A Posteriori Error Estimates for an Optimal Control Problem with a Bilinear State Equation

In this work we are interested in the design and analysis of a posteriori error estimates for an optimal control problem governed by an elliptic partial differential equation (PDE) as state equation. Our main source of difficulty here is that the control variable enters the state equation as a coefficient; control constraints are also considered. Let us make this discussion precise. Let $\Omega\subset\mathbb{R}^{d}$, with $d\in\{2,3\}$, be an open and bounded polygonal/polyhedral domain with Lipschitz boundary $\partial\Omega$ (MR2424078, Chapter 4). Given a desired state $y_{\Omega}\in L^{2}(\Omega)$ and a regularization parameter $\alpha>0$, let us introduce the cost functional

$$J(y,u):=\frac{1}{2}\|y-y_{\Omega}\|_{L^{2}(\Omega)}^{2}+\frac{\alpha}{2}\|u\|_{L^{2}(\Omega)}^{2}.$$

(1)

We are thus interested in the following optimal control problem: Find

$$\min{J(y,u)}$$

(2)

subject to the elliptic PDE

$$-\Delta y+uy=f\text{ in }\Omega,\qquad y=0\text{ on }\partial\Omega,$$

(3)

where $f\in L^{2}(\Omega)$ denotes an external source, and the control constraints

$$u\in\mathbb{U}_{ad},\qquad\mathbb{U}_{ad}:=\{v\in L^{2}(\Omega):0<\texttt{a}\leq v\leq\texttt{b}\text{ a.e. in }\Omega\}.$$

(4)

Here, $\texttt{a},\texttt{b}\in\mathbb{R}^{+}$ satisfy $\texttt{a}<\texttt{b}$.

Let us set notation and describe the setting we shall operate with. Throughout this work $d\in\{2,3\}$ and $\Omega\subset\mathbb{R}^{d}$ is an open and bounded polygonal/polyhedral domain with Lipschitz boundary $\partial\Omega$ (MR2424078, Chapter 4). Notice that we do not assume that $\Omega$ is convex. If $\mathcal{X}$ and $\mathcal{Y}$ are normed vector spaces, we write $\mathcal{X}\hookrightarrow\mathcal{Y}$ to denote that $\mathcal{X}$ is continuously embedded in $\mathcal{Y}$. We denote by $\|\cdot\|_{\mathcal{X}}$ the norm of $\mathcal{X}$. The relation $a\lesssim b$ indicates that $a\leq Cb$, with a positive constant that depends neither on $a$, $b$ nor on the involved discretization parameters. The value of $C$ might change at each occurrence.

Let $\mathfrak{f}$ be a given forcing term in $L^{2}(\Omega)$ and $\mathfrak{u}$ be an arbitrary function in $\mathbb{U}_{ad}$. With this setting at hand, we introduce the following weak problem:

$$z\in H_{0}^{1}(\Omega):\quad(\nabla z,\nabla v)_{L^{2}(\Omega)}+(\mathfrak{u}z,v)_{L^{2}(\Omega)}=(\mathfrak{f},v)_{L^{2}(\Omega)}\quad\forall v\in H_{0}^{1}(\Omega).$$

(5)

Lax–Milgram Theorem immediately yields the well-posedness of problem (5). In particular, we have the following stability estimate $\|\nabla z\|_{L^{2}(\Omega)}\lesssim\|\mathfrak{f}\|_{L^{2}(\Omega)}.$

In this section, we introduce a basic finite element approximation for the weak problem (5) and review basic a posteriori error estimates. To accomplish this task, we first introduce some terminology and further basic ingredients.

We denote by $\mathscr{T}=\{T\}$ a conforming partition of $\overline{\Omega}$ into simplices $T$ with size $h_{T}=\text{diam}(T)$ and define $h_{\mathscr{T}}:=\max_{T\in\mathscr{T}}h_{T}$. We denote by $\mathscr{S}$ the set of internal $(d-1)$-dimensional interelement boundaries $S$ of $\mathscr{T}$. For $T\in\mathscr{T}$, we let $\mathscr{S}_{T}$ denote the subset of $\mathscr{S}$ which contains the sides of the element $T$. We denote by ${\color[rgb]{0,0,0}\mathcal{N}_{S}\subset\mathscr{T}}$ the subset that contains the two elements that have $S$ as a side, namely, $\mathcal{N}_{S}=\{T^{+},T^{-}\}$, where $T^{+},T^{-}\in\mathscr{T}$ are such that $S=T^{+}\cap T^{-}$. For $T\in\mathscr{T}$, we define the star associated with the element $T$ as

$$\mathcal{N}_{T}:=\left\{T^{\prime}\in\mathscr{T}:\mathscr{S}_{T}\cap\mathscr{S}_{T^{\prime}}\neq\emptyset\right\}.$$

(6)

In an abuse of notation, below we denote by $\mathcal{N}_{T}$ either the set itself or the union of its elements.

We define, for $T\in\mathscr{T}$, the shape coefficient $\sigma_{T}$ of $T$ as the ratio of the diameter, i.e., $\mathrm{diam}(T)=h_{T}$, and the inball diameter of $T$, i.e., $2\sup\{r>0:B_{r}(x)\subset T\textrm{ for }x\in T\}$. The shape coefficient of a triangulation $\mathscr{T}$ corresponds to the quantity $\sigma_{\mathscr{T}}:=\max\{T:T\in\mathscr{T}\}$. A sequence of triangulations $\mathbb{T}=\{\mathscr{T}\}$, that is obtained by subsequent refinements of an initial mesh $\mathscr{T}_{0}$, is shape regular if $\sup\{\sigma_{\mathscr{T}}:\mathscr{T}\in\mathbb{T}\}\leq C$; see (Nochetto_etal2009, section 3.2.1) for details.

Given a mesh $\mathscr{T}\in\mathbb{T}$, we define the finite element space of continuous piecewise linear functions as

$$\mathbb{V}(\mathscr{T}):=\{v_{\mathscr{T}}\in C(\overline{\Omega}):v_{\mathscr{T}}|_{T}\in\mathbb{P}_{1}(T)\ \forall T\in\mathscr{T}\}\cap H_{0}^{1}(\Omega).$$

(7)

Given a discrete function $v_{\mathscr{T}}\in\mathbb{V}(\mathscr{T})$, we define, for any internal side $S\in\mathscr{S}$, the jump or interelement residual $\llbracket\nabla v_{\mathscr{T}}\cdot\boldsymbol{\nu}\rrbracket$ by

$$\llbracket\nabla v_{\mathscr{T}}\cdot\boldsymbol{\nu}\rrbracket:=\boldsymbol{\nu}^{+}\cdot\nabla v_{\mathscr{T}}|_{T^{+}}+\boldsymbol{\nu}^{-}\cdot\nabla v_{\mathscr{T}}|_{T^{-}},$$

where $\boldsymbol{\nu}^{+},\boldsymbol{\nu}^{-}$ denote the unit normals to $S$ pointing towards $T^{+}$, $T^{-}\in\mathscr{T}$, respectively. Here, $T^{+}$, $T^{-}\in\mathscr{T}$ are such that $T^{+}\neq T^{-}$ and $\partial T^{+}\cap\partial T^{-}=S$.

With these ingredients at hand, we introduce a Galerkin approximation to problem (5) as follows:

$$z_{\mathscr{T}}\in\mathbb{V}(\mathscr{T}):\quad(\nabla z_{\mathscr{T}},\nabla v_{\mathscr{T}})_{L^{2}(\Omega)}+(\mathfrak{u}z_{\mathscr{T}},v_{\mathscr{T}})_{L^{2}(\Omega)}=(\mathfrak{f},v_{\mathscr{T}})_{L^{2}(\Omega)}$$

(8)

for all $v_{\mathscr{T}}\in\mathbb{V}(\mathscr{T})$. Here, $\mathfrak{f}\in L^{2}(\Omega)$ and $\mathfrak{u}\in\mathbb{U}_{ad}$. The existence and uniqueness of a solution $z_{\mathscr{T}}\in\mathbb{V}(\mathscr{T})$ of problem (8) is standard. In particular, we have the stability estimate $\|\nabla z_{\mathscr{T}}\|_{L^{2}(\Omega)}\lesssim\|\mathfrak{f}\|_{L^{2}(\Omega)}$.

We introduce the following local error indicators and a posteriori error estimator associated to the discretization (8) of problem (5):

$$\mathcal{E}_{T}^{2}:=h_{T}^{2}\|\mathfrak{f}-\mathfrak{u}z_{\mathscr{T}}\|_{L^{2}(T)}^{2}+h_{T}\|\llbracket\nabla z_{\mathscr{T}}\cdot\boldsymbol{\nu}\rrbracket\|_{L^{2}(\partial T\setminus\partial\Omega)}^{2},\qquad\mathcal{E}_{\mathscr{T}}^{2}:=\sum_{T\in\mathscr{T}}\mathcal{E}_{T}^{2}.$$

We present the following global reliability result.

Let $J$ be the cost functional defined in (1). We consider the following weak version of the optimization problem (2)–(4): Find

$$\min\{J(y,u):(y,u)\in H_{0}^{1}(\Omega)\times\mathbb{U}_{ad}\}$$

(9)

subject to the state equation

$$(\nabla y,\nabla v)_{L^{2}(\Omega)}+(uy,v)_{L^{2}(\Omega)}=(f,v)_{L^{2}(\Omega)}\quad\forall v\in H_{0}^{1}(\Omega).$$

(10)

The existence of an optimal solution $(\bar{y},\bar{u})\in H_{0}^{1}(\Omega)\times\mathbb{U}_{ad}$ for problem (9)–(10) follows standard arguments; see (MR2536007, Proposition 2.3).

Due to the fact that the optimal control problem (9)–(10) is not convex, we discuss optimality conditions under the framework of local solutions in $L^{2}(\Omega)$. To be precise, a control $\bar{u}\in\mathbb{U}_{ad}$ is said to be locally optimal in $L^{2}(\Omega)$ for (9)–(10) if there exists a constant $\delta>0$ such that $J(\bar{y},\bar{u})\leq J(y,u)$ for all $u\in\mathbb{U}_{ad}$ such that $\|u-\bar{u}\|_{L^{2}(\Omega)}\leq\delta$. Here, $\bar{y}$ and $y$ denote the states associated to $\bar{u}$ and $u$, respectively.

Let us introduce the set $\mathcal{U}:=\{u\in L^{\infty}(\Omega):\exists c>0\text{ such that }u(x)>c>0\text{ a.e. }x\in\Omega\}$. We immediately notice that $\mathbb{U}_{ad}\subset\mathcal{U}$. Having defined $\mathcal{U}$, we introduce the control-to-state map $\mathcal{S}$ as follows: given a control $u\in\mathcal{U}$, $\mathcal{S}$ associates to it a unique state $y=\mathcal{S}u\in H_{0}^{1}(\Omega)$ solving (10). With these ingredients at hand, we define the reduced cost functional $j:\mathcal{U}\to\mathbb{R}_{0}^{+}$ by

$$j(u)=J(\mathcal{S}u,u):=\frac{1}{2}\|\mathcal{S}u-y_{\Omega}\|_{L^{2}(\Omega)}^{2}+\frac{{\color[rgb]{0,0,0}\alpha}}{2}\|u\|_{L^{2}(\Omega)}^{2}.$$

We are now in position to formulate first order optimality conditions: if $\bar{u}$ is locally optimal for problem (9)–(10), then (MR2536007, Proposition 2.10)

$$j^{\prime}(\bar{u})(u-\bar{u})\geq 0\quad\forall u\in\mathbb{U}_{ad}.$$

(11)

Here, $j^{\prime}(\bar{u})$ denotes the Gateâux derivative of the functional $j$ at $\bar{u}$ in the direction $u-\bar{u}$. We notice that, for $u,v\in\mathbb{U}_{ad}$, $j^{\prime}(u)v=(\alpha u-yp,v)_{L^{2}(\Omega)}$ (MR2536007, equation (2.6)) and immediately comment that $\mathcal{S}$ and $j$ are not Fréchet differentiable with respect to the $L^{2}(\Omega)$-topology (MR2536007, Remark 2.8). To investigate the inequality (11), we introduce the adjoint variable $p\in H_{0}^{1}(\Omega)$ as the unique solution to the adjoint equation

$$(\nabla w,\nabla p)_{L^{2}(\Omega)}+(up,w)_{L^{2}(\Omega)}=(y-y_{\Omega},w)_{L^{2}(\Omega)}\quad\forall w\in H_{0}^{1}(\Omega),$$

(12)

where $y=\mathcal{S}u$ solves (10). Observe that problem (12) is well-posed.

With the previous ingredients at hand, we reformulate first order optimality conditions as follows; see (MR2536007, Proposition 2.10 and equation (2.6)).

Let us now introduce the projection operator $\Pi_{[\texttt{a},\texttt{b}]}:L^{1}(\Omega)\rightarrow\mathbb{U}_{ad}$ as

$$\Pi_{[\texttt{a},\texttt{b}]}(v):=\min\{\texttt{b},\max\{v,\texttt{a}\}\}\textrm{ a.e. in }\Omega.$$

(14)

This operator allows us to present the following projection formula (MR2536007, equation (2.7)): If $\bar{u}$ denotes a locally optimal control for (9)–(10), then

$$\bar{u}(x):=\Pi_{[\texttt{a},\texttt{b}]}(\alpha^{-1}\bar{y}(x)\bar{p}(x))\textrm{ a.e.}\leavevmode\nobreak\ x\in\Omega.$$

(15)

Let $\bar{u}\in\mathbb{U}_{ad}$ be a control that satisfies the first order necessary optimality condition (13). In what follows, we will assume that there exists a constant $\mu>0$ such that

$$j^{\prime\prime}(\bar{u})v^{2}\geq\mu\|v\|_{L^{2}(\Omega)}^{2}\quad\forall v\in L^{\infty}(\Omega);$$

(16)

see (MR2536007, Assumption 2.20). Here, for each $v\in L^{\infty}(\Omega)$, we have that

$$j^{\prime\prime}(\bar{u})v^{2}:=\|\mathcal{S}^{\prime}(\bar{u})v\|_{L^{2}(\Omega)}^{2}+(\bar{y}-y_{\Omega},\mathcal{S}^{\prime\prime}(\bar{u})v^{2})_{L^{2}(\Omega)}+\alpha\|v\|_{L^{2}(\Omega)}^{2},$$

where $\mathcal{S}^{\prime}(\bar{u})v$ and $\mathcal{S}^{\prime\prime}(\bar{u})v^{2}$ are defined as in (MR2536007, Lemma 2.9). We notice that assumption (16) is fulfilled if $\|\mathcal{S}\bar{u}-y_{\Omega}\|_{L^{2}(\Omega)}$ is sufficiently small or $\alpha>0$ is sufficiently large; see (MR2536007, Remark 2.21) for details.

The following result states that every control $\bar{u}$ that satisfies (13) and (16) is a local solution for problem (9)–(10); see (MR2536007, Theorem 2.24).

We conclude this section with the following estimate (MR2536007, Proposition 2.22): Let $u,v\in\mathbb{U}_{ad}$ and $w\in L^{\infty}(\Omega)$. Then, there exists $\mathfrak{C}>0$, depending on $\|f\|_{L^{2}(\Omega)}$ and $\|y_{\Omega}\|_{L^{2}(\Omega)}$, such that

$$|j^{\prime\prime}(u)w^{2}-j^{\prime\prime}(v)w^{2}|\leq\mathfrak{C}\|u-v\|_{L^{2}(\Omega)}\|w\|_{L^{2}(\Omega)}^{2}.$$

(17)

In this section, we introduce two finite element discretization schemes for our optimal control problem.