Optimal finite element error estimates for an optimal control problem governed by the wave equation with controls of bounded variation

This work discusses the finite element discretization of an optimal control problem for the linear wave equation with time-dependent controls of bounded variation. The main focus lies on the convergence analysis of the discretization method. The state equation is discretized by a space-time finite element method. The controls are not discretized. Under suitable assumptions optimal convergence rates for the error in the state and control variable are proven. Based on a conditional gradient method the solution of the semi-discretized optimal control problem is computed. The theoretical convergence rates are confirmed in a numerical example.

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1 Introduction

In this paper we derive a priori error estimates for a finite element discretization of the following optimal control problem governed by the linear wave equation:

 (P)⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩minu∈BV(0,T)m12∥yu−yd∥2L2(ΩT)+∑mj=1αj∥Dtuj∥M(I)=:J(y,u)subject to (W)⎧⎪ ⎪⎨⎪ ⎪⎩∂tty−Δy=f=∑mj=1ujgj in I×Ωy=0 on I×∂Ω(y,∂ty)=(y0,y1) in {0}×Ω,

where , with , is a convex, polygonal/polyhedral bounded domain. For we denote . The desired state is assumed to satisfy . The time depending controls are given by , and is endowed with the norm . Here is the space of Borel measures, endowed with the total variation norm . Further, let with pairwise disjoint supports and . The initial data is chosen as . Finally, we set .

In this work we focus on controls of bounded variation in time. By using the total variation norm in , sparsity in the derivative of the controls is promoted, resulting in locally constant controls. This is in particular the case if the derivative of the optimal control is a linear combination of Dirac functions. Optimal control problems with -controls are already analyzed for elliptic and parabolic state equations in [6, 4, 13, 7, 8, 10].

Since our article deals with a priori error estimates of a finite element discretization for the control problem , we briefly discuss previous works on error estimates for PDE control problems with -controls. In [4] the authors discretize the time-dependent controls by cellwise constant functions. The state equation is discretized by piecewise constant finite elements in time and linear continuous finite elements in space. Based on this discretization approach, the authors show that the optimal value of the cost functional and the states converge with an order of in time and linear in space. However, numerical experiments in [4] indicate better results. In [13] the authors analyze a finite element discretization of an elliptic control problem with -controls in a one dimensional setting. As in our case the controls are not discretized. The main contribution of this work is the derivation of optimal error estimates for the control variable in the -norm. Their analysis relies on the one dimensional setting and on structural assumption on the optimal adjoint state which guarantee that the optimal control is piecewise constant and has finitely many jumps. In our work we derive similar optimal error estimates also for the problem with a multi-dimensional wave equation and our analysis relies partially on techniques developed in the former work.

Next we briefly address the difficulties in the derivation of finite element error estimates for optimal control problems with PDEs and -controls. Standard techniques for the derivation of finite element error estimates, see e.g. [9], cannot be applied due to the non-smoothness of the cost functional and the non-reflexivity of . In the last years several papers concerning the derivation of finite element error estimates for optimal control problems with measure-valued controls appeared, see e.g. [15, 18]. Using the fact that for one dimensional controls, is isomorphic to , some techniques from these works are used to derive error estimates for -controls. Finally, we mention that the literature on finite element error estimates for optimal control problems governed by the wave equation is very limited. To our knowledge the only existing work in this context is [18] which uses the space-time finite element discretization developed and analyzed in [19]. Our work also relies on this discretization method for the state equation and its error analysis.
The main contribution of this work is the derivation of an optimal error estimate of the control variable in the -norm and of the state variable in the -norm. The state equation is discretized by a space-time finite element method with piecewise linear and continuous Ansatz- and test-functions from [19]. The weak formulation of the discrete state equation is augmented with a stabilization term involving the stabilization parameter . Stability of the method depends on the value of this parameter. Moreover, for certain values of this parameter the method is equivalent to wellknown time stepping schemes, like the Crank-Nicolson scheme or the Leap-Frog scheme. The -controls are not discretized. Due to fact that the controls are only time-dependent the problem under consideration can be reformulated as a measure-valued control problem. Based on the optimality conditions of the continuous and discrete optimal control problem the error in the state variable in the -norm can be represented in terms of the finite element error of the state and adjoint state equation in the -norm resp. the as well as the error in the control variable in the -norm. The convergence rates for the finite element error of the state and adjoint state are obtained from [19]. Under the assumption that the continuous and time depending functions

 ¯p1,i:t↦−∫Tt∫Ω¯¯¯pgi dx ds, for i=1,⋯,m,

where is the optimal adjoint state, is bounded by , is equal to at finitely many points in , and the second derivatives of do not vanish in these points (see (A1) and (A2)), it follows that the continuous optimal control is piecewise constant and has finitely many jumps. To obtain this information about the form of the optimal BV control, using , is particularly elementary because we consider controls in one dimension. Furthermore, it is proven that the solution of the discrete problem has the same number of jumps which are located close to the jumps of the continuous optimal control. Using these properties the error of the optimal control in the -norm is estimated in terms of the error of the state variable in the -norm. Using a bootstrapping argument optimal rates for the error in the state and control variable as well as for the optimal value of the cost are proven. These rates are confirmed by a numerical example with known solution.
This work has the following structure. Section 2 summarizes several needed results on the regularity of weak solutions of the wave equation. In section 3 the space-time finite element method from [19] is presented. Moreover, important stability results as well as a priori error estimates are stated. Section 4 deals with the reformulation of the -control problem as a measure-valued control problem and with the analysis of this problem. In particular, first order optimality conditions are derived. The next section 5 is concerned with discretization of the control problem. It is based on the mentioned space-time finite element method and the variational discretization concept. In section 6 the error estimates for the optimal state and control variable as well as the optimal functional value are derived. Finally, in section 7 a generalized conditional gradient method is introduced which applicable in the context of controls which are not discretized. Based on this method a problem with known solution is solved and the theoretical error estimates are confirmed.

2 Preliminaries on the Wave Equation

We consider , as convex, polygonal/polyhedral domain. Let

be the non-decreasing eigenvalues of the Laplace operator

with homogeneous boundary conditions and let

be the corresponding system of eigenfunctions, which are orthonormal complete in

, and orthogonal complete in . Hence, let us introduce for the Hilbert spaces

 Hα={w∈L2(Ω)∣∣∥w∥2Hα:=∑k≥1(λ(k))α⟨w,μk⟩2L2(Ω)<∞}.

For we get respectively . The convexity of implies that . In general holds for . We denote the dual space of by . Next we introduce the weak solution of the wave equation with the forcing function , initial displacement , and initial velocity .

Definition 1.

([14], Chap.IV, Sec.4)
Let . We call a function with a weak solution of , if

 ∫T0−(∂ty,∂tη)L2(Ω)+(∇y,∇η)L2(Ω) dt=(y1,η(0))L2(Ω)+∫T0(f,η)L2(Ω) dt (1)

for any such that , , and satisfies the initial condition .

For the following existence and regularity results of weak solutions of the wave equation we refer to [19, Proposition 1.1., 1.3.]:

Theorem 1.

For each there exists a unique weak solution of . Moreover, there exists a constant such that the weak solution satisfies

 ∥y∥C(¯¯I;Hα+1)+∥∂ty∥C(¯¯I;Hα)+∥∂tty∥Lκ(I;Hα−1)≤c(∥y0∥Hα+1+∥y1∥Hα+∥f∥Lκ(I;Hα)) (2)

provided and

 ∥y∥C(¯¯I;Hα+2)+∥∂ty∥C(¯¯I;Hα+1)+∥∂tty∥C(¯¯I;Hα)≤c(∥y0∥Hα+2+∥y1∥Hα+1+∥f∥W1,1(I;Hα)). (3)

provided with , .

Proof.

The proof can be found in [19, Proposition 1.3, Remark 1.2]. ∎

Definition 2.

Let us define the following continuous linear operators:

 L:L2(ΩT)→L2(ΩT), f↦y(f)andQ:H10(Ω)×L2(Ω)→L2(ΩT), (y0,y1)↦y(y0,y1)

The function denotes the weak solution of the wave equation with and forcing function . The function denotes the weak solution of the wave equation with initial datum and and .

Lemma 1.

The adjoint operator of is given by where is the weak solution of the backwards in time equation

 (W∗)⎧⎪⎨⎪⎩∂ttp−Δp=w in I×Ωp=0 on I×∂Ω(p,∂tp)=(0,0) in {T}×Ω. (4)

3 Approximation of the Wave Equation

In the following we introduce the space-time finite element method for the discretization of the wave equation. This method can be found in [19]. We consider a mesh consisting of a finite set of triangles (for ) or tetrahedra (for ) with , where denotes the diameter of . We assume that the family of meshes is admissible, shape regular and quasi-uniform. Since is polygonal and convex, we require that holds. We denote the space of piecewise linear and continuous finite elements based on the triangulation by and its nodal basis by.

3.1 Space-Time Finite Element Method

We discretize the time interval uniformly with the time nodes and the stepsize . We denote the set of time nodes by . Then we introduce the space of piecewise linear and continuous functions with respect to by

 Sτ:={w∈C(¯¯¯I)|w|[tk−1,tk] linear % , 1≤k≤M}.

The standard hat functions form a basis , of this discrete space. Finally, we use the notation with .

Definition 3.

Let . We call a discrete solution of (1) if satisfies:

 ∫T0−(∂tyϑ,∂tη)L2(Ω)−(σ−16)τ2(∇∂tyϑ,∇∂tη)L2(Ω)+(∇yϑ,∇η)L2(Ω) dt=(y1,η(0))L2(Ω)+∫T0(f,η)L2(Ω) dt (5)

for all with and initial condition , where is the Ritz projection on , i.e.

 (∇Rhy0,∇φ)L2(Ω)=(∇y0,∇φ)L2(Ω)∀φ∈Sh.
Remark 1.

Here plays the role of a stabilization parameter. With an increasing value of the method becomes more stable. For the method is unconditionally stable, see [19].

3.2 A Priori Error Estimates for the Space-Time Finite Element Method

Next we make an assumption on the relationship between and which ensures stability of the method for .

Assumption 1.

Let be arbitrary and fixed. Moreover, let be the smallest constant in the inverse inequality for all . Moreover, let a be the constant in this a priori estimate for the Ritz projection . From now on it is assumed that

1. ,

2. ,

3. .

Remark 2.

This space-time finite element method is related to well-known time-stepping schemes. For it is related to the explicit Leap-Frog-method and for to the Crank-Nicolson scheme, see also [17, Remark 5.1, 5.4]. A more detailed discussion can be found in [19].

Lemma 2.

The solution of (5) for satisfies the following inequality

 ∥yϑ∥C(¯¯I;L2(Ω))≤c(∥y0∥H10(Ω)+∥y1∥L2(Ω)+∥f∥L1(I;L2(Ω))) (6)

with a constant independent of , , and .

Proof.

The result follows directly from [19, Theorem 2.1, Remark 2.1]. ∎

Theorem 2.

The following error estimates hold:

 (7)

provided as well as

 (8)

provided .

Proof.

The result follows directly from [19, Theorem 4.1., 4.3. and comments in its proof]. ∎

4 Equivalent Problem (~P)

In this section we introduce a specific isomorphism between and . Based on this isomorphism is equivalently formulated as a measure valued control problem. First of all we prove existence and uniqueness of a solution to .

Theorem 3.

Problem has a unique solution in .

Proof.

Utilizing the fact, that the forward mapping is continuous from to , the proof can be carried out along the line of [4, Theorem 3.1]. ∎

Next we introduce several linear and continuous operators and discuss its properties. The operator is given by

 (v,c)↦∑mj=1(∫t0 dvj(s)−1T∫T0∫t0 dvj(s) ds+cj)gj. (9)

The measures are the derivatives of the generated BV-function and are the mean values. Next, we define the predual operator of given by

 B∗:q↦(w′1,…,w′m,∫T0∫Ωqg1 dx dt,…,∫T0∫Ωqgm dx dt)

where solves

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩−w′′j=∫Ωq(⋅,x)gj(x) dx−1T∫T0∫Ωq(t,x)gj(x) dx dtin (0,T)w′j(0)=w′j(T)=0% with∫T0wj(t) dt=0forj=1,…,m. (10)
Proposition 1.

The operator is well defined and the predual of , i.e. the following holds

 ∫ΩTB(v,c)q dx dt=⟨(v,c),B∗(q)⟩

for all and for all .

Proof.

The equation (10) has a unique solution , since and has zero mean. Moreover, we have . Thus, the operator is well defined. Moreover, there holds

 ⟨(v,c),B∗(q)⟩=∑mj=1∫T0w′j dvj+∑mj=1cj∫T0∫Ωqgj dx dt=∑mj=1∫T0−w′′j∫t0 dvj dt+∑mj=1cj∫T0∫Ωqgj dx dt=∑mj=1∫T0(∫Ωqgj dx−1T∫T0∫Ωqgj dx dt)∫t0 dvj dt+∑mj=1cj∫T0∫Ωqgj dx dt=∫T0∫Ωq∑mj=1(∫t0 dvj−1T∫T0∫t0 dvj dt+cj)gj dx dt=∫ΩTB(v,c)q dx dt

for all and for all . The use of integration by parts is justified by the density of in . ∎

Proposition 2.

Let , be the solution of (10). Then there holds

 w′j(t)=∫Tt∫Ωq(s,x)gj(x) dx ds+(t−T)T∫T0∫Ωq(t,x)gj(x) dx dt.
Proposition 3.

The operator is an isomorphism.

Proof.

The inverse of is given by

 B−1:∑mj=1ujgj↦(u′1,…,u′m,1T∫T0u1 dt,…,1T∫T0um dt).

Using we can rewrite into the equivalent problem

with defined by .

4.1 First-Order optimality condition of (~P)

In the following a necessary and sufficient first-order optimality condition of is presented as well as sparsity results for the derivative of the optimal control. Let be the unique optimal pair. We define the quantities and by

 ¯¯¯p1,i:=−∫Tt∫Ω¯¯¯pgi dx ds

for .

Theorem 4.

The pair is an optimal control of if and only if

 ¯¯¯p1,i ∈αi∂∥¯¯¯vi∥M(I)i=1,…,m, (11) ¯¯¯p1(0) =0. (12)

Equivalently it holds

 ⟨v−¯¯¯vi,¯¯¯p1,i⟩M(I),C0(I)+αi∥¯¯¯vi∥M(I)≤αi∥v∥M(I)∀v∈M(I) and i=1,…,m (13)

and .

Proof.

The proof of Theorem 4 is done along the lines of the proof of
[4, Theorem 3.3]. By the convexity of we have, that is an optimal control of if and only if

 0∈∂(12∥S(¯¯¯v,¯¯c)−yd∥2L2(ΩT)+∑mj=1αj∥¯¯¯vj∥M(I))⊆(M(I)m×Rm)∗=(M(I)∗)m×Rm.

Define the following function for . Its Gateaux derivative has the form

 DF(v,c)(v,c)=B∗L∗(S(v,c)−yd)∈C0(I)m×Rm

According to the theory of convex analysis, e.g. [11, Proposition 5.6], we have

 0∈DF(v,c)(¯¯¯v,¯¯c)+∂(∑mi=1αi∥¯¯¯vi∥M(I)). (14)

Using

 ∂(∑mi=1αi∥¯¯¯vi∥M(I))=((αi∂∥¯¯¯vi∥M(I))mi=10)⊆(M(I)∗)m×Rm

and (14) as well as Proposition 2 imply

 ¯¯¯p1,i∈αi∂∥¯¯¯vi∥M(I)∀i=1,…,m,¯¯¯p1(0)=0. (15)

The following proposition is a consequence of [5, Proposition 3.2.]:

Proposition 4.

Let be an optimal control of , then for all and given in (11) holds

• ,

• ,

• , where is the Jordan decomposition of .

Remark 3.

Let us note that the boundary property of , i.e. , and the continuity of imply with Proposition 4, c), that there exists a such that

5 The Variationally Discretized Problem

In this section we introduce a discretized version of and discuss its properties. We use the concept of variational discretization in which the control is not discretized. In particular, we consider the problem :

 (~Psemiϑ){minv∈M(I)mc∈Rm12∥Sϑ(v,c)−yd∥2L2(ΩT)+∑mj=1αj∥vj∥M(I)=:Jϑ(v,c)

with defined by Here is defined by , where solves (5) for a source f and . The operator is defined by , where solves (5) with as initial datum and .

Remark 4.

We can represent the adjoint of in the form with , and . This is true since and and thus and can be used in (5) as test functions for the forwards and backwards equation. Hence, Theorem 8 and Lemma 2 are valid for as well.

Theorem 5.

The problem has a solution in .

Proof.

The existence of an optimal control for can be similarly shown as in the proof of Theorem 3. ∎

Note, that a BV-representation of the solutions , of , respectively are defined by

 ¯¯¯u(t):=∫t0d¯¯¯v(s)−1T∫T0∫t0d¯¯¯v(s) dt+¯¯c,and¯¯¯uϑ(t):=∫t0d¯¯¯vϑ(s)−1T∫T0∫t0d¯¯¯vϑ(s) dt+¯¯cϑ. (16)

Next we define the quantities and

 ¯¯¯p1,ϑ,j:=−∫Tt∫Ω¯¯¯pϑgj dx dsforj=1,…,m,

which is continuously differentiable and piecewise quadratic in time.

Theorem 6.

The pair is a optimal control of if and only if

 ¯¯¯p1,ϑ,i ∈