Finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state

Abstract

We investigate $C^1$ finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state formulated as fourth order variational inequalities for the state variable. For the problem with Dirichlet boundary conditions, we use an existing $H^{\frac{5}{2}-\epsilon }$ regularity result for the optimal state to derive $O(h^{\frac{1}{2}-\epsilon })$ convergence for the approximation of the optimal state in the $H^2$ norm. For the problem with mixed Dirichlet and Neumann boundary conditions, we show that the optimal state belongs to $H^3$ under appropriate assumptions on the data and obtain O(h) convergence for the approximation of the optimal state in the $H^2$ norm.

Appendices

Appendix A. KKT conditions for the Dirichlet boundary conditions

First we note that

$$\begin{aligned} {\mathscr {A}}\ne [-1,1] \end{aligned}$$

(A.1)

since $\int _I y'dx=0$ and $\int _I\psi \,dx>0$, and also

$$\begin{aligned} \{y':\,y\in V\}=\left\{ v\in H^1(I):\,\int _I v\,dx=0\right\} =H^1(I)/{\mathbb {R}}. \end{aligned}$$

(A.2)

Let ${\mathscr {K}}=\{v\in H^1(I)/{\mathbb {R}}:\,v\le \psi \;$ in $I\}$. We can rewrite (2.4) in the form of

$$\begin{aligned} \int _I \Phi (q-p)dx+\int _I(p'+f)(q'-p')dx\ge 0\quad \forall \,q \in {\mathscr {K}}, \end{aligned}$$

(A.3)

where

$$\begin{aligned} p={\bar{y}}' \end{aligned}$$

(A.4)

and the function $\Phi \in H^1(I)/{\mathbb {R}}$ is defined by

$$\begin{aligned} \beta \Phi '=y_d-{\bar{y}}. \end{aligned}$$

(A.5)

Let the bounded linear functional $L:H^1(I)/{\mathbb {R}}\longrightarrow {\mathbb {R}}$ be defined by

$$\begin{aligned} L v=\int _I\Phi v\,dx+\int _I(p'+f)v'dx. \end{aligned}$$

(A.6)

Observe that (A.3) implies

$$\begin{aligned} L v=0 \qquad \text {if }v\in H^1(I)/{\mathbb {R}}\text { and }{\mathscr {A}}\cap \mathrm {supp}\, v=\emptyset , \end{aligned}$$

(A.7)

since in this case $\pm \epsilon v+p\in {\mathscr {K}}$ for $0<\epsilon \ll 1$.

Since the active set ${\mathscr {A}}$ is a closed subset of [0, 1], according to (A.1) there exist two numbers $a,b\in I$ such that $a<b$ and $[a,b]\cap {\mathscr {A}}=\emptyset$. Let $G=(-1,a)\cup (b,1)$. Then we have (i) ${\mathscr {A}}\,\cap I\subset G$ and (ii) there exists a bounded linear extension operator $E_{\scriptscriptstyle G}:H^1(G)\longrightarrow H^1(I)/{\mathbb {R}}$.

Remark A.1

Observe that a bounded linear extension operator $E_G^*:H^1(G)\longrightarrow H^1(I)$ can be constructed by reflections (cf. Adams and Fournier 2003). The operator $E_G$ can then be defined by

$$\begin{aligned} E_G(v)=E_G^*(v)-\left( \int _I E_G^*(v)dx\right) \phi , \end{aligned}$$

where $\phi$ is a smooth function with compact support in (a, b) such that $\int _I \phi \,dx=1$.

We define a bounded linear map $T_{\scriptscriptstyle G}:H^1(G)\longrightarrow {\mathbb {R}}$ by

$$\begin{aligned} T_{\scriptscriptstyle G}v= L {\tilde{v}} \end{aligned}$$

(A.8)

where ${\tilde{v}}$ is any function in $H^1(I)/{\mathbb {R}}$ such that ${\tilde{v}}=v$ on G. $T_{\scriptscriptstyle G}$ is well-defined because the existence of ${\tilde{v}}$ is guaranteed by the extension operator $E_{\scriptscriptstyle G}$ and the independence of the choice of ${\tilde{v}}$ follows from (A.7).

Let $v\in H^1(G)$ be nonnegative. Then $-\epsilon {\tilde{v}}+p\in {\mathscr {K}}\,$ for $0<\epsilon \ll 1$ because $p\le \psi$ on G and $p<\psi$ on the compact set $[a,b]=I\backslash G$. Hence we have

$$\begin{aligned} -T_{\scriptscriptstyle G}v=\epsilon ^{-1}T_{\scriptscriptstyle G}(-\epsilon v)=\epsilon ^{-1}L(-\epsilon {\tilde{v}})\ge 0 \end{aligned}$$

(A.9)

by (A.3) and (A.6).

It follows from (A.9) and the Riesz-Schwartz Theorem (cf. Rudin 1966; Schwartz 1966) for nonnegative functionals that

$$\begin{aligned} T_{\scriptscriptstyle G}v=-\int _{[-1,a]\cup [b,1]} v\,d\mu _{\scriptscriptstyle \bar{G}}\quad \forall \,v\in H^1(G). \end{aligned}$$

(A.10)

where $\mu _{\scriptscriptstyle \bar{G}}$ is a nonnegative Borel measure on $[-1,a]\cup [b,1]$.

Because of (A.8) and (A.10), we have

$$\begin{aligned} -L v=-T(v\big |_G)=\int _{[-1,a]\cup [b,1]}v\,d\mu _{\scriptscriptstyle \bar{G}}\quad \forall \,v\in H^1(I)/{\mathbb {R}}, \end{aligned}$$

(A.11)

and the observation (A.7) implies that $\mu _{\scriptscriptstyle \bar{G}}$ is supported on ${\mathscr {A}}$.

We conclude from (A.6) and (A.11) that

$$\begin{aligned} \int _I\Phi v\,dx+\int _I(p'+f)v'dx+\int _{[-1,1]} v\, d{\tilde{\mu }}=0 \quad \forall \,v\in H^1(I)/{\mathbb {R}}, \end{aligned}$$

(A.12)

where ${\tilde{\mu }}$ is the trivial extension of $\mu _{\scriptscriptstyle \bar{G}}$ to $[-1,1]$. It follows that

$$\begin{aligned} \int _{[-1,1]} ({\bar{y}}'-\psi )d\mu =0, \end{aligned}$$

where $\mu =\beta {\tilde{\mu }}$, and in view of (A.2), (A.4), (A.5) and (A.12),

$$\begin{aligned} \int _I({\bar{y}}-y_d)z\,dx+\beta \int _I({\bar{y}}''+f)z''dx+\int _{[-1,1]} z'd\mu =0 \quad \forall \,z\in V. \end{aligned}$$

Appendix B. KKT conditions for the mixed boundary conditions

In this case we have, by (2.1b),

$$\begin{aligned} \{y':y\in V\}=\{v\in H^1(I):\,v(1)=0\}=H^1(I;1). \end{aligned}$$

(B.1)

Let ${\mathscr {K}}=\{v\in H^1(I;1):\,v\le \psi \quad \text {in I}\}$. We can rewrite (2.4) in the form of

$$\begin{aligned} \int _I \Phi (q-p)dx+\int _I (p'+f)(q'-p')dx\ge 0\quad \forall \,q \in {\mathscr {K}}, \end{aligned}$$

(B.2)

where

$$\begin{aligned} p={\bar{y}}'\in {\mathscr {K}}, \end{aligned}$$

(B.3)

and the function $\Phi \in H^1(I;1)$ is defined by

$$\begin{aligned} \beta \Phi '=y_d-{\bar{y}}. \end{aligned}$$

(B.4)

Note that $f\in H^1(I)$ by the assumption in (1.7). After integration by parts, the inequality (B.2) becomes

$$\begin{aligned}&- f(-1)[q(-1)-p(-1)]+\int _I (\Phi -f')(q-p)\,dx \nonumber \\&\quad +\int _I p'(q'-p')dx\ge 0\quad \forall \,q\in {\mathscr {K}}. \end{aligned}$$

(B.5)

The variational inequality defined by (B.3) and (B.5) is equivalent to a second order obstacle problem with mixed boundary conditions whose coincidence set is identical to the active set ${\mathscr {A}}$ in (2.10).

Since $\psi \in H^2(I)$ by the assumption in (1.7), we can apply the penalty method in Murthy and Stampacchia (1973) to show that

$$\begin{aligned} \text {the solution }p\text { of }(\hbox {B}.5)\text { belongs to } H^2(I), \end{aligned}$$

(B.6)

and, after integration by parts, we have

$$\begin{aligned}&-f(-1)q(-1)+\int _I (\Phi -f') q\,dx\nonumber \\&\quad +\int _I p'q'dx +\int _{[-1,1]} q\,d\nu =0 \quad \forall \,q\in H^1(I;1), \end{aligned}$$

(B.7)

where

$$\begin{aligned} d\nu =(p''+f'-\Phi )dx+(f(-1)+p'(-1))d\delta _{-1}, \end{aligned}$$

(B.8)

and $\delta _{-1}$ is the Dirac point measure at $-1$.

The variational inequality (B.5) is then equivalent to

$$\begin{aligned} p\le & {} \psi \qquad \text {in}\;I, \end{aligned}$$

(B.9a)

$$\begin{aligned} p''+f'-\Phi\ge & {} 0 \qquad \text {in}\;I,\end{aligned}$$

(B.9b)

$$\begin{aligned} f(-1)+p'(-1)\ge & {} 0, \end{aligned}$$

(B.9c)

$$\begin{aligned} \int _{[-1,1]}(p-\psi )d\nu= & {} 0. \end{aligned}$$

(B.9d)

Consequently the KKT conditions (2.7)–(2.9) hold for the Borel measure

$$\begin{aligned} \mu =\beta \nu . \end{aligned}$$

(B.10)

Remark B.1

In the special case where $f=0$ and $\psi$ is a positive constant, the condition (B.9d) implies $p'(-1)=0$ if $-1\not \in {\mathscr {A}}$, and the conditions (B.9a) and (B.9c) imply $p'(-1)=0$ if $-1\in {\mathscr {A}}$. Therefore we have $p'(-1)=0$ if $f=0$ and $\psi$ is a positive constant, in which case $\mu$ is absolutely continuous with respect to the Lebesgue measure. Hence it is necessary to choose $\gamma =p'(-1)=0$ in Example 2.10.