lambda }\rangle _{\mathbf{V}_\sigma '}, \end{aligned}$$
(50)

and the optimality condition

$$\begin{aligned} \int _{\varOmega }(\lambda _0\beta {\hat{\mathbf{f}}}-\varvec{\lambda }\eta (N{\hat{\mathbf{u}}}))(\mathbf{f}-{\hat{\mathbf{f}}})\ge 0, \end{aligned}$$
(51)

for all \((\mathbf{v},\mathbf{f})\in \mathbf{H}_\sigma \times \mathcal {U}\). Here, \(\eta '\) denotes the first derivative of \(\eta \).

Proof

Let \((\mathbf{u}^\varepsilon ,\mathbf{f}^\varepsilon )\in \mathbb {H}\) be a solution of problem (37) and \((\mathbf{v},\mathbf{f})\in \mathbb {H}\). For each \((\theta _1,\theta _2)\in [0,1]\times [0,1]\) we define the real function

$$\begin{aligned} \mathcal {H}(\theta _1,\theta _2)=J_\varepsilon (\mathbf{u}^\varepsilon +\theta _1\mathbf{v},\mathbf{f}^\varepsilon +\theta _2(\mathbf{f}-\mathbf{f}^\varepsilon )). \end{aligned}$$
(52)

Since \(\mathcal {H}\) attaints its minimum at (0, 0) and \(\mathcal {U}\) is a convex set, we obtain

$$\begin{aligned} \frac{\partial \mathcal {H}}{\partial \theta _1}(0,0)=0,\quad \frac{\partial \mathcal {H}}{\partial \theta _2}(0,0)\ge 0. \end{aligned}$$
(53)

Moreover, from Remark 2, we can consider the equivalent norms \(\Vert \varvec{\varphi }\Vert _{\mathbf{V}'_\sigma }\equiv \Vert \nabla (\tilde{\mathcal {A}}^{-1}\varvec{\varphi })\Vert \); thus, from (53) we deduce the optimality system for problem (37)

$$\begin{aligned}&\mu \langle \mathcal {A}\varvec{\lambda }^\varepsilon ,\mathbf{v}\rangle _{\mathbf{V}_\sigma '} +\langle \eta '(N\mathbf{u}^\varepsilon )(N\mathbf{v})\mathbf{u}^\varepsilon \cdot \nabla \mathbf{u}^\varepsilon ,\varvec{\lambda }^\varepsilon \rangle _{\mathbf{V}_\sigma '}+(\mathbf{u}^\varepsilon -{\hat{\mathbf{u}}},\mathbf{v})_{\mathbf{H}_\sigma }\nonumber \\&\quad +\langle \eta (N\mathbf{u}^\varepsilon )(\mathbf{u}^\varepsilon \cdot \nabla \mathbf{v}+\mathbf{v}\cdot \nabla \mathbf{u}^\varepsilon ),\varvec{\lambda }^\varepsilon \rangle _{\mathbf{V}_\sigma '}+\mu (D(\mathbf{u}^\varepsilon ),D(\mathbf{v}))_2+\langle \varvec{\xi }^\varepsilon ,\mathbf{v}\rangle _{\varGamma {\setminus }\varGamma _3}\nonumber \\&\quad =\langle \eta '(N\mathbf{u}^\varepsilon )(N\mathbf{v})\mathbf{f}^\varepsilon ,\varvec{\lambda }^\varepsilon \rangle _{\mathbf{V}_\sigma '}\ \forall \mathbf{v}\in \mathbf{H}_\sigma , \end{aligned}$$
(54)
$$\begin{aligned}&\qquad \beta \int _\varOmega \mathbf{f}^\varepsilon (\mathbf{f}-\mathbf{f}^\varepsilon )+\int _\varOmega (\mathbf{f}^\varepsilon -{\hat{\mathbf{f}}})(\mathbf{f}-\mathbf{f}^\varepsilon )-\int _\varOmega \varvec{\lambda }^\varepsilon \eta (N\mathbf{u}^\varepsilon )(\mathbf{f}-\mathbf{f}^\varepsilon )\ge 0 \ \forall \mathbf{f}\in \mathcal {U},\nonumber \\ \end{aligned}$$
(55)

where

$$\begin{aligned} \left\{ \begin{array}{rcl} \varvec{\lambda }^\varepsilon &{}:=&{}\dfrac{1}{\varepsilon }\tilde{\mathcal {A}}^{-1}(\mu \mathcal {A}{} \mathbf{u}^\varepsilon +\mathcal {B}(\mathbf{u}^\varepsilon ,\mathbf{u}^\varepsilon ,\mathbf{u}^\varepsilon )-\mathcal {F}^\varepsilon (\mathbf{u}^\varepsilon )) \in \mathbf{V}_\sigma ,\\ \\ \varvec{\xi }^\varepsilon &{}:=&{}\dfrac{1}{\varepsilon }(\mathbf{u}^{\varepsilon }_{|_{\varGamma {\setminus }\varGamma _3}}-\mathbf{u}_\mathbf{g})\in \mathbf{L}^2(\varGamma {\setminus }\varGamma _3)\subset \mathbf{H}^{-1/2}_{00}(\varGamma {\setminus }\varGamma _3). \end{array} \right. \end{aligned}$$
(56)

Also, from (54) we easily infer that there exists a positive constant K, independently of \(\varepsilon \), such that

$$\begin{aligned} \Vert \varvec{\xi }^\varepsilon \Vert _{\mathbf{H}^{-1/2}_{00}(\varGamma {\setminus }\varGamma _3)}\le K+K\Vert \varvec{\lambda }^\varepsilon \Vert _{\mathbf{V}_\sigma }. \end{aligned}$$
(57)

From estimate (57) we have two possibilities:

A):

\(I:=\Vert \varvec{\lambda }^\varepsilon \Vert _{\mathbf{V}_\sigma }\le C\), with C independent of \(\varepsilon \).

In this case, we conclude that the sequence \(\{(\varvec{\lambda }^\varepsilon ,\varvec{\xi }^\varepsilon )\}_{\varepsilon >0}\) is bounded in \(\mathbf{V}_\sigma \times \mathbf{H}^{-1/2}_{00}(\varGamma {\setminus }\varGamma _3)\). Then, there exists \((\varvec{\lambda },\varvec{\xi })\in \mathbf{V}_\sigma \times \mathbf{H}^{-1/2}_{00}(\varGamma {\setminus }\varGamma _3)\) and a subsequence of \(\{(\varvec{\lambda }^\varepsilon ,\varvec{\xi }^\varepsilon )\}_{\varepsilon >0}\), which for simplicity still we denote by \(\{(\varvec{\lambda }^\varepsilon ,\varvec{\xi }^\varepsilon )\}_{\varepsilon >0}\), such that, as \(\varepsilon \rightarrow 0\), the following convergences hold

$$\begin{aligned} \varvec{\lambda }^\varepsilon\rightarrow & {} \varvec{\lambda } \text{ weakly } \text{ in } {} \mathbf{V}_\sigma \text{ and } \text{ strongly } \text{ in } {} \mathbf{L}^2(\varOmega ), \end{aligned}$$
(58)
$$\begin{aligned} \langle \varvec{\xi }^\varepsilon ,\mathbf{v}\rangle _{\varGamma {\setminus }\varGamma _3}\rightarrow & {} \langle \varvec{\xi },\mathbf{v}\rangle _{\varGamma {\setminus }\varGamma _3}\ \forall \mathbf{v}\in \mathbf{H}_\sigma . \end{aligned}$$
(59)

Therefore, from (58), (59), and taking into account (40) and (41), passing to the limit in (54) and (55), as \(\varepsilon \rightarrow 0\), we obtain (50), (51) with \(\lambda _0=1\).

B):

\(I:=\Vert \varvec{\lambda }^\varepsilon \Vert _{\mathbf{V}_\sigma }\rightarrow \infty \), as \(\varepsilon \rightarrow 0\).

We denote

$$\begin{aligned} \hat{\varvec{\lambda }}^\varepsilon :=\frac{{\varvec{\lambda }}^\varepsilon }{I},\quad \hat{\varvec{\xi }}^\varepsilon :=\frac{{\varvec{\xi }}^\varepsilon }{I}. \end{aligned}$$
(60)

Then, from (57) and (60) we deduce that

$$\begin{aligned} \Vert \hat{\varvec{\lambda }}^\varepsilon \Vert _{\mathbf{V}_\sigma }= & {} 1\ \forall \varepsilon >0, \end{aligned}$$
(61)
$$\begin{aligned} \Vert \hat{\varvec{\xi }}^\varepsilon \Vert _{\mathbf{H}^{-1/2}_{00}(\varGamma {\setminus }\varGamma _3)}\le & {} \frac{K}{I}+K\le \widehat{K}. \end{aligned}$$
(62)

Thus, (61) and (62) implies that the sequence \(\{(\hat{\varvec{\lambda }}^\varepsilon ,\hat{\varvec{\xi }}^\varepsilon )\}_{\varepsilon >0}\) is bounded in \(\mathbf{V}_\sigma \times \mathbf{H}^{-1/2}_{00}(\varGamma {\setminus }\varGamma _3)\), then there exists \(({\varvec{\lambda }},{\varvec{\xi }})\in \mathbf{V}_\sigma \times \mathbf{H}^{-1/2}_{00}(\varGamma {\setminus }\varGamma _3)\) and a subsequence of \(\{(\hat{\varvec{\lambda }}^\varepsilon ,\hat{\varvec{\xi }}^\varepsilon )\}_{\varepsilon >0}\), still denoted by \(\{(\hat{\varvec{\lambda }}^\varepsilon ,\hat{\varvec{\xi }}^\varepsilon )\}_{\varepsilon >0}\), such that

$$\begin{aligned} \left\{ \begin{array}{lll} \hat{\varvec{\lambda }}^\varepsilon &{}\rightarrow {\varvec{\lambda }} \text{ weakly } \text{ in } {} \mathbf{V}_\sigma \text{ and } \text{ strongly } \text{ in } {} \mathbf{L}^2(\varOmega ),\\ \langle \hat{\varvec{\xi }}^\varepsilon ,\mathbf{v}\rangle _{\varGamma {\setminus }\varGamma _3}&{}\rightarrow \langle {\varvec{\xi }},\mathbf{v}\rangle _{\varGamma {\setminus }\varGamma _3}\ \forall \mathbf{v}\in \mathbf{H}_\sigma , \end{array} \right. \end{aligned}$$
(63)

as \(\varepsilon \rightarrow 0\). Multiplying the terms of (54) and (55) by \(\frac{1}{I}\) we obtain

$$\begin{aligned}&\mu \langle \mathcal {A}\hat{\varvec{\lambda }}^\varepsilon ,\mathbf{v}\rangle _{\mathbf{V}_\sigma '}+\langle \eta '(N\mathbf{u}^\varepsilon )(N\mathbf{v})\mathbf{u}^\varepsilon \cdot \nabla \mathbf{u}^\varepsilon ,\hat{\varvec{\lambda }}^\varepsilon \rangle _{\mathbf{V}_\sigma '} +\frac{1}{I}(\mathbf{u}^\varepsilon -{\hat{\mathbf{u}}},\mathbf{v})_{\mathbf{H}_\sigma }\nonumber \\&\qquad +\langle \eta (N\mathbf{u}^\varepsilon )(\mathbf{u}^\varepsilon \cdot \nabla \mathbf{v}+\mathbf{v}\cdot \nabla \mathbf{u}^\varepsilon ),\hat{\varvec{\lambda }}^\varepsilon \rangle _{\mathbf{V}_\sigma '}+\frac{\mu }{I}(D(\mathbf{u}^\varepsilon ),D(\mathbf{v}))_2+\langle \hat{\varvec{\xi }}^\varepsilon ,\mathbf{v}\rangle _{\varGamma {\setminus }\varGamma _3}\nonumber \\&\quad =\langle \eta '(N\mathbf{u}^\varepsilon )(N\mathbf{v})\mathbf{f}^\varepsilon ,\hat{\varvec{\lambda }}^\varepsilon \rangle _{\mathbf{V}_\sigma '}\ \forall \mathbf{v}\in \mathbf{H}_\sigma , \end{aligned}$$
(64)
$$\begin{aligned}&\frac{\beta }{I}\int _\varOmega \mathbf{f}^\varepsilon (\mathbf{f}-\mathbf{f}^\varepsilon )+\frac{1}{I}\int _\varOmega (\mathbf{f}^\varepsilon -{\hat{\mathbf{f}}})(\mathbf{f}-\mathbf{f}^\varepsilon )-\int _\varOmega \hat{\varvec{\lambda }}^\varepsilon \eta (N\mathbf{u}^\varepsilon )(\mathbf{f}-\mathbf{f}^\varepsilon )\ge 0\, \forall \mathbf{f}\in \mathcal {U}.\nonumber \\ \end{aligned}$$
(65)

Therefore, using the convergences given in (63), and taking into account (40) and (41), passing to the limit in (64) and (65), as \(\varepsilon \rightarrow 0\), we obtain (50) and (51) with \(\lambda _0=0\).

We have to verify that \((\varvec{\lambda },\varvec{\xi })\ne (\mathbf{0},\mathbf{0})\). In fact, suppose that \(\varvec{\lambda }=\mathbf{0}\), then replacing \(\mathbf{v}=\hat{\varvec{\lambda }}^\varepsilon \) in (64) and passing to the limit as \(\varepsilon \rightarrow 0\), we can obtain

$$\begin{aligned} \mu \Vert \hat{\varvec{\lambda }}^\varepsilon \Vert ^2_{\mathbf{V}_\sigma }+\alpha \Vert \hat{\varvec{\lambda }}^\varepsilon \Vert ^2_{\mathbf{L}^2(\varGamma _3)}\rightarrow 0, \end{aligned}$$

and, then \(\Vert \hat{\varvec{\lambda }}^\varepsilon \Vert ^2_{\mathbf{V}_\sigma }\rightarrow 0\), which contradicts (61). Therefore \((0,\varvec{\lambda },\varvec{\xi })\ne (0,\mathbf{0},\mathbf{0})\), and so the proof of theorem is complete.\(\square \)

Corollary 1

Under of conditions of Theorem 4. If \(\lambda _0=1\), then the optimal control \({\hat{\mathbf{f}}}\) is characterized as the projection of \(\frac{\varvec{\lambda }}{\eta (N{\hat{\mathbf{u}}})\beta }\) onto \(\mathcal {U}\), that is,

$$\begin{aligned} {\hat{\mathbf{f}}}=\mathop {Proj}\limits _{\mathcal {U}} \left( \frac{\varvec{\lambda }}{\eta (N{\hat{\mathbf{u}}})\beta }\right) . \end{aligned}$$
(66)

Proof

Since \(\eta (N{\hat{\mathbf{u}}})>0\), then the optimality condition (51) is equivalent to

$$\begin{aligned} \int _\varOmega \frac{\beta }{\eta (N{\hat{\mathbf{u}}})}({\hat{\mathbf{f}}}-\varvec{\lambda })(\mathbf{f}-{\hat{\mathbf{f}}})\ge 0. \end{aligned}$$
(67)

Then since the set of controls \(\mathcal {U}\) is closed and convex, from (67) and (Brézis 2011, Theorem 5.2, p.132) we deduce (66).\(\square \)

References

  1. Abergel, C., Casas, F.: Some optimal control problems of multistate equations appearing in fluid mechanics. Math. Model. Numer. Anal. 27, 223–247 (1993)

    MathSciNet  Article  Google Scholar 

  2. Alekseev, G.V.: Solvability of stationary problems of boundary control for thermal convection equations. Sib. Math. J. 39(5), 844–858 (1998)

    MathSciNet  Article  Google Scholar 

  3. Ammar-Khodja, F., Santos, M.M.: 2D density-dependent Leray problem with a discontinuous density. Methods Appl. Anal. 13(4), 321–335 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Amrouche, C., Penel, P., Seloula, N.: Some remarks on the boundary conditions in the theory of Navier–Stokes equations. Annales Mathématiques Blaise Pascal 20, 37–73 (2013)

    MathSciNet  Article  Google Scholar 

  5. Antonsev, S.N., Kazhikov, A.V., Monakov, B.N.: Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. North-Holland, Amsterdam (1990)

    Google Scholar 

  6. Beirão da Veiga, H.: On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions. Commun. Pure Appl. Math. 48, 552–577 (2005)

    MathSciNet  Article  Google Scholar 

  7. Beirão da Veiga, H.: Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip type boundary conditions. Adv. Differ. Equ. 9(9–10), 1079–1114 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Borzì, A., Park, E.-J., Vallejos Lass, M.: Multigrid optimization methods for the optimal control of convection diffusion problems with bilinear control. J. Optim. Theory Appl. 168, 510–533 (2016)

    MathSciNet  Article  Google Scholar 

  9. Brézis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

    Google Scholar 

  10. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 2, p. 585. Springer, Berlin (2000)

    Google Scholar 

  11. De los Reyes, J.C., Kunisch, K.: A semi-smooth Newton method for control constrained boundary optimal control of the Navier–Stokes equations. Nonlinear Anal. 62, 1289–1316 (2005)

    MathSciNet  Article  Google Scholar 

  12. Frolov, N.N.: On the solvability of a boundary value problem of the motion of a nonhomogeneous fluid. Math. Notes 53(5–6), 650–656 (1993)

    MathSciNet  Article  Google Scholar 

  13. Frolov, N.N.: A boundary value problem that describes the motion of a nonhomogeneous fluid. Sib. Math. J. 37(2), 376–393 (1996)

    MathSciNet  Article  Google Scholar 

  14. Fujita, H.: A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. Res. Inst. Math. Sci. Kōkyūroku 888, 199–216 (1994)

    MathSciNet  MATH  Google Scholar 

  15. Fujita, H.: A coherent analysis of Stokes flows under boundary conditions of friction type. J. Comput. Appl. Math. 149, 57–69 (2002)

    MathSciNet  Article  Google Scholar 

  16. Gunzburger, M.D., Hou, L., Svobodny, T.P.: Boundary velocity control of incompressible flow with an application to viscous drag reduction. SIAM J. Control Optim. 30(1), 167–181 (1992)

    MathSciNet  Article  Google Scholar 

  17. Hettich, R., Kaplan, A., Tichatschke, R.: Regularized penalty methods for ill-posed optimal control problems with elliptic equations. Part I: distributed control with bounded control set and state constraints. Control Cybern. 26, 5–28 (1997)

    MATH  Google Scholar 

  18. Hettich, R., Kaplan, A., Tichatschke, R.: Regularized penalty methods for ill-posed optimal control problems with elliptic equations. Part II: distributed control with unbounded set and state constraints. Control Cybern. 26, 29–42 (1997)

    MATH  Google Scholar 

  19. Illarionov, A.I.: Optimal boundary control of steady-state flow of a viscous inhomogeneous incompressible fluid. Math. Notes 69(5), 614–624 (2001)

    MathSciNet  Article  Google Scholar 

  20. Jägger, W., Mikelić, A.: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Equ. 170, 96–122 (2001)

    MathSciNet  Article  Google Scholar 

  21. Joseph, D.D.: Fluids dynamics of mixtures of incompressible miscible liquids. In: Fitzgibbon, W., Kuznetsov, Y., Neittaanmäki, P., P’eriaux, J., Pironneau, O. (eds.) Applied and Numerical Partial Differential Equations. Computational Methods in Applied Sciences, vol. 15. Springer, Dordrecht (2010)

    Google Scholar 

  22. Kazhikov, A.V.: Solvability of the initial and boundary-value problem for the equations of motion of an inhomogeneous viscous incompressible fluid. Dokl. Akad. Nauh. 216, 1008–1010 (1974)

    MathSciNet  Google Scholar 

  23. Köner, A., Vexler, B.: A priori error estimates for elliptic optimal control problems with bilinear state equation. J. Comput. Appl. Mech. 230, 781–802 (2009)

    MathSciNet  Article  Google Scholar 

  24. Lee, H.-C., Imanuvilov, O.Y.: Analysis of Neumann boundary optimal control problems for the stationary Boussinesq equations including solid media. SIAM J. Control Optim. 2(39), 457–477 (2000)

    MathSciNet  Article  Google Scholar 

  25. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    Google Scholar 

  26. Lions, J.L.: Control of Distributed Singular Systems. Gauthier-Villars, Paris (1985)

    Google Scholar 

  27. Maxwell, J.C.: On stresses in rarefed gases arising from inequalities of temperature. Philos. Trans. R. Soc. 170, 231–256 (1879)

    Article  Google Scholar 

  28. Mallea-Zepeda, E., Ortega-Torres, E., Villamizar-Roa, É.J.: An optimal control problem for the steady nonhomogeneous asymmetric fluids. Appl. Math. Optim. (2017). https://doi.org/10.1007/s00245-017-9466-5

    MathSciNet  Article  Google Scholar 

  29. Mulone, G., Salemi, F.: On the existence of hydrodynamic motion in a domain with free boundary type conditions. Meccanica 18(3), 136–144 (1983)

    Article  Google Scholar 

  30. Mulone, G., Salemi, F.: On the hydrodynamic motion in a domain with mixed boundary conditions: existence, uniqueness, stability and linearization principle. Ann. Mat. Pura Appl. 139(4), 147–174 (1985)

    MathSciNet  Article  Google Scholar 

  31. Navier, C.L.M.H.: Sur le lois de l’équilibrie et du mouvement des corps élastiques. Mem. Acad. R. Sci. Inst. France 369 (1827)

  32. Rodríguez-Bellido, M.A., Rueda-Gómez, D.A., Villamizar-Roa, É.J.: On a distributed control problem for a coupled chemotaxis-fluid model. Discret. Contin. Dyn. Syst. Ser. B. 22, 557–571 (2017)

    MathSciNet  MATH  Google Scholar 

  33. Santos, M.M.: Stationary solution of the Navier-Stokes equations in a 2d bounded domain for incompressible flow with discontinuous density. Z. Angew. Math. Phys. 53, 661–675 (2002)

    MathSciNet  Article  Google Scholar 

  34. Schöberl, J., Simon, R., Zulehner, W.: A robust multigrid method for elliptic optimal control problems. SIAM J. Numer. Anal. 49(4), 1482–1503 (2011)

    MathSciNet  Article  Google Scholar 

  35. Simon, J.: Nonhomogeneous viscous incompressible fluids: existence of velocity, density and pressure. SIAM J. Math. Anal. 21, 1093–1117 (1990)

    MathSciNet  Article  Google Scholar 

  36. Simon, J.: Sur le fluides visqueaux incompressibles et non homogènes. CRAS Paris 309, 447–451 (1989)

    MATH  Google Scholar 

  37. Solonnikov, V.A., Scadilov, V.E.: A certain boundary value problem for the stationary system of Navier–Stokes equations. Trudy Mat. Inst. Steklov. 125, 196–210 (1973)

    MathSciNet  MATH  Google Scholar 

  38. Stavre, R.: A distributed control problem for two coupled fluids in a porous medium. SIAM J. Control Optim. 53(1), 313–335 (2015)

    MathSciNet  Article  Google Scholar 

  39. Vallejos Lass, M., Borzì, A.: A robust multigrid method for elliptic optimal control problems. SIAM J. Numer. Anal. 49(4), 1482–1503 (2011)

    MathSciNet  Article  Google Scholar 

  40. Verfüth, R.: Finite element approximation of incompressible Navier–Stokes equations with slip boundary condition. Numer. Math. 50, 697–721 (1987)

    MathSciNet  Article  Google Scholar 

  41. Vitoriano, F.: On the steady viscous flow of a nonhomogeneous asymmetric fluid. Ann. Mat. Pura Appl. 192, 665–672 (2013)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

E. Mallea-Zepeda was supported by Proyecto UTA-Mayor 4740-18, Universidad de Tarapacá, Chile. E. Lenes was supported by the Departamento de Investigaciones of the Universidad del Sinú, Colombia.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Exequiel Mallea-Zepeda.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mallea-Zepeda, E., Lenes, E. & Rodríguez Zambrano, J. Bilinear Optimal Control Problem for the Stationary Navier–Stokes Equations with Variable Density and Slip Boundary Condition. Bull Braz Math Soc, New Series 50, 871–887 (2019). https://doi.org/10.1007/s00574-019-00131-6

Download citation

Keywords

  • Navier–Stokes equations
  • Variable density
  • Bilinear control problem
  • Optimality conditions

Mathematics Subject Classification

  • 49J20
  • 76D55
  • 35Q30