# Space-time finite element discretization of parabolic optimal control problems with energy regularization

We analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space L^2(0,T;H^-1(Ω)). By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babuška's theorem, we prove unique solvability in the continuous case. Furthermore, we establish the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Various numerical examples confirm the theoretical findings. We emphasize that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an L^2 regularization and with a sparse optimal control approach.

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## 1 Introduction

In this paper, we consider and analyze continuous space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of the following parabolic optimal control problem: For a given target function , we want to minimize the cost functional

 J(u,z):=12∫Q|u−ud|2dxdt+12ϱ∥z∥2L2(0,T;H−1(Ω)) (1)

subject to the linear parabolic state equation

 ∂tu−Δxu=zinQ,u=0onΣ,u=0onΣ0, (2)

where is the space-time domain with the lateral boundary , and . Moreover, , , is a bounded Lipschitz domain, is the final time, and is some regularization parameter.

In our recent paper [18], we have considered the related standard optimal control problem with regularization in , i.e.,

 J(u,z):=12∫Q|u−ud|2dxdt+12ϱ∥z∥2L2(Q), (3)

subject to (2). In both cases, we can numerically solve the corresponding parabolic forward-backward optimality systems at once. This allows not only for a more efficient solution of the global system, but also for parallelization in space and time, and for adaptive discretizations simultaneously in space and time. In contrast to classical time-stepping methods or discontinuous Galerkin (dG) methods which are defined with respect to time slices or slabs, see, e.g., the monographs [16] and [31], and the review article [9] on parallel-in-time methods, we use fully unstructured simplicial space-time meshes for the numerical solution of the parabolic state equation (2), see the recent review article [29] and the related references therein.

The standard approach for distributed control problems is to consider the control in . There is a huge number of publications on the standard setting (3) with -regularization. We here only refer to the monographs [5, 12, 32], to the more recent papers [10, 21, 22] on discontinuous (dG) and continuous Galerkin time-slice finite element methods, [23] on full space-time dG finite element methods, [11] on space-time adaptive wavelet methods, [13, 17] on multiharmonic methods, [1] on proper orthogonal decomposistion, [7]

on low-rank tensor method, and to our very recent paper

[18] on completely unstructured space-time finite element methods for optimal control of parabolic equations based on -regularization, and the references given therein. However, since the state is well defined as the solution of the forward heat equation for , we may also consider the tracking type functional as given in (1). Applying integration by parts also in time to derive a variational formulation for the adjoint equation, we end up, in contrast to the case of

regularization, with a positive definite but skew-symmetric bilinear form describing the optimality system. In this paper, we provide a complete numerical analysis for both the continuous and discrete system.

The rest of the paper is structured as follows. In Section 2, we introduce some notation and state some preliminary results on the solvability and numerical analysis of the parabolic initial-boundary value problem that serves as state equation in the optimal control problem. In Section 3, we analyze the unique solvability of the continuous optimality system, whereas Section 4 is devoted to the numerical analysis of the space-time finite element approximation. Numerical results are presented in Section 5 Finally, some conclusions are drawn in Section 6.

## 2 Preliminaries

In this section, we introduce basic notations, and summarize some recent results on space-time finite element methods for the numerical solution of the state equation (2). For the mathematical analysis of parabolic initial boundary value problems in space-time Sobolev spaces, see [14, 15], and [20, 33] for Bochner spaces of abstract functions, mapping the time interval to some Hilbert or Banach space.

Following the latter approach, we define

 X := L2(0,T;H10(Ω))∩H10,(0,T;H−1(Ω)) = {v∈L2(0,T;H10(Ω)):∂tv∈L2(0,T;H−1(Ω)),v=0 on Σ0}, Y := L2(0,T;H10(Ω)),Y∗:=L2(0,T;H−1(Ω)),

using the standard Sobolev spaces and its dual . Note that we have as used in [20]. The related norms are given by

 ∥u∥X:=[∥wu∥2Y+∥u∥2Y]1/2,∥v∥Y:=∥∇xv∥L2(Q),

where is the unique solution of the variational formulation [27]

 ∫Q∇xwu⋅∇xvdxdt=⟨∂tu,v⟩Q,∀v∈Y. (4)

The standard weak formulation of the initial boundary value problem (2) reads as follows: Given , find such that

 b(u,v)=⟨z,v⟩Q,∀v∈Y, (5)

with the bilinear form ,

 b(u,v):=∫Q[∂tuv+∇xu⋅∇xv]dxdt,∀(u,v)∈X×Y, (6)

and the linear form with the duality pairing as extension of the inner product in . Similarly, the first integral in (6) has to be understood as duality pairing as well.

The bilinear form is bounded,

 |b(u,v)|≤√2∥u∥X∥v∥Y,∀(u,v)∈X×Y, (7)

and satisfies the inf-sup stability condition [27, Theorem 2.1]

 inf0≠u∈Xsup0≠v∈Yb(u,v)∥u∥X∥v∥Y≥12√2. (8)

Moreover, for , we define

 ˜u(x,t)=∫t0v(x,s)ds,(x,t)∈Q

to obtain

 b(˜u,v)=∥v∥2L2(Q)+12∥∇x˜u(T)∥2L2(Ω)>0.

Hence, we can apply the Nec̆as-Babuška theorem [2, 24] to conclude that the variational problem (5) is well-posed, see also [3, 6, 8].

For the finite element discretization of the variational formulation (5), we introduce conforming space-time finite element spaces and , where we assume . In particular, we may use spanned by continuous and piecewise linear basis functions which are defined with respect to some admissible decomposition of the space-time domain into shape regular simplicial finite elements , and which are zero at the initial time and at the lateral boundary , where denotes a suitable mesh-size parameter, see, e.g., [6, 8, 27]. Then the finite element approximation of (5) is to find such that

 b(uh,vh)=⟨z,vh⟩Q,∀vh∈Yh. (9)

When replacing (4) by its finite element approximation to find such that

 ∫Q∇xwu,h⋅∇xvhdxdt=∫Q∂tuvhdxdt,∀vh∈Yh, (10)

we can define a discrete norm

 ∥u∥Xh:=[∥wu,h∥2Y+∥u∥2Y]1/2.

As in the continuous case, see (8), we can prove a discrete inf-sup condition, see [27, Theorem 3.1],

 12√2∥uh∥Xh≤sup0≠vh∈Yhb(uh,vh)∥vh∥Y,∀uh∈Xh. (11)

Hence, we conclude unique solvability of the Galerkin scheme (9), and we obtain the following quasi-optimal error estimate, see [27, Theorem 3.2]:

 ∥u−uh∥X0,h≤5infzh∈X0,h∥u−zh∥X0. (12)

In particular, when assuming , this finally results in the energy error estimate, see [27, Theorem 3.3],

 ∥u−uh∥L2(0,T;H10(Ω))≤ch|u|H2(Q). (13)

## 3 The first-order optimality system

We now consider the optimal control problem to minimize (1) subject to the heat equation (2). As in (4), we define as the unique solution of the variational problem

 ∫Q∇xwz⋅∇xvdxdt=⟨z,v⟩Q∀v∈Y, (14)

to conclude

 ∥z∥2L2(0,T;H−1(Ω))=∥∇xwz∥2L2(Q)=⟨z,wz⟩Q.

Now, using standard arguments, we can write the first-order optimality system as the primal problem

 ∂tu−Δxu=zinQ,u=0onΣ,u=0onΣ0,

 −∂tp−Δxp=u−udinQ,p=0onΣ,p=0onΣT, (15)

 p+ϱwz=0inQ. (16)

Using the variational formulation (5) of the primal problem, inserting the definition (14), and the gradient equation (16), we get a first variational equation to find such that

 1ϱ∫Q∇xp⋅∇xvdxdt+∫Q[∂tuv+∇xu⋅∇xv]dxdt=0,∀v∈Y.

On the other hand, when considering the variational formulation of the adjoint problem and integrating by parts also in time, we arrive at the second variational equation

 −∫Q[p∂tq+∇xp⋅∇xq]dxdt+∫Quqdxdt=∫Qudqdxdt,∀q∈X.

Hence, we end up with a variational problem to find such that

 B(u,p;v,q)=⟨ud,q⟩L2(Q),∀(v,q)∈Y×X, (17)

with

 B(u,p;v,q):=1ϱa(p,v)+b(u,v)−b(q,p)+c(u,q). (18)

Here, the bilinear form is the same as used in Section 2,

 a(p,v):=∫Q∇xp⋅∇xvdxdt,andc(u,q):=∫Quqdxdt.

Note that and .

###### Theorem 1.

For , the variational problem (17) admits a unique solution satisfying the a priori estimates

 ∥u∥X≤8ϱ∥ud∥X∗,∥p∥Y≤√28∥ud∥X∗.
###### Proof.

Using the Riesz representation theorem, we introduce operators , , and , satisfying, for and ,

 ⟨Ap,v⟩Q=a(p,v),⟨Bu,v⟩Q=b(u,v),⟨Cu,q⟩Q=c(u,q).

Hence, we can write the variational problem (17) as operator equation

 (1ϱAB−B∗C)(pu)=(0ud).

Since the operator is bounded and elliptic, we can determine to obtain the Schur complement system

 [C+ϱB∗A−1B]u=udinX∗. (19)

For , we first have

 ∥¯¯¯p∥2Y=∫Q|∇x¯¯¯p|2dxdt=a(¯¯¯p,¯¯¯p)=⟨A¯¯¯p,¯¯¯p⟩Q=⟨B∗A−1Bu,u⟩Q.

From the stability condition (8) for the state equation, see Section 2, we immediately get

 12√2∥u∥X≤sup0≠v∈Y⟨Bu,v⟩Q∥v∥Y=sup0≠v∈Y⟨A¯¯¯p,v⟩Q∥v∥Y≤∥¯¯¯p∥Y.

Hence, we have

 ⟨(C+ϱB∗A−1B)u,u⟩Q≥18ϱ∥u∥2Xfor allu∈X.

Thus, we conclude unique solvability of the Schur complement system (19), and from

 18ϱ∥u∥2X≤⟨(C+ϱB∗A−1B)u,u⟩Q=⟨ud,u⟩Q≤∥ud∥X∗∥u∥X

we obtain the first estimate. Now, the boundedness of , i.e., the boundedness (7) of the bilinear form yields

 ∥p∥2Y=a(p,p)=−ϱb(u,p)≤√2ϱ∥u∥X∥p∥Y,

i.e.,

 ∥p∥Y≤√2ϱ∥u∥X≤√28∥ud∥X∗.

Although unique solvability of the variational problem (17) already implies a related stability condition for the bilinear form , we will present an alternative proof for this stability condition in order to be able to derive related results for the Galerkin discretization of (17).

###### Lemma 1.

The bilinear form (18) satisfies the stability condition

 116ϱ[∥u∥2X+∥p∥2Y]1/2≤sup0≠(v,q)∈Y×XB(u,p;v,q)[∥v∥2Y+∥q∥2X]1/2 (20)

for all , when assuming .

###### Proof.

For , let be the unique solution of the variational problem (4). For and arbitrary , we have . Choosing , we obtain

 B(u,p;v,q) = 1ϱa(p,u+wu+αp)+b(u,u+wu+αp)−b(αu,p)+c(u,αu) ≥ αϱ∥p∥2Y+1ϱa(p,u+wu)+b(u,u+wu).

Following the proof of [27, Theorem 2.1], we use

 b(u,u+wu) = ∫Q[∂tu(u+wu)+∇xu⋅∇x(u+wu)]dxdt =12∥u(T)∥2L2(Ω)+∫Q∇xwu⋅∇xwudxdt+∥u∥2Y+∫Q∇xu⋅∇xwudxdt ≥∥wu∥2Y+∥u∥2Y−∥u∥2Y∥wu∥2Y ≥12[∥wu∥2Y+∥u∥2Y]=12∥u∥2X.

Moreover, for , we have

 a(p,u+wu) = ∫Q∇xp⋅∇x(u+wu)dxdt≥−∥p∥Y∥u+wu∥Y ≥ −12γ∥p∥2Y−12γ∥u+wu∥2Y ≥ −12γ∥p∥2Y−γ(∥u∥2Y+∥wu∥2Y)=−12γ∥p∥2Y−γ∥u∥2X.

Choosing and , we get

 B(u,p;v,q) ≥ 1ϱ(α−12γ)∥p∥2Y+(12−γϱ)∥u∥2X = 14[∥u∥2X+∥p∥2Y].

On the other hand, we have

 ∥q∥2X+∥v∥2Y = α2∥u∥2X+∥u+wu+αp∥2Y ≤ α2∥u∥2X+(∥u∥Y+∥wu∥Y+α∥p∥Y)2 ≤ α2∥u∥2X+(2+α2)(∥u∥2Y+∥wu∥2Y+∥p∥2Y) ≤ 2(1+α2)[∥u∥2X+∥p∥2Y] = 2(2+116ϱ2+4ϱ2)[∥u∥2X+∥p∥2Y] ≤ (4+18+8)1ϱ2[∥u∥2X+∥p∥2Y]≤16ϱ2[∥u∥2X+∥p∥2Y],

provided that . Therefore,

follows. This concludes the proof. ∎

## 4 Discretization

As before, let and be some conforming space-time finite element spaces satisfying . Again, we choose , but now we use . By construction, we have .

Instead of (10), we now consider the variational formulation to find such that

 ∫Q∇xwh⋅∇xvhdxdt=∫Q∂tuvhdxdt,∀vh∈Yh, (21)

to define the discrete norm

 ∥u∥X0,h:=[∥wu,h∥2Y+∥u∥2Y]1/2.

The space-time finite element discretization of the variational formulation (17) is to find such that

 B(uh,ph;vh,qh)=⟨ud,qh⟩L2(Q),∀(vh,qh)∈Yh×X0,h. (22)

As in the continuous case, see Theorem 1, and following [27, Section 3], we can confirm a discrete inf-sup condition for the bilinear form .

###### Lemma 2.

The bilinear form (18) satisfies the discrete stability condition

 116ϱ[∥uh∥2X0,h+∥ph∥2Y]1/2≤sup0≠(vh,qh)∈Yh×X0,hB(uh,ph;vh,qh)[∥vh∥2Y+∥qh∥2X0,h]1/2 (23)

for all , when assuming and .

###### Proof.

Since the proof follows the lines of the proof of Theorem 1, we only sketch the most important steps.

For , let be the unique finite element solution of the variational problem (21). For , and due to , we then have , . Moreover, set .

As in the proof of Theorem 1, we now conclude

 B(uh,ph;vh,qh)≥14[∥ph∥2Y+∥wuh,h∥2Y+∥uh∥2Y]=14[∥uh∥2X0,h+∥ph∥2Y].

On the other hand, and as in the proof of Theorem 1, we have

 ∥qh∥2X0,h+∥vh∥2Y = α2∥uh∥2X0,h+∥uh+wuh,h+αph∥2Y ≤ 16ϱ2[∥uh∥2X0,h+∥ph∥2Y].

Now the assertion follows as in the continuous case. ∎

The discrete inf-sup condition (23) implies unique solvability of the space-time finite element scheme (22). By combining (22) with (17) and by using the inclusions and , we also conclude the Galerkin orthogonality

 B(u−uh,p−ph;vh,qh)=0,∀(vh,qh)∈Yh×X0,h. (24)
###### Theorem 2.

Let and be the unique solutions of the variational problems (17) and (22), respectively, where and . Furthermore, assume that . Then there holds the discretization error estimate

 ϱ[∥u−uh∥2X0,h+∥p−ph∥2Y]1/2≤ch[1ϱ∥p∥H2(Q)+∥u∥H2(Q)].
###### Proof.

For arbitrary , the discrete inf-sup condition (23) and the Galerkin orthogonality (24) immediately yield the estimates

 116ϱ[∥uh−zh∥2X0,h+∥ph−rh∥2Y]1/2 ≤sup0≠(vh,qh)∈Yh×X0,hB(uh−zh,ph−rh;vh,qh)[∥vh∥2Y+∥qh∥2X0,h]1/2 =sup0≠(vh,qh)∈Yh×X0,hB(u−zh,p−rh;vh,qh)[∥vh∥2Y+∥qh∥2X0,h]1/2.

Now we consider

 B(u−zh,p−rh;vh,qh)[∥vh∥2Y+∥qh∥2X0,h]1/2 =1ϱa(p−rh,vh)+b(u−zh,vh)−b(qh,p−rh)+c(u−zh,qh)[∥vh∥2Y+∥qh∥2X0,h]1/2 ≤1ϱa(p−rh,vh)+b(u−zh,vh)∥vh∥Y−b(qh,p−rh)[∥vh∥2Y+∥qh∥2X0,h]1/2+c(u−zh,qh)∥qh∥X0,h ≤1ϱ∥p−rh∥Y+√2∥u−zh∥X0+c∥u−zh∥L2(Q)−b(qh,p−rh)[∥vh∥2Y+∥qh∥2X0,h]1/2.

Integrating by parts in time and using in , we obtain

 b(qh,p−rh) = ∫Q[∂tqh(p−rh)+∇xqh⋅∇x(p−rh)]dxdt =∫Ωqh(T)(p(T)−rh(T))dx+∫Q[−qh∂t(p−rh)+∇xqh⋅∇x(p−rh)]dxdt ≤∥qh(T)∥L2(Ω)∥p(T)−rh(T)∥L2(Ω)+√2∥qh∥Y[∥∂t(p−rh)∥Y∗+∥p−rh∥Y].

Therefore, the inequalities

 b(qh,p−rh)[∥vh∥2Y+∥qh∥2X0,h]1/2 ≤ b(qh,p−rh)∥qh∥Y ≤∥qh(T)∥L2(Ω)∥qh∥Y∥p(T)−rh(T)∥L2(Ω)+√2[∥∂t(p−rh)∥Y∗+∥p−rh∥Y]

follow. Let with or be a shape regular simplicial finite element with mesh size . For a piecewise linear finite element function, we then have the equivalence

 ∫τμℓ[vh(x)]2dx≃hμℓμ+1∑k=1v2ℓk,

where the are the local nodal values of . Hence, we can write

 ∥qh(T)∥2L2(Ω)=∑τnℓ∈ΣT∥qh(T)∥2L2(τnℓ)≃∑τnℓ∈ΣThnℓn+1∑k=1v2ℓk

as well as

 ∥qh∥2L2(Q)=∑τn+1ℓ∈Q∥vh∥2L2(τn+1ℓ)≃∑τn+1ℓ∈Qhn+1ℓn+2∑k=1v2ℓk.

Thus, we conclude

 ∥qh(T)∥2L2(Ω)≤ch−1T∥qh∥2L2(Q),

where is the globally quasi-uniform mesh size of all space-time finite elements sharing . Since we have on , we finally obtain

 ∥qh(T)∥L2(Ω)