Optimal a priori error estimates of parabolic optimal control problems with a moving point control

TL;DR

This paper establishes optimal a priori error estimates for a parabolic optimal control problem involving a moving point source, correcting previous flawed analysis and providing new error bounds with logarithmic factors.

Contribution

It offers the first correct proof of optimal error estimates for this problem, including global and local error analysis on a curve, improving upon prior flawed results.

Findings

Optimal convergence rates achieved in discretization

Error estimates include logarithmic factors

Corrected proof addresses previous flaws in analysis

Abstract

In this paper we consider a parabolic optimal control problem with a Dirac type control with moving point source in two space dimensions. We discretize the problem with piecewise constant functions in time and continuous piecewise linear finite elements in space. For this discretization we show optimal order of convergence with respect to the time and the space discretization parameters modulo some logarithmic terms. Error analysis for the same problem was carried out in the recent paper [17], however, the analysis there contains a serious flaw. One of the main goals of this paper is to provide the correct proof. The main ingredients of our analysis are the global and local error estimates on a curve, that have an independent interest.

Equations

$$\underset{q,u}{\min }J(q,u):=\frac{1}{2}{\int }_0^T\Vert u(t)-\hat{u}(t){\Vert }_{L^2(\Omega )}^{2}dt+\frac{\alpha }{2}{\int }_0^T|q(t){|}^{2}dt$$

$$u_t(t,x)-\Delta u(t,x)$$

$$u(t,x)$$

$$u(0,x)$$

$$q_a\le q(t)\le q_b\text{a.e. in }I.$$

$$\Vert \bar{q}-{\bar{q}}_{kh}{\Vert }_{L^2(I)}\le C\left(|\ln h{|}^3(k+h^2)+C_{\gamma }|\ln h|k\right)\left(\Vert \bar{q}{\Vert }_{L^2(I)}+\Vert \hat{u}{\Vert }_{L^2(I;L^{\infty }(\Omega ))}\right).$$

$$v_t(t,x)-\Delta v(t,x)$$

$$v(t,x)$$

$$v(0,x)$$

$$v\in L^2(I;H_0^1(\Omega ))\cap H^1(I;H^{-1}(\Omega )).$$

$$v\in L^2(I;H^2(\Omega )\cap H_0^1(\Omega ))\cap H^1(I;L^2(\Omega )),$$

$$\Vert v{\Vert }_{L^2(I;H^2(\Omega ))}+\Vert v_t{\Vert }_{L^2(I;L^2(\Omega ))}\le C\Vert f{\Vert }_{L^2(I;L^2(\Omega ))},$$

$$\Vert v{\Vert }_{L^2(I;W^{1,s}(\Omega ))}\le Cs\Vert v{\Vert }_{L^2(I;H^2(\Omega ))}\le Cs\Vert f{\Vert }_{L^2(I;L^2(\Omega ))}.$$

$$\Vert v{\Vert }_{L^2(I;C(\Omega ))}\le C_p\Vert f{\Vert }_{L^2(I;L^p(\Omega ))},$$

$$\Vert v_t{\Vert }_{L^2(I;L^p({\Omega }_0))}+\Vert v{\Vert }_{L^2(I;W^{2,p}({\Omega }_0))}\le Cp(\Vert f{\Vert }_{L^2(I;L^p({\Omega }_1))}+\Vert f{\Vert }_{L^2(I;L^2(\Omega ))}).$$

$$⟨u,\phi {⟩}_{L^2(I;L^p(\Omega )),L^2(I;L^{p^{\prime }}(\Omega ))}={\int }_{I}w(t,\gamma (t))q(t)dt,$$

$$-w_t(t,x)-\Delta w(t,x)$$

$$w(t,x)$$

$$w(T,x)$$

$$\Vert u{\Vert }_{L^2(I;L^p(\Omega ))}\le Cp\Vert q{\Vert }_{L^2(I)}.$$

$$\Vert u{\Vert }_{L^2(I;L^p(\Omega ))}=\underset{\Vert \phi {\Vert }_{L^2(I;L^{p^{\prime }}(\Omega ))}=1}{\sup }(u,\phi {)}_{I\times \Omega },\text{where}\frac{1}{p}+\frac{1}{p^{\prime }}=1.$$

$$\Vert w{\Vert }_{L^2(I;C(\Omega ))}\le \frac{C}{p^{\prime }-1}\Vert \phi {\Vert }_{L^2(I;L^{p^{\prime }}(\Omega ))}=\frac{C}{p^{\prime }-1}\le Cp,\text{as}p\to \infty .$$

$$\Vert u{\Vert }_{L^2(I;L^p(\Omega ))}$$

$$={\int }_{I}q(t)w(t,\gamma (t))dt\le \Vert q{\Vert }_{L^2(I)}\Vert w{\Vert }_{L^2(I;C(\Omega ))}\le Cp\Vert q{\Vert }_{L^2(I)}.$$

$$u\in L^2(I;W_0^{1,s}(\Omega ))\text{and}{u}_t\in L^2(I;W^{-1,s}(\Omega )).$$

$$⟨u_t,\phi ⟩+(\nabla u,\nabla \phi )={\int }_{I}q(t)\phi (t,\gamma (t))dt\text{for all }\phi \in L^2(I;W_0^{1,s^{\prime }}(\Omega )),$$

$$u\in L^2(I;W_0^{1,s}(\Omega ))\text{and}{u}_t\in L^2(I;W^{-1,s}(\Omega )).$$

$$j(q)=J(q,u(q)),$$

$$\min j(q),q\in Q_{\text{ad}},$$

$$Q_{\text{ad}}=\{q\in Q|q_a\le q(t)\le q_b\text{a. e. in }I\}.$$

$$j^{\prime }(\bar{q})(\partial q-\bar{q})\ge 0\text{for all }\partial q\in Q_{\text{ad}}.$$

$$j^{\prime }(q)(\partial q)={\int }_I\left(\alpha q(t)+z(t,\gamma (t))\right)\partial q(t)dt,$$

$$-z_t(t,x)-\Delta z(t,x)$$

$$z(t,x)$$

Full text

1 Introduction

In this paper we provide numerical analysis for the following optimal control problem:

[TABLE]

subject to the second order parabolic equation

[TABLE]

and subject to pointwise control constraints

[TABLE]

Here $I=(0,T)$, $\Omega\subset\mathbb{R}^{2}$ is a convex polygonal domain and $\delta_{\gamma(t)}$ is the Dirac delta function at point $x_{t}=\gamma(t)$ at each $t$. We will assume:

Assumption 1

• •

$\gamma\in C^{1}(\bar{I})$* and $\max_{t\in\bar{I}}|\gamma^{\prime}(t)|\leq C_{\gamma}$.*

Assumption 2

• •

$\gamma(t)\subset\overline{\Omega}_{0}\subset\subset\Omega_{1}$, for any $t\in I$, with $\overline{\Omega}_{1}\subset\subset\Omega$.

The parameter $\alpha$ is assumed to be positive and the desired state $\hat{u}$ fulfills $\hat{u}\in L^{2}(I;L^{\infty}(\Omega))$. The control bounds $q_{a},q_{b}\in\mathbb{R}\cup\{\pm\infty\}$ fulfill $q_{a}<q_{b}$. The precise functional-analytic setting is discussed in the next section.

For the discretization, we consider the standard continuous piecewise linear finite elements in space and piecewise constant discontinuous Galerkin method in time. This is a special case ($r=0$, $s=1$) of so called dG($r$)cG($s$) discretization, see e.g. ErikssonK_JohnsonC_ThomeeV_1985 for the analysis of the method for parabolic problems and e.g. MeidnerD_VexlerB_2008a ; MeidnerD_VexlerB_2008b for error estimates in the context of optimal control problems. Throughout, we will denote by $h$ the spatial mesh size and by $k$ the size of time steps, see Section 3 for details.

The main result of the paper is the following.

Theorem 1.1

Let $\bar{q}$ be optimal control for the problem (1)-(2) and $\bar{q}_{kh}$ be the optimal dG(0)cG(1) solution. Then there exists a constant $C$ independent of $h$ and $k$ such that

[TABLE]

We would also like to point out that in addition to the optimal order estimate, modulo logarithmic terms, our analysis does not require any relationship between the sizes of the space discretization $h$ and the time steps $k$.

The problem with fixed location of the point source (i.e. with $\delta_{x_{0}}(x)$ for some fixed $x_{0}\in\Omega$) starting with the work of Lions LionsJL_1971 , was investigated in a number of publications, see AmourouxM_BabaryJP_1978 ; BanksHT_1992 ; ChryssoverghiI_1981 ; DroniouJ_RaymondJP_2000 ; NguyenPA_RaymondJP_2011 for the continuous problem and GongW_HinzeM_ZhouZ_2014 ; LeykekhmanD_VexlerB_2013 ; LeykekhmanD_VexlerB_2016c for the finite element approximation and error estimates. There is also a closely related problem of measured valued controls, which received a lot of attention lately CasasE_ClasonC_KunischK_2013 ; CasasE_KunischK_2016 ; CasasE_VexlerB_ZuazuaE_2015 ; CasasE_ZuazuaE_2013 ; KunischK_PieperK_VexlerB_2014 .

The problem with moving Dirac was considered in CastroC_ZuazuaE_2004a ; NguyenPA_RaymondJP_2001 on a continuous level. The error analysis was carried out in the recent paper GongW_YanN_2016a . However, the analysis there contains a serious flaw. The last inequality in the estimate $(3.33)$ in GongW_YanN_2016a is not correct. One of the main goals of this paper is to provide the correct proof. The main ingredients of our analysis are the global and local error estimates on a curve, Theorem 3.1 and Theorem 3.2, respectively. These results are new and have an independent interest.

Throughout the paper we use the usual notation for Lebesgue and Sobolev spaces. We denote by $(\cdot,\cdot)_{\Omega}$ the inner product in $L^{2}(\Omega)$ and by $(\cdot,\cdot)_{\tilde{I}\times\Omega}$ the inner product in $L^{2}(\tilde{I}\times\Omega)$ for any subinterval $\tilde{I}\subset I$.

The rest of the paper is organized as follows. In Section 2 we discuss the functional analytic setting of the problem, state the optimality system and prove regularity results for the state and for the adjoint state. In Section 3 we establish important global and local best approximation results along the curve for the heat equation. Finally in Section 4 we prove our main result.

2 Optimal control problem and regularity

In order to state the functional analytic setting for the optimal control problem, we first introduce the auxiliary problem

[TABLE]

with a right-hand side $f\in L^{2}(I;L^{p}(\Omega))$ for some $1<p<\infty$. This equation possesses a unique solution

[TABLE]

Due to the convexity of the polygonal domain $\Omega$ the solution $v$ possesses an additional regularity for $p=2$:

[TABLE]

with the corresponding estimate

[TABLE]

see, e.g., EvansLC_2010 . From the Sobolev embedding $H^{2}(\Omega)\hookrightarrow W^{1,s}(\Omega)$ for any $s<\infty$ in two space dimensions and the previous lemma we can establish the following result for $s>2$,

[TABLE]

The exact form of the constant can be traced, for example, from the proof of (AltHW_2016, , Thm. 10.8). In addition, there holds the following regularity result (see LeykekhmanD_VexlerB_2013 ).

Lemma 1

If $f\in L^{2}(I;L^{p}(\Omega))$ for an arbitrary $p>1$, then $v\in L^{2}(I;C(\Omega))$ and

[TABLE]

where $C_{p}\sim\frac{1}{p-1}$, as $p\to 1$.

We will also need the following local regularity result (see LeykekhmanD_VexlerB_2013 ).

Lemma 2

Let $\Omega_{0}\subset\subset\Omega_{1}\subset\subset\Omega$ and $f\in L^{2}(I;L^{2}(\Omega))\cap L^{2}(I;L^{p}(\Omega_{1}))$ for some $2\leq p<\infty$. Then $v\in L^{2}(I;W^{2,p}(\Omega_{0}))\cap H^{1}(I;L^{p}(\Omega_{0}))$ and there exists a constant $C$ independent of $p$ such that

[TABLE]

To introduce a weak solution of the state equation (2) we use the method of transposition, (cf. LionsJL_MagenesE_Vol2 ). For a given control $q\in Q=L^{2}(I)$ we denote by $u=u(q)\in L^{2}(I;L^{p}(\Omega))$ with $2\leq p<\infty$ a weak solution of (2), if for all $\varphi\in L^{2}(I;L^{p^{\prime}}(\Omega))$ with $\frac{1}{p}+\frac{1}{p^{\prime}}=1$ there holds

[TABLE]

where $w\in L^{2}(I;W^{2,p^{\prime}}(\Omega)\cap H^{1}_{0}(\Omega))\cap H^{1}(I;L^{p^{\prime}}(\Omega))$ is the weak solution of the adjoint equation

[TABLE]

The existence of this weak solution $u=u(q)$ follows by duality using the embedding $L^{2}(I;W^{2,p^{\prime}}(\Omega))\hookrightarrow L^{2}(I;C(\Omega))$ for $p^{\prime}>1$. Using Lemma 1 we can prove additional regularity for the state variable $u=u(q)$.

Proposition 2.1

Without lose of generality we assume $2\leq p<\infty$. Let $q\in Q=L^{2}(I)$ be given and $u=u(q)$ be the solution of the state equation (2). Then $u\in L^{2}(I;L^{p}(\Omega))$ for any $p<\infty$ and the following estimate holds for $p\to\infty$ with a constant $C$ independent of $p$,

[TABLE]

Proof

To establish the result we use a duality argument. There holds

[TABLE]

Let $w$ be the solution to (7) for $\varphi\in L^{2}(I;L^{p^{\prime}}(\Omega))$ with $\lVert\varphi\rVert_{L^{2}(I;L^{p^{\prime}}(\Omega))}=1$. From Lemma 1, $w\in L^{2}(I;C(\Omega))$ and the following estimate holds

[TABLE]

Thus,

[TABLE]

Remark 1

We would like to note that the above regularity requires only Assumption 2 on $\gamma$. Higher regularity of $\gamma$ is needed for optimal order error estimates only.

A further regularity result for the state equation follows from ElschnerJ_RehbergJ_SchmidtG_2007 .

Proposition 2.2

Let $q\in Q=L^{2}(I)$ be given and $u=u(q)$ be the solution of the state equation (2). Then for each $1<s<2$ there holds

[TABLE]

Moreover, the state $u$ fulfills the following weak formulation

[TABLE]

where $\frac{1}{s^{\prime}}+\frac{1}{s}=1$ and $\langle\cdot,\cdot\rangle$ is the duality product between $L^{2}(I;W^{-1,s}(\Omega))$ and $L^{2}(I;W^{1,s^{\prime}}_{0}(\Omega))$.

Proof

For $s<2$ we have $s^{\prime}>2$ and therefore $W^{1,s^{\prime}}_{0}(\Omega)$ is embedded into $C(\bar{\Omega})$. Therefore the right-hand side $q(t)\delta_{\gamma(t)}$ of the state equation can be identified with an element in $L^{2}(I;W^{-1,s}(\Omega))$. Using the result from (ElschnerJ_RehbergJ_SchmidtG_2007, , Theorem 5.1) on maximal parabolic regularity and exploiting the fact that $-\Delta\colon W^{1,s}_{0}(\Omega)\to W^{-1,s}(\Omega)$ is an isomorphism, see JerisonD_KenigCE_1995 , we obtain

[TABLE]

Given the above regularity the corresponding weak formulation is fulfilled by a standard density argument.

As the next step we introduce the reduced cost functional $j\colon Q\to\mathbb{R}$ on the control space $Q=L^{2}(I)$ by

[TABLE]

where $J$ is the cost function in (1) and $u(q)$ is the weak solution of the state equation (2) as defined above. The optimal control problem can then be equivalently reformulated as

[TABLE]

where the set of admissible controls is defined according to (3) by

[TABLE]

By standard arguments this optimization problem possesses a unique solution $\bar{q}\in Q=L^{2}(I)$ with the corresponding state $\bar{u}=u(\bar{q})\in L^{2}(I;L^{p}(\Omega))$ for all $p<\infty$, see Proposition 2.1 for the regularity of $\bar{u}$. Due to the fact, that this optimal control problem is convex, the solution $\bar{q}$ is equivalently characterized by the optimality condition

[TABLE]

The (directional) derivative $j^{\prime}(q)(\partial q)$ for given $q,\partial q\in Q$ can be expressed as

[TABLE]

where $z=z(q)$ is the solution of the adjoint equation

[TABLE]

and $u=u(q)$ on the right-hand side of (11a) is the solution of the state equation (2). The adjoint solution, which corresponds to the optimal control $\bar{q}$ is denoted by $\bar{z}=z(\bar{q})$.

The optimality condition (10) is a variational inequality, which can be equivalently formulated using the projection

[TABLE]

The resulting condition reads:

[TABLE]

In the next proposition we provide regularity results for the solution of the adjoint equation.

Proposition 2.3

Let $q\in Q$ be given, let $u=u(q)$ be the corresponding state fulfilling (2) and let $z=z(q)$ be the corresponding adjoint state fulfilling (11). Then,

• (a)

$z\in L^{2}(I;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\cap H^{1}(I;L^{2}(\Omega))$* and the following estimate holds*

[TABLE]

• (b)

If $\Omega_{0}\subset\subset\Omega$, then $z\in L^{2}(I;W^{2,p}(\Omega_{0}))\cap H^{1}(I;L^{p}(\Omega_{0}))$ for all $2\leq p<\infty$ and the following estimate holds

[TABLE]

Proof

• (a)

The right-hand side of the adjoint equation fulfills $u-\hat{u}\in L^{2}(I;L^{p}(\Omega))$ for all $1<p<\infty$, see Proposition 2.1. Due to the convexity of the domain $\Omega$ we directly obtain $z\in L^{2}(I;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\cap H^{1}(I;L^{2}(\Omega))$ and the estimate

[TABLE]

The result from Proposition 2.1 leads directly to the first estimate.

• (b)

From Lemma 2 for $p\geq 2$ we have

[TABLE]

Hence, by the triangle inequality and Proposition 2.1 we obtain

[TABLE]

That completes the proof.

3 Discretization and the best approximation type results

3.1 Space-time discretization and notation

For discretization of the problem under the consideration we introduce a partitions of $I=[0,T]$ into subintervals $I_{m}=(t_{m-1},t_{m}]$ of length $k_{m}=t_{m}-t_{m-1}$, where $0=t_{0}<t_{1}<\cdots<t_{M-1}<t_{M}=T$. We assume that

[TABLE]

The maximal time step is denoted by $k=\max_{m}k_{m}$. The semidiscrete space $X^{0}_{k}$ of piecewise constant functions in time is defined by

[TABLE]

where $\mathcal{P}_{0}(I;V)$ is the space of constant functions in time with values in Banach space $V$. We will employ the following notation for functions in $X^{0}_{k}$

[TABLE]

Let $\mathcal{T}$ denote a quasi-uniform triangulation of $\Omega$ with a mesh size $h$, i.e., $\mathcal{T}=\{\tau\}$ is a partition of $\Omega$ into triangles $\tau$ of diameter $h_{\tau}$ such that for $h=\max_{\tau}h_{\tau}$,

[TABLE]

hold. Let $V_{h}$ be the set of all functions in $H^{1}_{0}(\Omega)$ that are linear on each $\tau$, i.e. $V_{h}$ is the usual space of continuous piecewise linear finite elements. We will require the modified Clément interpolant $i_{h}\colon L^{1}(\Omega)\to V_{h}$ and the $L^{2}$-projection $P_{h}\colon L^{2}(\Omega)\to V_{h}$ defined by

[TABLE]

To obtain the fully discrete approximation we consider the space-time finite element space

[TABLE]

We will also need the following semidiscrete projection $\pi_{k}\colon C(\bar{I};H^{1}_{0}(\Omega))\to X^{0}_{k}$ defined by

[TABLE]

and the fully discrete projection $\pi_{kh}\colon C(\bar{I};L^{1}(\Omega))\to X^{0,1}_{k,h}$ defined by $\pi_{kh}=i_{h}\pi_{k}$.

To introduce the dG(0)cG(1) discretization we define the following bilinear form

[TABLE]

where $\langle\cdot,\cdot\rangle_{I_{m}\times\Omega}$ is the duality product between $L^{2}(I_{m};W^{-1,s}(\Omega))$ and $L^{2}(I_{m};W^{1,s^{\prime}}_{0}(\Omega))$. We note, that the first sum vanishes for $v\in X^{0}_{k}$. Rearranging the terms, we obtain an equivalent (dual) expression for $B$:

[TABLE]

In the two following theorems we establish global and local best approximation type results along the curve for the error between the solution $v$ of the auxiliary equation (4) and its dG(0)cG(1) approximation $v_{kh}\in X^{0,1}_{k,h}$ defined as

[TABLE]

Since dG(0)cG(1) method is a consistent discretization we have the following Galerkin orthogonality relation:

[TABLE]

3.2 Discretization of the curve and the weight function

To define fully discrete optimization problem we will also require a discretization of the curve $\gamma$. We define $\gamma_{k}=\pi_{k}\gamma$ by

[TABLE]

i.e., $\gamma_{k}$ is a piecewise constant approximation of $\gamma$. Next we introduce a weight function

[TABLE]

and a discrete piecewise constant in time approximation

[TABLE]

Define

[TABLE]

One can easily check that $\sigma$ and $\sigma_{k}$ satisfy the following properties for any $(t,x)\in I\times\Omega$,

[TABLE]

3.3 Global error estimate along the curve

In this section we prove the following global approximation result.

Theorem 3.1 (Global best approximation)

Assume $v$ and $v_{kh}$ satisfy (4) and (20) respectively. Then there exists a constant $C$ independent of $k$ and $h$ such that for any $1\leq p\leq\infty$,

[TABLE]

Proof

To establish the result we use a duality argument. First, we introduce a smoothed Delta function, which we will denote by $\tilde{\delta}_{\gamma_{k}}$. This function on each $I_{m}$ is defined as $\tilde{\delta}_{\gamma_{k,m}}$ and supported in one cell, which we denote by $\tau^{0}_{m}$, i.e.

[TABLE]

In addition we also have (see (SchatzAH_WahlbinLB_1995, ,  Appendix))

[TABLE]

Thus in particular $\|\tilde{\delta}_{\gamma_{k}}\|_{L^{1}(\Omega)}\leq C$, $\|\tilde{\delta}_{\gamma_{k}}\|_{L^{2}(\Omega)}\leq Ch^{-1}$, and $\|\tilde{\delta}_{\gamma_{k}}\|_{L^{\infty}(\Omega)}\leq Ch^{-2}$.

We define $g$ to be a solution to the following backward parabolic problem

[TABLE]

There holds

[TABLE]

Let $g_{kh}\in X^{0,1}_{k,h}$ be dG(0)cG(1) solution defined by

[TABLE]

Then using that dG(0)cG(1) method is consistent, we have

[TABLE]

where we have used the dual expression (19) for the bilinear form $B$ and the fact that the last term in (19) can be included in the sum by setting $g_{kh,M+1}=0$ and defining consequently $[g_{kh}]_{M}=-g_{kh,M}$. The first sum in (19) vanishes due to $g_{kh}\in X^{0,1}_{k,h}$. For each $t$, integrating by parts elementwise and using that $g_{kh}$ is linear in the spacial variable, by the Hölder’s inequality we have

[TABLE]

where $[\![\partial_{n}g_{kh}]\!]$ denotes the jumps of the normal derivatives across the element faces.

From Lemma 2.4 in RannacherR_1991a we have

[TABLE]

where $\Delta_{h}\colon V_{h}\to V_{h}$ is the discrete Laplace operator, defined by

[TABLE]

To estimate the term involving the jumps in (29), we first use the Hölder’s inequality and the inverse estimate to obtain

[TABLE]

Now we use the fact that the equation (28) can be rewritten on the each time level as

[TABLE]

or equivalently as

[TABLE]

where $P_{h}$ is the $L^{2}$-projection, see (15). From (32) by the triangle inequality, we obtain

[TABLE]

Using that the $L^{2}$-projection is stable in $L^{1}$-norm (cf. CrouzeixM_ThomeeV_1987a ), we have

[TABLE]

Inserting the above estimate into (31) and using (25a), we obtain

[TABLE]

Combining (29) and (30) with the above estimates we have

[TABLE]

To complete the proof of the theorem it is sufficient to show

[TABLE]

Then from (33) and (34) it would follow that

[TABLE]

Then using that the dG(0)cG(1) method is invariant on $X^{0,1}_{k,h}$, by replacing $v$ an $v_{kh}$ with $v-\chi$ and $v_{kh}-\chi$ for any $\chi\in Xkh$, we obtain Theorem 3.1.

The estimate (34) will follow from the series of lemmas. The first lemma treats the term $\lVert\sigma_{k}\Delta_{h}g_{kh}\rVert^{2}_{L^{2}(I;L^{2}(\Omega))}$.

Lemma 3

For any $\varepsilon>0$ there exists $C_{\varepsilon}$ such that

[TABLE]

where $\sigma_{k}$ and $\sigma_{k,m}$ are defined in (23) and (24), respectively.

Proof

The equation (28) for each time interval $I_{m}$ can be rewritten as (32). Multiplying (32) with $\varphi=-\sigma^{2}_{k}\Delta_{h}g_{kh}$ and integrating over $I_{m}\times\Omega$, we have

[TABLE]

We have

[TABLE]

By the Cauchy-Schwarz inequality and using (25b) we get

[TABLE]

On the other hand we have

[TABLE]

Using the identity

[TABLE]

we have

[TABLE]

By the Cauchy-Schwarz inequality, we obtain

[TABLE]

where in the last step we used that from (25d)

[TABLE]

for some $\tilde{t}\in I_{m}$. Using the Young’s inequality for $J_{11}$, neglecting $-\frac{1}{2}\|[\sigma_{k}\nabla g_{kh}]_{m}\|^{2}_{L^{2}(\Omega)}$, and using the assumption on the time steps $k_{m}\leq\kappa k_{m+1}$ and that $\sigma_{k}\leq C$, we obtain

[TABLE]

To estimate $J_{2}$, first by the Cauchy-Schwarz inequality and the approximation theory we have

[TABLE]

Using that $g_{kh}$ is piecewise linear we have

[TABLE]

There holds $\partial_{ij}(\sigma^{2})=2(\partial_{i}\sigma)(\partial_{j}\sigma)+2\sigma\partial_{ij}\sigma$ and $\nabla(\sigma^{2})=2\sigma\nabla\sigma$. Thus by the properties of $\sigma$ (25b) and (25c), we have

[TABLE]

Same estimates hold for $\sigma_{k}$. Using these estimates, the fact that $h\leq\sigma_{k}$ and the inverse inequality (in view of (25e) the inverise inequality is valid with $\sigma$ inside the norm), we obtain

[TABLE]

To estimate $J_{3}$ we first notice that

[TABLE]

The proof is identical to the proof of $(3.21)$ in LeykekhmanD_VexlerB_2013 .

By the Cauchy-Schwarz inequality, (38), and the Young’s inequality, we obtain

[TABLE]

Using the estimates (36), (37), and (39) we have

[TABLE]

Summing over $m$ and using that $g_{kh,M+1}=0$ we obtain the lemma.

The second lemma treats the term involving jumps.

Lemma 4

There exists a constant $C$ such that

[TABLE]

Proof

We test (32) with $\varphi\rvert_{I_{m}}=\sigma^{2}_{k,m}[g_{kh}]_{m}$ and obtain

[TABLE]

The first term on the right hand side of (40) using the Young’s inequality can be estimated as

[TABLE]

The last term on the right hand side of (40) can easily be estimated using (38) as

[TABLE]

Combining the above two estimates we obtain

[TABLE]

Summing over $m$ we obtain the lemma.

Lemma 5

There exists a constant $C$ such that

[TABLE]

Proof

Adding the primal (18) and the dual (19) representation of the bilinear form $B(\cdot,\cdot)$ one immediately arrives at

[TABLE]

see e.g., MeidnerD_VexlerB_2008a . Applying this inequality together with the discrete Sobolev inequality, see (BrennerSC_ScottLR_2008, ,  Lemma 4.9.2), results in

[TABLE]

This gives the desired estimate.

We proceed with the proof of Theorem 3.1. From Lemma 3, Lemma 4, and Lemma 5. It follows that

[TABLE]

Taking $\varepsilon$ sufficiently small we have (34). From (33) we can conclude that

[TABLE]

for some constant $C$ independent of $h$ and $k$. Using that dG(0)cG(1) method is invariant on $X^{0,1}_{k,h}$, by replacing $v$ and $v_{kh}$ with $v-\chi$ and $v_{kh}-\chi$ for any $\chi\in X^{0,1}_{k,h}$, we obtain

[TABLE]

By the triangle inequality and the above estimate we deduce

[TABLE]

Taking the infimum over $\chi$, we obtain Theorem 3.1.

3.4 Interior error estimate

To obtain optimal error estimates we will also require the following interior result.

Theorem 3.2 (Interior approximation)

Let $B_{d,m}:=B_{d}(\gamma(t_{m}))$ denote a ball of radius $d$ centered at $\gamma(t_{m})$. Assume $v$ and $v_{kh}$ satisfy (4) and (20) respectively and let $d>4h$. Then there exists a constant $C$ independent of $h$, $k$ and $d$ such that for any $1\leq p\leq\infty$

[TABLE]

Proof

To obtain the interior estimate we introduce a smooth cut-off function $\omega$ in space and piecewise constant in time, such that $\omega_{m}:=\omega\rvert_{I_{m}}$,

[TABLE]

As in the proof of Theorem 3.1 we obtain by (29) that

[TABLE]

where $g_{kh}$ is the solution of (28). Note that $\omega v$ is discontinuous in time. The first term can be estimated using the global result from Theorem 3.1. To this end we introduce the solution $\tilde{v}_{kh}\in X^{0,1}_{k,h}$ defined by

[TABLE]

There holds

[TABLE]

Applying Theorem 3.1 for the second term, we have

[TABLE]

From (43), canceling $\frac{1}{2}\int_{0}^{T}\lvert v_{kh}(t,\gamma_{k}(t))\rvert^{2}\,dt$ and using the above estimate, we obtain

[TABLE]

It remains to estimate the term $B((1-\omega)v,g_{kh})$. Using the dual expression (19) of the bilinear form $B$ we obtain

[TABLE]

To estimate $J_{1}$ we define $\psi=(1-\omega)v$ and proceed using the Ritz projection $R_{h}\colon H^{1}_{0}(\Omega)\to V_{h}$ defined by

[TABLE]

There holds

[TABLE]

Using the estimate

[TABLE]

where in the last step we used (25a), we obtain

[TABLE]

By the interior pointwise error estimates from Theorem 5.1 in SchatzAH_WahlbinLB_1977 , we have for each $t\in I_{m}$,

[TABLE]

since the support of $\psi_{m}=(1-\omega_{m})v$ is contained in $\Omega\setminus B_{d/2,m}$. On $\Omega\setminus B_{d/4,m}$ there holds $\sigma_{k,m}\geq d/4$ and therefore for each $t\in I_{m}$,

[TABLE]

Inserting the last two estimates into (47) we get

[TABLE]

Using a standard elliptic estimate and recalling $\psi=(1-\omega)v$ we have

[TABLE]

where in the last step we used $\lvert\nabla\omega(t)\rvert\leq cd^{-1}\leq ch^{-1}$. This results in

[TABLE]

Therefore, we get

[TABLE]

For $J_{2}$ we obtain

[TABLE]

where we used that $supp(1-\omega_{m})v_{m}\subset\Omega\setminus B_{d/2,m}$ and $\sigma_{k,m}\geq d/2$ on this set as well as the definition of $\pi_{k}$ (17). Inserting the estimate (48) for $J_{1}$ and the estimate (49) for $J_{2}$ into (45) we obtain

[TABLE]

Using the estimate (34) and Lemma 4

[TABLE]

Inserting this inequality into (44) we obtain

[TABLE]

Using that the dG([math])cG($1$) method is invariant on $X^{0,1}_{k,h}$, by replacing $v$ and $v_{kh}$ with $v-\chi$ and $v_{kh}-\chi$ for any $\chi\in X^{0,1}_{k,h}$, we obtain the estimate in Theorem 3.2.

4 Discretization of the optimal control problem

In this section we describe the discretization of the optimal control problem (1)-(2) and prove our main result, Theorem 1.1. We start with discretization of the state equation. For a given control $q\in Q$ we define the corresponding discrete state $u_{kh}=u_{kh}(q)\in X^{0,1}_{k,h}$ by

[TABLE]

Using the weak formulation for $u=u(q)$ from Proposition 2.2 we obtain the perturbed Galerkin orthogonality,

[TABLE]

Note, that the jump terms involving $u$ vanish due to the fact that

[TABLE]

and $\varphi_{kh,m}\in W^{1,\infty}(\Omega)$.

Similarly to the continuous problem, we define the discrete reduced cost functional $j_{kh}\colon Q\to\mathbb{R}$ by

[TABLE]

where $J$ is the cost function in (1). The discretized optimal control problem is then given as

[TABLE]

where $Q_{\text{ad}}$ is the set of admissible controls (9). We note, that the control variable $q$ is not explicitly discretized, cf. HinzeM_2005a . With standard arguments one proves the existence of a unique solution $\bar{q}_{kh}\in Q_{\text{ad}}$ of (52). Due to convexity of the problem, the following condition is necessary and sufficient for the optimality,

[TABLE]

As on the continuous level, the directional derivative $j^{\prime}_{kh}(q)(\partial q)$ for given $q,\partial q\in Q$ can be expressed as

[TABLE]

where $z_{kh}=z_{kh}(q)$ is the solution of the discrete adjoint equation

[TABLE]

The discrete adjoint state, which corresponds to the discrete optimal control $\bar{q}_{kh}$ is denoted by $\bar{z}_{kh}=z(\bar{q}_{kh})$. The variational inequality (53) is equivalent to the following pointwise projection formula, cf. (12),

[TABLE]

or

[TABLE]

on each $I_{m}$. Due to the fact that $\bar{z}_{kh}\in X^{0,1}_{k,h}$, we have $\bar{z}_{kh}(t,\gamma_{k}(t))$ is piecewise constant and therefore by the projection formula also $\bar{q}_{kh}$ is piecewise constant. As a result no explicit discretization of the control variable is required.

To prove Theorem 1.1 we first need estimates for the error in the state and in the adjoint variables for a given (fixed) control $q$. Due to the structure of the optimality conditions, we will have to estimate the error $\lVert z(\cdot,\gamma(\cdot))-z_{kh}(\cdot,\gamma_{k}(\cdot))\rVert_{I}$, where $z=z(q)$ and $z_{kh}=z_{kh}(q)$. Note, that $z_{kh}$ is not the Galerkin projection of $z$ due to the fact that the right-hand side of the adjoint equation (11) involves $u=u(q)$ and the right-hand side of the discrete adjoint equation (54) involves $u_{kh}=u_{kh}(q)$. To obtain an estimate of optimal order, we will first estimate the error $u-u_{kh}$ with respect to the $L^{2}(I;L^{1}(\Omega))$ norm. Note, that an $L^{2}$ estimate would not lead to an optimal result.

Theorem 4.1

Let $q\in Q$ be given and let $u=u(q)$ be the solution of the state equation (2) and $u_{kh}=u_{kh}(q)\in X^{0,1}_{k,h}$ be the solution of the discrete state equation (50). Then there holds the following estimate

[TABLE]

Proof

We denote by $e=u-u_{kh}$ the error and consider the following auxiliary dual problem

[TABLE]

where

[TABLE]

and the corresponding discrete solution $w_{kh}\in X^{0,1}_{k,h}$ defined by

[TABLE]

Using (51) for $e=u-u_{kh}$ and the Galerkin orthogonality for $w-w_{kh}$ we obtain,

[TABLE]

Using the local estimate from Theorem 3.2 with $B_{d,m}\subset\Omega_{1}$ for any $m=1,\dots,M,$ where $\Omega_{0}\subset\subset\Omega_{1}\subset\subset\Omega$, we obtain

[TABLE]

We take $\chi=i_{h}\pi_{k}w$, where $i_{h}$ is the modified Clément interpolant and $\pi_{k}$ is the projection defined in (17). Thus, by the triangle inequality, approximation theory, inverse inequality and the stability of the Clément interpolant in $L^{p}$ norm, we have

[TABLE]

$J_{2}$ can be estimated similarly since for $\chi=i_{h}\pi_{k}w$ by the triangle inequality we have

[TABLE]

As a result

[TABLE]

Using Lemma 2, we obtain

[TABLE]

and hence

[TABLE]

For the terms $J_{3}$ and $J_{4}$ we obtain using an $L^{2}$-estimate from MeidnerD_VexlerB_2008a

[TABLE]

$J_{5}$ can be estimated similarly since by the triangle inequality

[TABLE]

On the other hand using that $w\in L^{2}(I;W^{2,p}(\Omega_{0}))$ for $p>2$ and that $W^{2,p}(\Omega_{0}))\hookrightarrow C^{1}(\Omega_{0})$ for $p>2$, and using Assumption 1, we have

[TABLE]

where in the last two steps we used (56). Combining the estimate for $J_{1}$, $J_{2}$, $J_{3}$, $J_{4}$, $J_{5}$ and the above estimate and inserting them into (55) we obtain:

[TABLE]

Setting $p=|\ln h|$ completes the proof.

In the following theorem we provide an estimate of the error in the adjoint state for fixed control $q$.

Theorem 4.2

Let $q\in Q$ be given and let $z=z(q)$ be the solution of the adjoint equation (11) and $z_{kh}=z_{kh}(q)\in X^{0,1}_{k,h}$ be the solution of the discrete adjoint equation (54). Then there holds the following estimate

[TABLE]

Proof

First by the triangle inequality

[TABLE]

Using Proposition 2.3 and the assumptions on $\gamma$, we have similarly to Theorem 4.1

[TABLE]

Setting $p=|\ln h|$, we obtain

[TABLE]

Next, we introduce an intermediate adjoint state $\widetilde{z}_{kh}\in X^{0,1}_{k,h}$ defined by

[TABLE]

where $u=u(q)$ and therefore $\widetilde{z}_{kh}$ is the Galerkin projection of $z$. By the local best approximation result of Theorem 3.2 for any $\chi\in X^{0,1}_{k,h}$ we have

[TABLE]

The terms $J_{1}$, $J_{2}$, $J_{3}$, $J_{4}$ and $J_{5}$ can be estimated the same way as in the proof of Theorem 4.1 using the regularity result for the adjoint state $z$ from Proposition 2.3. This results in

[TABLE]

Setting $p=|\ln h|$ and taking square root, we obtain

[TABLE]

It remains to estimate the corresponding error between $\widetilde{z}_{kh}$ and $z_{kh}$. We denote $e_{kh}=\widetilde{z}_{kh}-z_{kh}\in X^{0,1}_{k,h}$. Then we have

[TABLE]

As in the proof of Lemma 5 we use the fact that

[TABLE]

holds for all $v\in X^{0,1}_{k,h}$. Applying this inequality together with the discrete Sobolev inequality, see BrennerSC_ScottLR_2008 , results in

[TABLE]

Therefore

[TABLE]

Using Theorem 4.1 we obtain

[TABLE]

Combining this estimate with (59) we complete the proof.

Using the result of Theorem 4.2 we proceed with the proof of Theorem 1.1.

Proof

Due to the quadratic structure of discrete reduced functional $j_{kh}$ the second derivative $j^{\prime\prime}_{kh}(q)(p,p)$ is independent of $q$ and there holds

[TABLE]

Using optimality conditions (10) for $\bar{q}$ and (53) for $\bar{q}_{kh}$ and the fact that $\bar{q},\bar{q}_{kh}\in Q_{\text{ad}}$ we obtain

[TABLE]

Using the coercivity (60) we get

[TABLE]

Applying Theorem 4.2 completes the proof.