Optimal finite element error estimates for an optimal control problem governed by the wave equation with controls of bounded variation
Sebastian Engel Philip Trautmann Boris Vexler
math.OC cs.NA math.NA 26A45, 49J20, 49M25, 65N15, 65N30


































	





Optimal finite element error estimates for an optimal control problem governed by the wave equation with controls of bounded variation.
    S. Engel
[
Technical University of Munich, Chair for Optimal Control, Department of Mathematics,
Boltzmannstr. 3, 85748 Garching b. München, Germany,
(< 	sebastian.engel@ma.tum.de>).
Funding: The first author's work was supported by the International Research Training Group IGDK, funded by the German Science Foundation (DFG) and the Austrian Science Fund (FWF)],
P. Trautmann
[
University of Graz, Institute of Mathematics and Scientific Computing,
Heinrichstraße 36, 8010 Graz, Austria
(<philip.trautmann@uni-graz.at>)],
B. Vexler
[
Technical University of Munich, Chair for Optimal Control, Department of Mathematics,
Boltzmannstr. 3, 85748 Garching b. München, Germany,
(<vexler@ma.tum.de>)],

    December 30, 2023
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AMS subject classifications: 26A45, 49J20, 49M25, 65N15, 65N30.



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.
BV-Functions; optimal control of a wave equation; error bounds; finite elements.

























§ 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){   min_u∈ BV(0,T)^m1/2 y_u-y_d _L^2(Ω_T)^2 + ∑_j=1^m α_j D_tu_j_M(I)=:J(y,u)
    
    subject to     (𝒲) {∂_tt y - Δ y = f = ∑_j=1^m u_jg_j      in    I ×Ω
    
    y = 0      on    I×∂Ω
    
    (y,∂_t y) =   (y_0 , y_1)      in    { 0 }×Ω,
    .
    .

where Ω⊂ℝ^n, with n∈{ 1,2,3}, is a convex, polygonal/polyhedral bounded domain. For T∈ (0,∞) we denote I=(0,T). The desired state y_d is assumed to satisfy y_d∈ C^1(I;L^2(Ω)). The time depending controls u are given by u=(u_1,⋯,u_m)∈ BV(0,T)^m, and BV(0,T)^m is endowed with the norm u_BV(I)^m=∑_j=1^m (u_j_L^1(I)^m+D_tu_j_M(I)). Here M(I) is the space of Borel measures, endowed with the total variation norm ·_M(I). Further, let (g_j)_j=1^m ⊂ L^∞(Ω)∖{ 0 } with pairwise disjoint supports and α_j>0. The initial data is chosen as (y_0,y_1)∈ H^1_0(Ω)× L^2(Ω). Finally, we set Ω_T:=I×Ω.




In this work we focus on controls of bounded variation in time.
By using the total variation norm in (P), 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 BV-controls are already analyzed for elliptic and parabolic state equations in <cit.>.

Since our article deals with a priori error estimates of a finite element discretization for the control problem (P), we briefly discuss previous works on error estimates for PDE control problems with BV-controls. In <cit.> the authors discretize the time-dependent BV-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 √(τ) in time and linear in space. However, numerical experiments in <cit.> indicate better results. In <cit.> the authors analyze a finite element discretization of an elliptic control problem with BV-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 L^1-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 BV-controls. Standard techniques for the derivation of finite element error estimates, see e.g. <cit.>, cannot be applied due to the non-smoothness of the cost functional and the non-reflexivity of BV(I). 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. <cit.>. Using the fact that for one dimensional controls, BV(I) is isomorphic to M(I)×ℝ, some techniques from these works are used to derive error estimates for BV-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 <cit.> which uses the space-time finite element discretization developed and analyzed in <cit.>. 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 L^1(I)-norm and of the state variable in the L^2(Ω_T)-norm. The state equation is discretized by a space-time finite element method with piecewise linear and continuous Ansatz- and test-functions from <cit.>. The weak formulation of the discrete state equation is augmented with a stabilization term involving the stabilization parameter σ. 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 BV-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 L^2(Ω_T)-norm can be represented in terms of the finite element error of the state and adjoint state equation in the L^2(Ω_T)-norm resp. the L^∞(I;L^2(Ω)) as well as the error in the control variable in the L^1(I)-norm. The convergence rates for the finite element error of the state and adjoint state are obtained from <cit.>. Under the assumption that the continuous and time depending functions

    p̅_1,i:t↦-∫^T_t∫_Ωpg_i  dx  ds,  for i=1,⋯,m,

where p is the optimal adjoint state, p̅_1,i is bounded by ±α_i, is equal to α_i at finitely many points in I, and the second derivatives of p̅_1,i 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 p̅_1,i, 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 L^1(I)-norm is estimated in terms of the error of the state variable in the L^2(Ω_T)-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 <ref> summarizes several needed results on the regularity of weak solutions of the wave equation. In section <ref> the space-time finite element method from <cit.> is presented. Moreover, important stability results as well as a priori error estimates are stated. Section <ref> deals with the reformulation of the BV-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 <ref> 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 <ref> the error estimates for the optimal state and control variable as well as the optimal functional value are derived. Finally, in section <ref> 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.





§ PRELIMINARIES ON THE WAVE EQUATION

We consider Ω⊂ℝ^d, d=1,2,3 as convex, polygonal/polyhedral domain. Let {λ_k }_k∈ℕ be the non-decreasing eigenvalues of the Laplace operator -Δ with homogeneous boundary conditions and let {μ_k }_k∈ℕ be the corresponding system of eigenfunctions, which are orthonormal complete in L^2(Ω), and orthogonal complete in H^1_0(Ω). Hence, let us introduce for α≥ 0
the Hilbert spaces

    ℍ^α={ w∈ L^2(Ω) |w_ℍ^α^2:=∑_k≥ 1(λ^(k))^α⟨ w,μ_k⟩_L^2(Ω)^2<∞. }.

For α=0,1 we get L^2(Ω) respectively H^1_0(Ω). The convexity of Ω implies that ℍ^2= H^2(Ω)∩ H^1_0(Ω). In general holds ℍ^α↪ℍ^β for α≥β. We denote the dual space of ℍ^α by ℍ^-α.
Next we introduce the weak solution of the wave equation with the forcing function f, initial displacement y_0, and initial velocity y_1.
(<cit.>, Chap.IV, Sec.4)

Let (f,y_0,y_1)∈ L^1(I;L^2(Ω))× H^1_0(Ω)× L^2(Ω). We call a function y∈ C(I;H^1_0(Ω)) with ∂_t y ∈ C(I;L^2(Ω)) a weak solution of (𝒲) , if

    ∫_0^T-(∂_t  y ,∂_t η)_L^2(Ω)+(∇ y,∇η )_L^2(Ω) dt=(y_1,η(0) )_L^2(Ω)+∫_0^T(f,η)_L^2(Ω) dt

for any η∈ L^1(I;H^1_0(Ω)) such that ∂_t η∈ L^1(I;L^2(Ω)), η|_t=T=0, and y satisfies the initial condition y|_t=0=y_0. 

For the following existence and regularity results of weak solutions of the wave equation we refer to <cit.>:

For each (f,y_0,y_1)∈ L^1(0,T;L^2(Ω))× H^1_0(Ω) × L^2(Ω) there exists a unique weak solution y of (𝒲). Moreover, there exists a constant c>0 such that the weak solution y satisfies

    y_C(I;ℍ^α+1) + ∂_t y_C(I;ℍ^α)+∂_tty_L^κ(I;ℍ^α-1)≤ c(y_0_ℍ^α+1  +y_1_ℍ^α+ f_L^κ(I;ℍ^α))

provided (f,y_0,y_1)∈ L^κ(I;ℍ^α) ×ℍ^α+1×ℍ^α and

    y_C(I;ℍ^α +2) +∂_t y_C(I;ℍ^α+1) + ∂_tt y_C(I;ℍ^α)
    
    ≤ c(y_0_ℍ^α +2+y_1_ℍ^α+1+f_W^1,1(I;ℍ^α)).

provided (f,y_0,y_1)∈ W^1,1(I;ℍ^α) ×ℍ^α+2×ℍ^α+1 with 0≤α, 1≤κ≤∞.


The proof can be found in <cit.>.


Let us define the following continuous linear operators:

    L L^2(Ω_T) →  L^2(Ω_T), f ↦ y(f)  and   Q H^1_0(Ω)× L^2(Ω) → L^2(Ω_T), (y_0,y_1) ↦ y(y_0,y_1)

The function y(f) denotes the weak solution of the wave equation with y_0=y_1=0 and forcing function f. The function
y(y_0,y_1) denotes the weak solution of the wave equation with initial datum y_0 and y_1 and f=0.


The adjoint operator L^*:L^2(Ω_T)→ L^2(Ω_T) of L is given by w ↦ p(w) where p(w)∈ C(I;H^1_0(Ω))∩ C(I;L^2(Ω)) is the weak solution of the backwards in time equation

    (𝒲^*) {∂_tt p - Δ p = w     in    I ×Ω
    
    	p = 0      on    I×∂Ω
    
    	(p,∂_t p) =   (0 , 0)      in    { T }×Ω.
    	.




§ 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 <cit.>.
We consider a mesh 𝒯_h consisting of a finite set of triangles (for d=2) or tetrahedra (for d=3) K with h=max_K∈𝒯_hρ(K), where ρ(K) denotes the diameter of K.
We assume that the family of meshes (𝒯_h)_h is admissible, shape regular and  quasi-uniform.

Since Ω is polygonal and convex, we require that Ω=⋃_K∈𝒯_hK holds. We denote the space of piecewise linear and continuous finite elements based on the triangulation 𝒯_h by S_h⊂ H^1_0(Ω)∩ C(Ω) and its nodal basis by(φ_i)_i=1^N.


 §.§ Space-Time Finite Element Method

We discretize the time interval I uniformly with the time nodes 0=t_0 < ... < t_M=T and the stepsize τ=T/M. We denote the set of time nodes by w^τ={ t_0,...,t_M}.

Then we introduce the space of piecewise linear and continuous functions with respect to w^τ by

    S_τ:={ w∈ C(I)|   w|_[t_k-1,t_k] linear , 1≤ k ≤ M  }.

The standard hat functions form a basis e_m(t_k)=δ_mk, m,k=0,…,M of this discrete space. Finally, we use the notation ϑ:=(τ,h) with τ,h>0.

Let σ≥ 0. We call y_ϑ∈Ŝ_ϑ:=span{ v_h· v_τ| v_h∈ S_h, v_τ∈ S_τ} a discrete solution of (<ref>) if y_ϑ satisfies:

    ∫_0^T-(∂_t y_ϑ,∂_tη)_L^2(Ω)-(σ -1/6)τ^2( ∇∂_t y_ϑ,∇∂_tη)_L^2(Ω)+ (∇ y_ϑ,∇η)_L^2(Ω) dt
    
    = (y_1,η(0))_L^2(Ω) + ∫_0^T(f,η)_L^2(Ω) dt

for all η∈Ŝ_ϑ with η(T)=0 and initial condition y_ϑ(0):=R_hy_0, where R_h is the Ritz projection on S_h, i.e.

    (∇ R_hy_0,∇φ)_L^2(Ω)=(∇ y_0,∇φ)_L^2(Ω)  ∀φ∈ S_h.


Here σ plays the role of a stabilization parameter. With an increasing value of σ the method becomes more stable. For σ≥1/4 the method is unconditionally stable, see <cit.>.




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

Next we make an assumption on the relationship between τ and h which ensures stability of the method for 0≤σ<1/4.

Let ε_0∈ (0,1] be arbitrary and fixed. Moreover, let c_1 be the smallest constant in the inverse inequality ∇φ_L^2(Ω)≤ c_1 h^-1φ_L^2(Ω) for all φ_h∈ S_h. Moreover, let a c_2 be the constant in this a priori estimate for the Ritz projection w-R_hw_L^2(Ω)≤ c_2h∇ w_L^2(Ω). From now on it is assumed that

  
  * σ≥1/4-c_1h^2(1-ε_0^2)/τ^2,
  
  * σ≥1+ε_0^2/4-c_1h^2/τ^2,
  
  * |σ|τ^2 ≤ 2(c_2h^2 +τ^2).



This space-time finite element method is related to well-known time-stepping schemes. For σ=0 it is related to the explicit Leap-Frog-method and for σ=1/4 to the Crank-Nicolson scheme, see also <cit.>.
A more detailed discussion can be found in <cit.>.


The solution y_ϑ of (<ref>) for (f,y_0,y_1)∈ L^1(I,L^2(Ω))× H^1_0(Ω)× L^2(Ω) satisfies the following inequality

    y_ϑ_C(I;L^2(Ω))≤ c(y_0_H^1_0(Ω)+y_1_L^2(Ω)+f_L^1(I;L^2(Ω)))

with a constant c independent of h, f, y_0 and y_1.


The result follows directly from <cit.>.


The following error estimates hold:

    y-y_ϑ_C(I;L^2(Ω))≤
    c(h^2+τ^2)(y_0_ℍ^3+y_1_ℍ^2+f_L^∞(I;ℍ^2))

provided (f,y_0,y_1)∈ L^∞(I;ℍ^2)×ℍ^3×ℍ^2 as well as

    y-y_ϑ_C(I;L^2(Ω))≤
    c(h^2+τ^2)(y_0_ℍ^3+y_1_ℍ^2+f_W^1,1(I;H^1_0(Ω)))

provided (f,y_0,y_1)∈ W^1,1(I;H^1_0(Ω))×ℍ^3×ℍ^2.


The result follows directly from <cit.>.



§ EQUIVALENT PROBLEM (P̃) 

In this section we introduce a specific isomorphism between BV(I)^m⊗{(g_j)_j=1^m} and M(I)^m×ℝ^m. Based on this isomorphism (P) is equivalently formulated as a measure valued control problem. First of all we prove existence and uniqueness of a solution to (P).
 Problem (P) has a unique solution in BV(I)^m.


Utilizing the fact, that the forward mapping is continuous from L^2(I)^m to L^2(Ω_T), the proof can be carried out along the line of <cit.>.

Next we introduce several linear and continuous operators and discuss its properties. The operator B  M(I)^m ×ℝ^m  →  L^2(Ω_T) is given by

    (v,c)  ↦∑_j=1^m ( ∫_0^t dv_j(s) -1/T∫_0^T∫_0^t dv_j(s) ds + c_j )g_j.

The measures v_j are the derivatives of the generated BV-function and c_j are the mean values.
Next, we define the predual operator of B given by B^* L^2(Ω_T)→ C_0(I)^m×ℝ^m

    B^* q↦(w'_1,…,w'_m,∫_0^T∫_Ω qg_1 dx dt,…,∫_0^T∫_Ω qg_m  dx dt)

where w∈ H^2(I) solves

    {
    -w”_j   =∫_Ω q(·,x)g_j(x) dx-1/T∫_0^T∫_Ω q(t,x)g_j(x) dx dt  in (0,T)
    
    w'_j(0)   =w'_j(T)=0  with  ∫_0^Tw_j(t) dt=0  for   j=1,…,m.
    .


The operator B^* L^2(Ω_T)→ C_0(I)^m×ℝ^m is well defined and the predual of B, i.e. the following holds

    ∫_Ω_TB(v,c)q dx dt=⟨ (v,c),B^*(q)⟩

for all (v,c)∈ M(I)^m×ℝ^m and for all q∈ L^2(Ω_T).


The equation (<ref>) has a unique solution w_j∈ H^2(I), since ∫_Ω q(·,x)g_j dx-1/T∫_0^T∫_Ω q(t,x) dx dt∈ L^2(I) and has zero mean. Moreover, we have w'_j∈ H^1_0(I)↪ C_0(I). Thus, the operator B^* is well defined.
Moreover, there holds

    ⟨ (v,c),B^*(q)⟩=∑_j=1^m∫_0^Tw'_j  dv_j+∑_j=1^mc_j∫_0^T∫_Ω qg_j  dx  dt
    
    =∑_j=1^m∫_0^T-w”_j∫_0^t  dv_j  dt+∑_j=1^mc_j∫_0^T∫_Ω qg_j  dx  dt
    
    =∑_j=1^m∫_0^T(∫_Ω qg_j  dx-1/T∫_0^T∫_Ω qg_j  dx  dt)∫_0^t  dv_j  dt+∑_j=1^mc_j∫_0^T∫_Ω qg_j  dx  dt
    
    =∫_0^T∫_Ω q∑_j=1^m(∫_0^t  dv_j-1/T∫_0^T∫_0^t  dv_j  dt+c_j)g_j  dx  dt
    =∫_Ω_TB(v,c)q  dx  dt

for all (v,c)∈ M(I)^m×ℝ^m and for all q∈ L^2(Ω_T). The use of integration by parts is justified by the density of C_c^∞(I) in H^1_0(I).


Let w_j∈ H^2(I), j=1,⋯,m be the solution of (<ref>). Then there holds

    w'_j(t)=∫_t^T∫_Ω q(s,x)g_j(x)  dx  ds+(t-T)/T∫_0^T∫_Ω q(t,x)g_j(x)  dx  dt.



The operator B M(I)^m×ℝ^m→ BV(I)^m⊗{(g_j)_j=1^m} is an isomorphism.


The inverse of B is given by

    B^-1∑_j=1^mu_jg_j↦(u'_1,…,u'_m,1/T∫_0^Tu_1  dt,…, 1/T∫_0^T u_m  dt ).


Using B we can rewrite (P) into the equivalent problem

    (P̃){   min_
    v∈ M(I)^m 
    
    c∈ℝ^m
    1/2 S(v,c)-y_d _L^2(Ω_T)^2 + ∑_j=1^m α_j v_j_M(I)=:J̃(v,c),
    .

with S  M(I)^m ×ℝ^m  →  L^2(Ω_T) defined by (v,c) ↦ L(B(v,c))+Q(y_0,y_1).


 §.§ First-Order optimality condition of (P̃) 

In the following a necessary and sufficient first-order optimality condition of (P̃) is presented as well as sparsity results for the derivative of the optimal control.
Let (v,c) be the unique optimal pair. We define the quantities p=L^*(S(v,c)-y_d) and p_1∈ C(I)^m by

    p_1,i:=-∫^T_t∫_Ωpg_i  dx  ds

for i=1,…,m.

The pair (v,c)∈ M(I)^m×ℝ^m is an optimal control of (P̃) if and only if

    p_1,i   ∈α_i ∂v_i_M(I)   i=1,…, m,
    p_1(0)   =0.

Equivalently it holds

    ⟨ v-v_i,p_1,i⟩_M(I),C_0(I)+ α_i v_i_M(I)≤α_i v_M(I)  ∀ v∈ M(I) and i=1,…,m
	
and p_1(0)=0.


The proof of Theorem <ref> is done along the lines of the proof of

<cit.>.
By the convexity of (P̃) we have, that (v,c)∈ M(I)^m×ℝ^m is an optimal control of (P̃) if and only if

    0∈∂(
    1/2 S(v,c)-y_d _L^2(Ω_T)^2 + ∑_j=1^m α_j v_j_M(I))⊆ (M(I)^m×ℝ^m)^*=(M(I)^∗)^m×ℝ^m.

Define the following function F(v,c):=1/2 S(v,c)-y_d _L^2(Ω_T)^2 for (v,c)∈ M(0,T)^m×ℝ^m.
Its Gateaux derivative has the form

    DF_(v,c)(v,c)=B^∗ L^∗(S(v,c)-y_d)∈ C_0(I)^m×ℝ^m

According to the theory of convex analysis, e.g. <cit.>, we have

    0∈
    DF_(v,c)(v,c) + ∂(∑_i=1^m α_i v_i_M(I)).

Using

    ∂(∑_i=1^m α_i v_i_M(I))=[ ( α_i ∂v_i _M(I))_i=1^m;                       0 ]⊆ (M(I)^∗)^m×ℝ^m

and (<ref>) as well as Proposition <ref> imply

    p_1,i∈α_i ∂v_i _M(I)  ∀ i=1,…,m,  p_1(0)=0.


The following proposition is a consequence of <cit.>:

Let (v,c)∈ M(I)^m×ℝ^m be an optimal control of (P̃), then for all i=1,⋯,m and p_1=(p_1,i)_i=1^m given in (<ref>) holds


  a)p_1,i_C_0(I)≤α_i,

  b)∫_0^T -p_1,i/α_idv_i=∫_0^T d|v_i|=v_i_M(I),

  c)(v_i^±)⊆{ t ∈ I |p_1,i(t)=±α_i},
where v_i=v_i^+-v_i^- is the Jordan decomposition of v_i.



Let us note that the boundary property of p_1, i.e. p_1(0)=p_1(T)=0, and the continuity of p_1 imply with Proposition <ref>, c), that there exists a ϵ_i>0 such that dist((v_i^±),{ 0, T})>ϵ_i.



§ THE VARIATIONALLY DISCRETIZED PROBLEM

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

    (P̃_ϑ^semi){   min_
    	v∈ M(I)^m 
    
    	c∈ℝ^m
    	1/2 S_ϑ(v,c)-y_d _L^2(Ω_T)^2 + ∑_j=1^m α_j v_j_M(I)=:J_ϑ(v,c)
    .

with S_ϑ  M(I)^m ×ℝ^m  ⟶  L^2(Ω_T) defined by
(v,c) ↦  L_ϑ(B(v,c))+Q_ϑ(y_0,y_1).
Here L_ϑ  L^2(Ω_T)  →   L^2(Ω_T) is defined by f↦ y_ϑ(f), where y_ϑ(f) solves (<ref>) for a source f and (y_0,y_1)=(0,0). The operator Q_ϑ  H^1_0(Ω)× L^2(Ω)  →   L^2(Ω_T) is defined by (y_0,y_1)↦ y_ϑ(y_0,y_1), where y_ϑ(y_0,y_1) solves (<ref>) with (y_0,y_1) as initial datum and f=0.

We can represent the adjoint of L_ϑ in the form  w↦ L_ϑ^*(w)(t,x)=L_ϑ(w∘ϕ)(ϕ(t,x)) with ϕ(t,x)=(T-t,x), and w∈ L^2(Ω_T). This is true since L_ϑ(f)(0)=0 and L_ϑ^*(w)(T)=0 and thus L_ϑ(f) and L_ϑ^*(w) can be used in (<ref>) as test functions for the forwards and backwards equation.  Hence, Theorem <ref> and Lemma <ref> are valid for L_ϑ^*(w) as well.


The problem (P̃_ϑ^semi) has a solution in M(I)^m ×ℝ^m.


The existence of an optimal control for (P̃_ϑ^semi) can be similarly shown as in the proof of Theorem <ref>.

Note, that a BV-representation of the solutions (v, c), (v_ϑ,c_ϑ) of (P̃), respectively (P̃_ϑ^semi) are defined by

    u(t):=∫_0^t dv(s)-1/T∫_0^T∫_0^t dv(s) dt+c,  and  u_ϑ(t):=∫_0^t dv_ϑ(s)-1/T∫_0^T∫_0^t dv_ϑ(s) dt+c_ϑ.

Next we define the quantities p_ϑ=L^*_ϑ(S_ϑ(v_ϑ,c_ϑ)-y_d) and

    p_1,ϑ,j:=-∫^T_t∫_Ωp_ϑ g_j  dx  ds  for   j=1,…,m,

which is continuously differentiable and piecewise quadratic in time.

The pair (v_ϑ,c_ϑ)∈ M(I)^m×ℝ^m is a optimal control of (P̃_ϑ^semi) if and only if

    p_1,ϑ,i   ∈α_i ∂v_ϑ,i_M(I)   i=1,…, m,
    p_1,ϑ(0)   =0.

Equivalently it holds

    ⟨ v-v_i,p_1,ϑ,i⟩_M(I),C_0(I)+α_i v_i_M(I)⩽α_i v_M(I)  ∀ v∈ M(I), i=1,…,m,
	
and p_1,ϑ(0)=0.

 The proof is similar to Theorem <ref>.

Due to Theorem <ref>, we can show that Proposition <ref> holds analogiously for (P̃^semi_ϑ).



§ A PRIORI ERROR ESTIMATES

In this section error estimates of problem (P̃_ϑ^semi) for the optimal control, optimal state and optimal cost functional value are presented. Under specific assumptions, we proof optimal rates for the optimal control, state and cost.
For reason of convenience, the following notation is introduced.
For an optimal control (v_ϑ,c_ϑ) ∈  M(I)^m×ℝ^m of (P_ϑ^semi) and the optimal control (v,c)∈ M(I)^m×ℝ^m of (P) we introduce the corresponding optimal states by
y_ϑ:=S_ϑ(v_ϑ,c_ϑ) and
y:=S(v,c).
Further, we define the mixed state by ŷ_ϑ:=L_ϑ(B(v,c))+Q_ϑ(y_0,y_1).
The mixed adjoint state is chosen as p̂_ϑ:=L_ϑ^*(y-y_d).
In the proofs of following the Lemmata and  Theorem, we use similar steps as in the proof of <cit.>.

There holds

    ⟨p_1,ϑ- p_1,v_ϑ-v⟩≤ 0

with (v,c) as the optimal control of (P̃) and (v_ϑ,c_ϑ) as a solution of (P̃_ϑ^semi).


Inequality (<ref>) follows from monotonicity of the subdifferential.


Consider optimal control (v,c) of (P̃),  and (v_ϑ,c_ϑ) of (P̃^semi_ϑ), as well as their BV-representations u, and u_ϑ. For the optimal states y and y_ϑ of problem (P), respectively (P̃^semi_ϑ), we have

    y_ϑ-y_L^2(Ω_T)≤ cy-ŷ_ϑ_L^2(Ω_T)+cu_ϑ-u_L^1(I)^m^1/2p -   p̂_ϑ_L^∞(I;L^2(Ω))^1/2

with a constant c>0 depending on g.


Lemma <ref>, the  properties of B and B^* and the fact that p_1(0)=p_1,ϑ(0)=0 imply the following

    0≥⟨ p_1,ϑ-p_1,v_ϑ-v⟩=⟨ B^*(p_ϑ-p),(v_ϑ-v,c_ϑ-c)⟩
    = (p_ϑ-p̂_ϑ,(u_ϑ-u)· g)_L^2(Ω_T)
    
    +(p̂_ϑ-p,(u_ϑ-u)· g)_L^2(Ω_T)
    
    =(y_ϑ-y,L_ϑ((u_ϑ-u)g))_L^2(Ω_T)+(p̂_ϑ-p,(u_ϑ-u)· g)_L^2(Ω_T)
    
    =y_ϑ-y_L^2(Ω_T)^2+⟨y_ϑ-y,y-ŷ_ϑ⟩_L^2(Ω_T)+⟨p̂_ϑ-p,(
    u_ϑ-u)· g⟩_L^2(Ω_T).

From these calculations we obtained (<ref>) by

    y_ϑ-y_L^2(Ω_T)^2 ≤(y-y_ϑ,y-ŷ_ϑ)_L^2(Ω_T)+(p - p̂_ϑ,(u_ϑ-u)· g )_L^2(Ω_T)
    ≤1/2y-y_ϑ_L^2(Ω_T)^2+1/2y-ŷ_ϑ_L^2(Ω_T)^2+cu_ϑ-u_L^1(I)^mp - p̂_ϑ_L^∞(I;L^2(Ω)).



The sequence of the BV representatives ( u_ϑ)_ϑ of the optimal controls of (P̃_ϑ^semi) are bounded in BV(I)^m with respect to ϑ→ 0.


At first, we show that

    u_ϑ=∫_0^t dv_ϑ(s)-1/T∫_0^T∫_0^t dv_ϑ(s) ds+c_ϑ=û_ϑ+c_ϑ

is bounded in BV(I)^m for ϑ→ 0. Due to the optimality of u_ϑ, holds the inequality J_ϑ(u_ϑ)≤ J_ϑ(0) for all considered ϑ. Define y_ϑ:=S_ϑ(0,0) and y=S(0,0).
Using Lemma <ref> we have

    y_ϑ_C(I;L^2(Ω))≤ c(y_0_H^1_0(Ω)+y_1_L^2(Ω)).

Thus, the discrete states y_ϑ are bounded in L^2(Ω_T). Hence { J_ϑ(0)}_ϑ is bounded in ℝ. This implies that J_ϑ(u_ϑ) is bounded and thus, ( y_ϑ)_ϑ and ( D_tu_ϑ=v_ϑ )_ϑ are bounded in L^2(Ω_T), and M(I)^m respectively.
Now it suffices to show that c_ϑ∈ℝ^m is bounded in order to get the boundedness of (u_ϑ )_ϑ in BV(I)^m. Assume that c_ϑ∈ℝ^m is unbounded for ϑ→ 0. It holds

    ∑_j=1^mα_j D_tû_ϑ,j_M(I)=∑_j=1^mα_j D_tu_ϑ,j_M(I)≤ J_ϑ(u_ϑ)≤ J_ϑ(0)

and with the Poincare inequality for BV(I) functions (<cit.>), we get that ( û_ϑ )_ϑ is bounded in BV(I)^m. Consider z_ϑ=y_ϑ-y_ϑ with y_ϑ=L_ϑ(û_ϑ· g)+Q_ϑ(y_0,y_1). The BV boundedness of (û_ϑ)_ϑ, and therefore the boundedness in L^2(I)^m, implies by Lemma <ref> that (y_ϑ)_ϑ is bounded in L^2(Ω_T).
The boundedness of (y_ϑ)_ϑ and (y_ϑ)_ϑ lead to the boundedness of (z_ϑ)_ϑ in L^2(Ω_T).
The linearity of L_ϑ(B(·,·)), implies z_ϑ=L_ϑ(B(0,c_ϑ)). Consider now p_ϑ:=max_1≤ j ≤ m|c_ϑ,j|, with c_ϑ→∞, ξ_ϑ:=1/p_ϑ z_ϑ,  and a_ϑ=c_ϑ/p_ϑ. There exists a ϑ_0>0 such that for all ϑ<ϑ_0 the sequence a_ϑ is bounded by definition in ℝ^m. Thus, let us now consider ϑ≤ϑ_0 for all sequences in this proof. Hence, there exists a subsequence of a_ϑ, which converges to some a. Denote this converging subsequence again by a_ϑ. The linear structure of L_ϑ(B(·,·)) gives us ξ_ϑ=L_ϑ(B(0,a_ϑ)).
Define by ξ_ϑ the solution L_ϑ(B(0,a)). Next we show that Lemma <ref> leads to
ξ_ϑ - ξ_ϑ_L^2(Ω_T)→ 0.
Thus, we have

    ξ_ϑ - ξ_ϑ_L^2(Ω_T)=L_ϑ(B(0,a_ϑ-a))_L^2(Ω_T)≤ c |a_ϑ-a|_ℝ^m→ 0.

Define ξ by L(B(0,a)). Then we have

    ξ-ξ_ϑ_C(I;L^2(Ω))=L(B(0,a)-L_ϑ(B(0,a))_C(I;L^2(Ω))→ 0

according to Theorem <ref>.
With the boundedness of z_ϑ in L^2(Ω_T), the unboundedness of | p_ϑ|, and the definition of ξ_ϑ=z_ϑ/p_ϑ, we can deduce that ξ_ϑ→ 0 in L^2(Ω_T). Hence, ξ_ϑ= ξ_ϑ + (ξ_ϑ -ξ_ϑ)→ 0 in L^2(Ω_T).
Thus, we obtain that ξ=0, which implies ∑_j=1^m a_jg_j=0. Because g_j ∈ L^∞(Ω)∖{ 0 } have pointwise disjoint supports, we get that a_j=0, which is a contradiction. Thus, it is shown that c_ϑ is bounded, and hence ( u_ϑ)_ϑ is bounded in BV(I)^m.

The next theorem states an a priori error estimate for the optimal state. Under additional assumptions on the structure of optimal adjoint state an improved rate for the optimal state is proven. Furthermore, an optimal convergence for the control in the L^1(I)-norm is proven. In order to obtain optimal convergence rates we assume the following regularity on the data:

We assume that

  
  * y_d ∈ C^1(I;H^1_0(Ω))
  
  * g∈ (ℍ^2)^m
  
  * (y_0,y_1)∈ℍ^3×ℍ^2



There holds: B(v,c)∈ L^∞(I;ℍ^2) for any (v,c)∈ M(I)^m×ℝ^m


This follows from the definition of B(v,c) and the embedding of BV(I) in L^∞(I).


The following non-optimal error rate holds:

    y-y_ϑ_L^2(Ω_T)≤ c(h+τ)(y_0_ℍ^3 +y_1_ℍ^2+y_d_C^1(I;H^1_0(Ω)))).



By Theorem <ref> we have y-y_d ∈ C^1(I;H^1_0(Ω)). Using (<ref>), implies

    p-p̂_ϑ_L^∞(I;L^2(Ω))^1/2≤
    c(h^2 +τ^2)^1/2y-y_d_W^1,1(I;H^1_0(Ω))^1/2.

Corollary (<ref>) and <ref> lead to y-ŷ_ϑ_L^2(Ω_T)= O(τ^2+h^2). Hence, Lemma <ref> and (<ref>) imply (<ref>).


For the optimal control (v,c) of (P̃) and solutions (v_ϑ,c_ϑ) of (P̃^semi_ϑ) the following a priori error estimates hold:

    | J(v,c)-J_ϑ(v_ϑ,c_ϑ)| = O(τ^2+τ^2)


    ∑_i=1^m|v_i_M(I)-v_ϑ,i_M(I)| = O(τ+h)


The proof of Theorem <ref> is a modified version of the proof of <cit.>.


Optimality leads to the following two inequalities

    J(v,c)≤ J(v_ϑ,c_ϑ)  and   J_ϑ(v_ϑ,c_ϑ)≤ J_ϑ(v,c).

This implies
J(v,c)-J_ϑ(v,c)≤ J(v,c)-J_ϑ(v_ϑ,c_ϑ)≤ J(v_ϑ,c_ϑ)-J_ϑ(v_ϑ,c_ϑ).
So it remains to estimate the error with respect to the cost functionals for a fixed (v,c), i.e. (v,c) and (v_ϑ,c_ϑ). Common calculations lead to the following estimate:

    |J(v,c)-J_ϑ(v,c)|
    
    =1/2S(v,c)-S_ϑ(v,c)_L^2(Ω_T)^2+|(S(v,c)-S_ϑ(v,c),S(v,c)-y_d))_L^2(Ω_T)|
    ≤1/2S(v,c)-S_ϑ(v,c)_L^2(Ω_T)^2+S(v,c)-S_ϑ(v,c)_L^2(Ω_T)S(v,c)-y_d_L^2(Ω_T)

Then Corollary <ref> and <ref> implies the first assertion. Finally, Theorem <ref> implies (<ref>).



 §.§ Optimal Convergence Rates for the Optimal Controls of (P̃_ϑ^semi)

Under certain assumptions we show that the BV-representations u_ϑ of the optimal controls of (P̃_ϑ^semi), with respect to ϑ, converge with a specific rate in the L^1-norm to the solution u of (P).
Further, define the following functions:

    z(t)   :=∂_t p_1(t)= ∫_Ω L^* ( S(v,c)-y_d )(t) g dx
    
    z_ϑ(t)   :=∂_t p_1,ϑ(t)= ∫_Ω L_ϑ^* ( S_ϑ(v_ϑ,c_ϑ)-y_d )(t) g dx

with (v,c) as the optimal control of (P) and (v_ϑ,c_ϑ) as optimal control of (P_ϑ^semi). Due to Proposition <ref> and Remark <ref>, it holds that (v_i)⊆{ t | z_i(t)=0} and (v_ϑ,i)⊆{ t | z_ϑ,i(t)=0}.

The matrix G:=(⟨ Lg_i,Lg_j⟩_L^2(Ω_T))_i,j=1^m∈ℝ^m× m is symmetric and positive definite.


the matrix G is a Gramian-matrix, which is a consequence of the uniqueness of solutions of the wave equation the fact that {g_i}_i=1^m is a linear independent system.


u_ϑ converges weakly* in BV(0,T)^m to the solution u for ϑ→ 0.


Let (ϑ_n)_n=1^∞=(τ_n,h_n) be a null sequence such that (τ_n)_n=1^∞⊂ℝ^+, (h_n)_n=1^∞⊂ℝ^+.
Lemma <ref> implies that (u_ϑ_n)_n=1^∞ is a bounded sequence in BV(I)^m where (v_ϑ_n,c_ϑ_n) are optimal controls of (P^semi_ϑ_n).
The weak* compactness of closed and bounded sets in BV(I)^m implies the existence of a subsequence (u_ϑ_n_k)_k which converges weakly* to some u∈ BV(I)^m. Hence, (u_ϑ_n_k)_k converges in L^2(I)^m to u and D_t u_ϑ_n_k=v_ϑ_n_k converges weakly* in M(I)^m to D_tu. There exists a unique element (v,c)∈ M(I)^m×ℝ^m such that u=∫_0^· dv(s)-1/T∫_0^T∫_0^t  dv(s) dt+c. Due to the weak* l.s.c. of ·_M(I) in M(I),
we get

    lim inf_k →∞∑_i=1^m α_i v_ϑ_n_k,i_M(I)≥∑_i=1^m α_i D_tu_i_M(I).

Let us show that

    lim_k →∞S_ϑ_n_k(v_ϑ_n_k,c_ϑ_n_k)-y_d_L^2(Ω_T)^2= S(v,c)-y_d_L^2(Ω_T)^2

holds. Theorem <ref>, the stability of L_ϑ_n_k, see Lemma <ref> and the strong convergence of u_ϑ_k in L^2(I) lead to

    S_ϑ_n_k(v_ϑ_n_k,c_ϑ_n_k)-S(v,c)_L^2(Ω_T)≤S_ϑ_n_k(v_ϑ_n_k,c_ϑ_n_k)-S_ϑ_n_k(v,c)_L^2(Ω_T)
    
    +S_ϑ_n_k(v,c)-S(v,c)_L^2(Ω_T)≤ cu_ϑ_k-u_L^2(I)^m
    
    +c(h_n_k^2+τ_n_k^2)
    (y_0_ℍ^3+y_1_ℍ^2+B(v,c)_L^∞(I;ℍ^2))

This leads to (<ref>). With (<ref>), (<ref>) and Theorem <ref> we get

    J(v,c)=lim inf_k →∞ J_ϑ_k(v_ϑ_k,c_ϑ_k)≥ J(v,c).

The uniqueness of the optimal control of (P) leads to the desired result.


There holds u_ϑ→u in L^2(I) for ϑ→ 0.

Next we prove pointwise convergence of z_ϑ and ∂_t z_ϑ.

For ϑ→ 0 we have z_ϑ - z_L^∞(I)^m→ 0.


By Theorem <ref> and Definition <ref>, we have that z_ϑ∈ C(I) and z∈ C^1(I). Hence, z_ϑ - z_L^∞(I)^m is well-defined. There holds that

    z_ϑ - z_L^∞(I)^m≤ csup_t∈IL^*_ϑ(S_ϑ(v_ϑ,c_ϑ)-y_d)(t,·)-L^*(S(v,c)-y_d)(t,·)_L^1(Ω)
    ≤ c [L^*_ϑ(S_ϑ(v_ϑ,c_ϑ )) -L^*(S(v,c))_C(I;L^2(Ω))+L^*_ϑ(y_d) -L^*(y_d)_C(I;L^2(Ω))].

Next we show that z_ϑ - z_L^∞(I)^m→ 0 holds. According to Theorem <ref> we obtain

    L^*_ϑ(y_d) -L^*(y_d)_C(I;L^2(Ω))≤ c(h^2+τ^2)^1/3y_d_C^1(I̅;H^1_0(Ω)).

Furthermore, we have

    L^*_ϑ(S_ϑ(v_ϑ,c_ϑ)) -L^*(S(v,c))_C(I;L^2(Ω))≤L^*_ϑ(S_ϑ(v_ϑ,c_ϑ)) -L_ϑ^*(S(v,c))_C(I;L^2(Ω))
    
    +L^*_ϑ(S(v,c)) -L^*(S(v,c))_C(I;L^2(Ω))

For the second term on the right hand side of (<ref>), we have using Theorem <ref>:

    L^*_ϑ(S(v,c)) -L^*(S(v,c))_C(I;L^2(Ω))≤	c(h^2+τ^2)S(v,c)_C^1(I̅;H^1_0(Ω)).

For the first term on the right hand side of (<ref>), we use the stability of L_ϑ and L^∗_ϑ from Lemma <ref> to obtain

    L^*_ϑ(S_ϑ(v_ϑ,c_ϑ)) -	L_ϑ^*(S(v,c))_C(I;L^2(Ω))≤ c S_ϑ(v_ϑ,c_ϑ)-S(v,c)_C(I;L^2(Ω))
    ≤ c (S_ϑ(v_ϑ,c_ϑ)-S_ϑ(v,c)_C(I;L^2(Ω))+S_ϑ(v,c)-S(v,c)_C(I;L^2(Ω))) 
    ≤ cu_ϑ-u_L^2(I)
    +S_ϑ(v,c)-S(v,c)_C(I;L^2(Ω)).

The strong convergence of u_ϑ to u in L^2(I), see Corollary <ref>, and Theorem <ref> imply that L^*_ϑ(S_ϑ(v_ϑ,c_ϑ)) -
L_ϑ^*(S(v,c))_C(I;L^2(Ω)) converges to 0 for ϑ→ 0.  Hence, z_ϑ - z_L^∞(I)^m converges to 0 for ϑ→ 0.

Let us further define u_C_τ(I;L^2(Ω)):=max_t_i∈w^τu(t_i)_L^2(Ω) for all u∈ C(I;L^2(Ω)) and the mesh operators
∂_t w_m=w_m-w_m-1/τ.

There exists a constant c>0 such that the following inequality holds for all f∈ L^1(I;L^2(Ω))

    ∂_t
    L^*_ϑ(f)
    _C_τ(I;L^2(Ω))≤
    cf_L^1(I;L^2(Ω)).



This follows directly from <cit.>.


The following a priori error estimate

    ∂_t(L^∗(f)-L^*_ϑ(f))_C_τ(I;L^2(Ω))≤ (h^2+τ^2)^1/3f_W^1,1(I;L^2(Ω))

holds for all f∈ W^1,1(I;L^2(Ω)).


This follows directly from <cit.>.


We have

    ∂_t(z_ϑ - z)_C_τ(I)^m 0  for  ϑ→ 0.



Lemma <ref> implies

    ∂_t(z_ϑ - z)_C_τ(I)^m
    =∑_ℓ=1^msup_t_i∈w^τ|∂_t∫_Ω(
    L^*_ϑ(S_ϑ(v_ϑ,c_ϑ)-y_d)(t_i)-L^*(S(v,c)-y_d)(t_i)
    )g_ℓ dx
    |
    ≤ c∂_t(
    L^*_ϑ(S_ϑ(v_ϑ,c_ϑ)-y_d)-L^*(S(v,c)-y_d)
    )_C_τ(I;L^2(Ω))
    ≤
    c∂_t(
    L^*_ϑ(S_ϑ(v_ϑ,c_ϑ)-y_d)-L^*_ϑ(S(v,c)-y_d)
    )_C_τ(I;L^2(Ω))
    
    +c∂_t(
    L^*_ϑ(S(v,c)-y_d)-L^*(S(v,c)-y_d)
    )_C_τ(I;L^2(Ω))
    ≤ cS_ϑ(v_ϑ,c_ϑ)-S(v,c)_L^1(I;L^2(Ω))
    
    +c∂_t(
    L^*_ϑ(S(v,c)-y_d)-L^*(S(v,c)-y_d)
    )_C_τ(I;L^2(Ω))

The first term on the right hand side of the last inequality converges to 0 for ϑ→ 0, e.g. see (<ref>). Because y_d and S(v,c)∈ C^1(I;L^2(Ω)) holds Lemma <ref> implies that the last term in the last inequality converges to 0 for ϑ→ 0. This proves the assertion.


We have

    ∂_t(z_ϑ - z)_L^∞(I)^m→ 0  for ϑ→ 0.



At first we define a cell-wise discretization of the derivative of z as follows

    δ_tz:= ∑_i=1^M z(t_i)-z(t_i-1)/τ1_I_i,  with I_i:=(t_i-1,t_i), i=1,⋯,M.

Then we proceed with

    ∂_t (z_ϑ-z)_L^∞(I)^m≤∂_t z_ϑ - δ_tz_L^∞(I)^m+δ_tz - ∂_t z_L^∞(I)^m.

Using the disjoint supports of the characteristic functions in the definition of δ_tz leads to

    ∂_t z_ϑ - δ_tz_L^∞(I)^m=∑_j=1^mmax_i=1,⋯,M|z_ϑ^j(t_i)-z_ϑ^j(t_i-1)/τ -z_j(t_i)-z_j(t_i-1)/τ|=∂_t(z_ϑ -z)_C_τ(I)^m
	
which converges to 0 under the consideration of (<ref>).
Further, calculations show that

    δ_tz - ∂_t z_L^∞(I)^m=∑_i=1^M (z(t_i)-z(t_i-1)/τ - ∂_tz(t))1_I_i(t)_L^∞(I)^m
    ≤∑_i=1^M (z(t_i)-z(t_i-1)/τ - ∂_t z(t_i))1_I_i(t)_L^∞(I)^m+
    ∑_i=1^M (∂_t z(t_i) - ∂_t z(t))1_I_i(t)_L^∞(I)^m
    
    =∑_j=1^m(max_i=1,⋯,M|z_j(t_i)-z_j(t_i-1)/τ - ∂_t z_j(t_i)|+
    max_i=1,⋯,Msup_t∈ I_i|∂_t z_j(t_i) - ∂_t z_j(t)|).

In the last equation, we directly see that the first term converges to 0 due to <cit.>. The second term converges to 0 due to the uniform continuity of ∂_tz(t) in I.
Hence, the result follows for ϑ→ 0, which implies the claim.


	The convergence p_1,ϑ - p_1_L^∞(I)^m→ 0 holds for ϑ→ 0.


This follows directly from

    p_1,ϑ - p_1_L^∞(I)^m≤ cL_ϑ^*(S_ϑ(v_ϑ,c_ϑ)-y_d)-L^*(S(v,c)-y_d)_C(I;L^2(Ω)),

 which converges to 0 using the same steps as in Lemma <ref>.

In order to proof a priori error estimates for the control in the L^1(I)-norm and higher convergence rates for the state variable we have to make the following assumption.



  (A1) { t∈ I||p_1,i(t)|=α} = { t_1,i, ⋯ , t_m_i,i} for m_i∈ℕ, with i=1,⋯, m.

  (A2) ∂_t z_i(t_j,i)≠ 0, for j=1,⋯,m_i and i=1,⋯,m.



The assumption (A1) enforces finitely many jumps for the optimal control of (P), i.e. it holds (D_t u_i)⊆{ t_1,i, ⋯ , t_m_i,i} for u∈ BV(I)^m.

 Let (v_ϑ,c_ϑ) be an optimal control of (P^semi_ϑ).
Under the assumptions (A1) and (A2) above, there exists a δ >0, and ϑ_0 >0 such that for all 0<ϑ≤ϑ_0 holds

    v_ϑ,i=∑_l=1^m_i c_l,ϑ^iδ_t_l,ϑ^i with c_l,ϑ^i∈ℝ and 
    t_l,ϑ^i∈ B_δ(t_l,i),

where B_δ(t_l,i) are pairwise disjoint for a fixed i=1,⋯,m with respect to the index l=1,⋯,m_i and with 0<ϑ→ 0. The coefficients in front of the Dirac measures of v_ϑ,i, i.e. c_l,ϑ^i for l=1,⋯,m_i, are possibly 0.


Let us begin with the case m=1, m_1=1, i.e. { t∈ I || p_1(t)|=α}={t̃_1}. First of all we know that |p_1(t)|≤α for all t∈I holds and since p_1∈ C^1(I) as well as that t̃_1 is an interior point follows z(t̃_1)=-∂_tp_1(t̃_1)=0. Moreover, due to (A2) there exists a δ >0 and c_1>0 such that |∂_t z(t)|>c_1 for all t∈ B_δ(t̃_1)⊂ I. Since ∂_t z is continuous, ∂_t z does not change its sign on B_δ(t̃) and hence z is strictly monotone in B_δ(t̃_1). Therefore t̃_1 is the only root of z in B_δ(t̃_1). Moreover, there exist t_-,t_+∈ B_δ(t̃_1) with z(t_-)<0<z(t_+). By Lemma <ref> there exists a ϑ_0=(τ_0,h_0) such that z_ϑ(t_-)<0<z_ϑ(t_+) for all ϑ<ϑ_0. Since z_ϑ is continuous there exists a t_ϑ∈ (t_-,t_+) such that z_ϑ(t_ϑ)=0 for all ϑ<ϑ_0. Next we show that there exists a ϑ_0≤ϑ_0 such that t_ϑ is the only root of z_ϑ in B_δ(t̃_1) for all ϑ<ϑ̃_0. Lemma <ref> implies existence of a ϑ̃_0<ϑ_0 that ∂_t z_ϑ is either strictly positive or strictly negative on B_δ(t_1). Now let t̂_ϑ be a second root of z_ϑ in B_δ(t̃_1). Then it holds

    0=z_ϑ(t_ϑ)-z_ϑ(t̂_ϑ)=∫_t̂_ϑ^t_ϑ∂_tz_ϑ(t)  dt≠ 0.

Hence, there is no second root of z_ϑ in B_δ(t̃_1). Next we show that t≠ t_ϑ and z_ϑ(t)=0 imply the existence of ϑ̂_0<ϑ̃_0 such that |p_1,ϑ(t)|<α for all ϑ<ϑ̂ and thus v_ϑ=c_1,ϑδ_t_ϑ with c_1 possibly zero. Such a t can only exist in I∖ B_δ(t̃_1). Due to Assumption <ref> and the condition |p_1,ϑ(t)|≤α for all t∈I there exists a ε>0 such that |p_1(t)|<α-ε for all t∈I∖ B_δ(t̃_1). Lemma <ref> implies the existence of a ϑ̂_0<ϑ̃ with |p_1,ϑ(t)|<α-ε/2 for all ϑ < ϑ̂_0 and t∈I∖ B_δ(t̃_1).
In the case of m=1 and { t∈ I || p_1(t)|=α}={t̃_1,⋯ , t̃_m_1} with m_1>1, we can find for each t̃_i a δ_i>0 with ⋂_i=1^m_1B_δ_i(t̃_i)=∅ and a ϑ_i such that there exists a t_ϑ^i∈ B_δ_i(t̃_i) with v_ϑ|_B_δ_i(t̃_i)=c_i,ϑδ_t_ϑ^i and v_ϑ|_I∖∪_i=1^m_1B_δ_i(t̃_i)=0. Then choose ϑ_0<min_i=1,…,m_1ϑ_i.
In the case of m>1, one has to consider the same proof as above with respect to an additional subindex i=1,⋯,m, and the smallest ϑ_0 and δ used in the proofs of each component i=1,⋯,m.

From now on, we will assume that ϑ≤ϑ_0 holds with ϑ_0 from Lemma <ref>.
Furthermore, without loss of generality, we assume that δ>0 in Lemma <ref> is considered to be small enough such that there exists a δ>0 for which δ< dist(B_δ(t_j_1,i),{ 0,T }), and δ< dist(B_δ(t_j_1,i),B_δ(t_j_2,i)), j_1,j_2=1,⋯,m_i, j_1≠ j_2 for i=1,⋯,m. Let us note that Remark <ref> and Lemma <ref> guarantee that such a δ>0 exists.
Under these assumptions, we can work with the following definition.

Let us define the BV representations of the optimal controls of (P) and (P^semi_ϑ) in a more explicit form

    u_i=c_i+∑_j=1^m_i c_j^i(1_(t_j,i,T](t)-T-t_j,i/T),  u_i,ϑ=c_i,ϑ+
    ∑_j=1^m_i c_j,ϑ^i(1_(t_j,ϑ^i,T](t)-T-t_j,ϑ^i/T)

for i=1,…,m.


The following inequality holds

    u-u_ϑ_L^1(I)^m≤ c(∑_i=1^m (|c_i-c_i,ϑ|+ ∑_j=1^m_i |c_j^i|·|t_j,i-t_j,ϑ^i|+|c_j^i-c_j,ϑ^i|))

for some constant c which depends only on T.

We can prove (<ref>) by using 1_(t_j,i,T]-1_(t_j,ϑ^i,T]_L^1(I)=| t_j,i - t_j,ϑ^i |.

For each t_j,i there is a function f_j,i∈ C^∞_c(Ω_T), with j=1,⋯,m_i and i=1,⋯,m, such that the function

    g_i^j(t,x):=∫_t^T  L^*(f_j,i)(s,x) ds∈ C_0(I;L^2(Ω))

fulfills the properties


  a) L^*(f_j,i)(t,x)=h_i,j(t) f_i(x) for some h_i,j∈ C^∞_c(I), f_i ∈ C^∞_c(Ω), and f_j,i=f_i∂_tth_i,j - h_i,jΔ f_i,

  b) 0≤∫_t^T h_i,j(s) ds<1 in [t_j,i-δ-δ/2,t_j,i+δ+δ/2] ∖ B_δ(t_j,i),

  c) ∫_t^T h_i,j(s) ds=1 in B_δ(t_j,i),

  d)
(∫_t^T h_i,j(s) ds)⊆[t_j,i-δ-δ/2,t_j,i+δ+δ/2]⊂⊂ I, i.e. ∫_Ω g_i^j dx∈ C_0(I),

  e)  ⟨ f_i,g_l⟩_L^2(Ω)=δ_i,l,

with δ_i,l=0, if l≠ 0 and 1 else.


For all I_j,i:=[t_j,i-δ-δ/2,t_j,i+δ+δ/2], j=1,⋯,m_i and i=1,⋯,m,  there exists p_j,i∈ C^∞_c(I) such that 0≤p_j,i≤ 1 with

    p_j,i=0  in  I∖ I_j,i,   p_j,i=1  in  [t_j,i-δ,t_j,i+δ].

Let us define h_i,j=-∂_tp_j,i. For each g_i, i=1,⋯,m, we can find a f_i∈ C^∞_c(Ω) such that (f_i , g_k )_L^2(Ω) is 0 for i≠ k and 1 else. One can show that L^*(f_j,i)=h_i,j f_i with f_j,i:=f_i∂_tth_i,j - h_i,jΔ f_i holds.






Hence, g_i^j, defined in (<ref>), fulfills the desired properties a)-e).


There exists a constant c>0 independent of ϑ such that

    | c_j^i-c_j,ϑ^i |≤ c(τ^2+h^2+y-y_ϑ_L^2(Ω_T))  with  j=1,⋯,m_i  and i=1,⋯,m.



Let i=1,⋯,m, j∈{ 1,⋯,m_i } and consider from Lemma <ref> the function g^j_i=∫^T_t h_i,jf_i ds=∫_t^T L^∗(f_j,i) ds. Hence, we have

    c_j^i-c_j,ϑ^i=∫_0^T ∫_t^T  h_i,j(s) ds dD_t(u_i-u_ϑ,i)(t)=∫_0^T h_i,j (u_i-u_ϑ,i) dt
    
    =∑_l=1^m ∫_0^T h_i,j (u_l-u_ϑ,l) dt ∫_Ω f_i g_l dx=∫_0^T∫_Ω
    -∂_t g^j_i (∑_l=1^m (u_l-u_ϑ,l)g_l) dt dx

Thus, it follows

    c_j^i-c_j,ϑ^i= ∫_0^T∫_Ω-∂_t g^j_i (∑_l=1^m (u_l-u_ϑ,l) g_l) dt dx= ∫_0^T ∫_Ω L^*(f_j,i)(B(v,c)-B(v_ϑ,c_ϑ)) dt dx
    
    =(L^*(f_j,i)-L_ϑ^*(f_j,i),B(v,c)-B(v_ϑ,c_ϑ))_L^2(Ω_T)+(L_ϑ^*(f_j,i),B(v,c)-B(v_ϑ,c_ϑ))_L^2(Ω_T).

By Theorem <ref> and the boundedness of (v_ϑ,c_ϑ) we obtain

    (L^*(f_j,i)-L_ϑ^*(f_j,i),B(v,c)-B(v_ϑ,c_ϑ))_L^2(Ω_T)
    ≤L^*(f_j,i)-L_ϑ^*(f_j,i)_L^2(Ω_T)B(v,c)-B(v_ϑ,c_ϑ)_L^2(Ω_T)=O(τ^2+h^2).

Moreover, there holds

    (L_ϑ^*(f_j,i),B(v,c)-B(v_ϑ,c_ϑ))_L^2(Ω_T)
    =(f_j,i,L_ϑ(B(v,c)) - L_ϑ(B(v_ϑ,c_ϑ)))_L^2(Ω_T)
    ≤f_j,i_L^2(Ω_T)S_ϑ(v,c) -S_ϑ(v_ϑ,c_ϑ)_L^2(Ω_T)
    ≤ cS_ϑ(v,c)-S(v,c)_L^2(Ω_T) + cS(v,c)-S_ϑ(v_ϑ,c_ϑ)_L^2(Ω_T)
    ≤ c(τ^2+h^2+y-y_ϑ_L^2(Ω_T))

according to Corollary <ref> and <ref>.


There holds that

    | t_j,1^i-t_j,ϑ^i |≤ c(τ^2+h^2+y-y_ϑ_L^2(Ω_T))  with j=1,⋯,m_i  and  i=1,⋯,m .



Using that z_i(t_j,i)=z_ϑ,i(t_j,ϑ^i)=0 and z_i=∂_t p_1,i∈ C^1(I) gives us
z_i(t_j,ϑ^i)-z_ϑ,i(t_j,ϑ^i)=z_i(t_j,ϑ^i)=z_i(t_j,i)+∂_t z_i(ξ)(t_j,i-t_j,ϑ^i)
for some ξ∈ B_δ(t_j,i). In the proof of Lemma <ref> we have shown that |∂_t z_i(ξ̃)|>0 for all ξ̃∈ B_δ(t_j,i) and therefore we have ∂_t z_i(ξ)≠ 0.
Then Lemma <ref> and Theorem <ref> imply

    | t_j,i-t_j,ϑ^i |≤ c z_i-z_ϑ,i_L^∞(I)≤ c z-z_ϑ_L^∞(I)^m
    ≤ c L^*(S(v,c)-y_d)-L_ϑ^*(S_ϑ(v_ϑ,c_ϑ)-y_d)_C(I;L^2(Ω))
    ≤ cL^*(S(v,c)-y_d)-L_ϑ^*(S(v,c)-y_d)_C(I;L^2(Ω))+cL_ϑ^*(S(v,c)-S_ϑ(v_ϑ,c_ϑ))_C(I;L^2(Ω))
    ≤ cL^*(S(v,c)-y_d)-L_ϑ^*(S(v,c)-y_d)_C(I;L^2(Ω))+cS(v,c)-S_ϑ(v_ϑ,c_ϑ)_L^1(I;L^2(Ω))
    ≤ c(τ^2+h^2+S(v,c)-S_ϑ(v_ϑ,c_ϑ)_L^2(Ω_T)).



Let ϑ_0 be small enough such that

    ((L_ϑ_0(g_i),L_ϑ_0(g_j))_L^2(Ω_T))_i,j={ 1,⋯, m }>0  holds.

Then we obtain

    | c_i-c_i,ϑ|≤ c(τ^2 +h^2 + y-y_ϑ_L^2(Ω_T)).



First, define the function h(t)=1_(t,T]-T-t/T. The optimality conditions of the continuous and discrete problem lead to

    0=p_1,i(0)=∑_l=1^m c_l∫_0^T∫_Ω L^* ( L(g_l))g_i dx dt
    
    +∑_l=1^m∑_j=1^m_l c_j^l∫_0^T∫_Ω L^* ( L(h(t_j,l)g_l))g_i dx dt
    +∫_0^T∫_Ω L^* (Q(y_0,y_1)-y_d )g_i dx dt

as well as in the discrete case to

    0=p_1,ϑ,i(0)=∑_l=1^m c_l,ϑ∫_0^T∫_Ω L_ϑ^* ( L_ϑ(g_l))g_i dx dt
    
    +∑_l=1^m∑_j=1^m_l c_j,ϑ^l∫_0^T∫_Ω L_ϑ^* ( L_ϑ(h(t_j,ϑ^i)g_l)) g_i dx dt
    +∫_0^T∫_Ω L_ϑ^* (Q_ϑ(y_0,y_1)-y_d ) g_i dx dt.

By taking the difference of the last two terms we get

    ∑_l=1^m(c_l,ϑ-c_l)∫_0^T∫_Ω L^* ( L(g_l))g_i dx dt=∑_l=1^m c_l,ϑ∫_0^T∫_Ω[L^* ( L(g_l)) -  L_ϑ^* ( L_ϑ(g_l))]g_i dx dt
    
    +∑_l=1^m∑_j=1^m_l (c_j^l-c_j,ϑ^l)∫_0^T∫_Ω L^* (L(h(t_j,l)g_l))g_i dx dt
    
    +∑_l=1^m∑_j=1^m_l c_j,ϑ^l(∫_0^T∫_Ω[ L^* ( L(h(t_j,l)g_l))-L_ϑ^* ( L_ϑ(h(t_j,ϑ^i)g_l))]g_i dx dt)
    
    +∫_0^T∫_Ω[L^* (Q(y_0,y_1)-y_d )-L_ϑ^* (Q_ϑ(y_0,y_1)-y_d )]g_i dx dt.

For the following we remark that u̅_ϑ is bounded BV(I), see Lemma <ref>. Then we consider the first term in (<ref>) on the righthand side. The regularity of g_l implies that L(g_l)∈ C^1(I;H^1_0(Ω)) according to Theorem <ref>. Thus, with (<ref>), Lemma <ref>, Corollary <ref> and <ref> it follows,

    ∑_l=1^m c_l,ϑ∫_0^T∫_Ω[L^* ( L(g_l)) -  L_ϑ^* ( L_ϑ(g_l))]g_i dx dt≤∑_l=1^m c L^*(L(g_l))-L_ϑ^*(L_ϑ(g_l))_L^2(Ω_T)
    ≤ c∑_l=1^m( L^*(L(g_l))-L_ϑ^*(L(g_l))_L^2(Ω_T)+ L(g_l)-L_ϑ(g_l)_L^2(Ω_T))=O(τ^2 + h^2).

By Lemma <ref> we obtain:

    ∑_l=1^m∑_j=1^m_l (c_j^l-c_j,ϑ^l)∫_0^T∫_Ω L^* (L(h(t_j,l)g_l))g_i dx dt
    ≤ c∑_l=1^m∑_j=1^m_l |c_j^l-c_j,ϑ^l|≤ c(τ^2+h^2+y-y_ϑ_L^2(Ω_T))

Now we consider the third term on the righthand side of (<ref>). The stability of L^∗_ϑ, see Lemma <ref>, imply

    ∑_l=1^m∑_j=1^m_l c_j,ϑ^l(∫_0^T∫_Ω[ L^* ( L(h(t_j,l)g_l))-L_ϑ^*(L_ϑ(h(t_j,ϑ^l)g_l))]g_i dx dt)
    ≤ c∑_l=1^m∑_j=1^m_lL^*(L(h(t_j,l)g_l))-L_ϑ^* (L_ϑ(h(t_j,ϑ^l)g_l))_L^1(Ω_T)
    ≤ c∑_l=1^m∑_j=1^m_lL^* ( L(h(t_j,l)g_l))-L_ϑ^* ( L(h(t_j,l^j)g_l))_L^1(Ω_T) +
    L(h(t_j,l)g_l)-L_ϑ(h(t_j,ϑ^l)g_l)_L^1(Ω_T)

Again, by Theorem <ref> we have L(h(t̃)g_l)∈ C^1(I;H^1_0(Ω)) and any t̃∈ I. Hence, with (<ref>) we get

    ∑_l=1^m∑_j=1^m_lL^* (L(h(t_j,l)g_l))-L_ϑ^*(L(h(t_j,l)g_l))_L^1(Ω_T) =O(τ^2 + h^2).

Next we consider the following inequality

    L(h(t_j,lg_l))-L_ϑ(h(t_j,ϑ^l)g_l)_L^1(Ω_T)
    ≤L(h(t_j,l)g_l)-L_ϑ(h(t_j,ϑ^l)g_l)_L^1(Ω_T)+L_ϑ(h(t_j,l)g_l)-L_ϑ(h(t_j,ϑ^l)g_l)_L^1(Ω_T).

Due to Corollary <ref> and <ref>, the first term on the right hand side of (<ref>) possess the asymptotic rate O(τ^2+h^2). By Lemma <ref>, and Lemma <ref> we obtain for the second term an estimate in terms of c(τ^2 +h^2 + y-y_ϑ_L^2(Ω_T)).
Finally, we consider the last term in (<ref>). We have

    ∫_0^T∫_Ω[L^* (Q(y_0,y_1)-y_d )-L_ϑ^* (Q_ϑ(y_0,y_1)-y_d )]g_i dx dt
    ≤ cL^* (Q(y_0,y_1)-y_d )-L_ϑ^* (Q(y_0,y_1)-y_d )_L^2(Ω_T)
    
    +cL_ϑ^* (Q(y_0,y_1)-y_d )-L_ϑ^* (Q_ϑ(y_0,y_1)-y_d )_L^2(Ω_T).

The first term converges in (<ref>) with a rate (τ^2+h^2) according to Theorem <ref> since Q(y_0,y_1)-y_d∈ C^1(I,H^1_0(Ω)). The prescribed regularity of (y_0,y_1), Lemma <ref> and the error estimates in (<ref>) give us an estimate in terms of order (τ^2 + h^2) of the last term in (<ref>). Thus, we have

    ∑_l=1^m(c_l-c_l,ϑ)∫_0^T∫_Ω L^* ( L(g_l))g_i dx dt ≤ c(τ^2 + h^2+y-y_ϑ_L^2(Ω_T)).

Next we recall the symmetric positive definiteness of the matrix G from Lemma <ref>. It holds

    G(c-c_ϑ)=(∑_l=1^m(c_l-c_l,ϑ)∫_0^T∫_Ω L^* ( L(g_l))g_i dx dt)_i=1^m.

Furthermore, we have

    G(c-c_ϑ)_ℝ^m≥λ_minc-c_ϑ_ℝ^m≥ cλ_minc-c_ϑ_∞≥ cλ_min|c_i-c_i,ϑ|

for i=1,⋯,m where λ_min>0 is the smallest eigenvalue of G.
Using (<ref>) and the convergence rates in (<ref>) gives us (<ref>).

From now on we assume that all assumptions in Lemma <ref> hold.

It holds that

    u-u_ϑ_L^1(I)≤ c(τ^2+h^2+y-y_ϑ_L^2(Ω_T)).


This corollary is a consequence of Lemma <ref>, <ref>, <ref>, <ref>.
Next we state the main result of this work.

The following convergence rates hold.

    u-u_ϑ_L^1(I)^m =O(τ^2 +h^2),      | c_i-c_i,ϑ| =O(τ^2 +h^2),
    | t_j,i-t_j,ϑ^i |=O(τ^2 +h^2),      | c_j^i-c_j,ϑ^i |=O(τ^2 +h^2)

with j=1,⋯,m_i, i=1,⋯,m.
Furthermore, we have for the optimal states of (P̃) and (P̃_ϑ^semi)

    y-y_ϑ_L^2(Ω_T) =O(τ^2 +h^2).



Using the inequality in (<ref>), Corollary <ref> and <ref> and Corollary <ref>, we obtain for some ϵ>0

    y_ϑ-y_L^2(Ω_T)≤ cy-ŷ_ϑ_L^2(Ω_T)+cu_ϑ-u_L^1(I)^m^1/2p -p̂_ϑ_C(I;L^2(Ω))^1/2
    ≤ c(τ^2+h^2)+cu_ϑ-u_L^1(I)^m^1/2p -   p̂_ϑ_C(I;L^2(Ω))^1/2
    ≤ c(τ^2+h^2)+cϵu_ϑ-u_L^1(I)^m + c/4 ϵp -p̂_ϑ_C(I;L^2(Ω))
    ≤ c(τ^2+h^2)+cϵy_ϑ-y_L^2(Ω_T) + c/4 ϵp - p̂_ϑ_C(I;L^2(Ω)).

Consider a ϵ>0 such that cϵ=1/2, then we have

    y_ϑ-y_L^2(Ω_T)≤ c(τ^2+h^2+  p -   p̂_ϑ_C(I;L^2(Ω))).

Then the a priori estimate (<ref>) implies

    p-p̂_ϑ_C(I;L^2(Ω))≤
    c(h^2 +τ^2)y-y_d_C^1(I;H^1_0(Ω)).

 So we have  y_ϑ-y_L^2(Ω_T)=O(τ^2+h^2) and thus the same rate for the control in the L^1(I)-norm. Using the optimal rates of y_ϑ-y_L^2(Ω_T) in (<ref>), (<ref>), and (<ref>)
implies the optimal quadratic convergence rates for c_i,ϑ, t_j,ϑ^i and c_j,ϑ^i. 


For the BV representations of the optimal controls of (P̃) and (P̃_ϑ^semi) hold

    |D_tu_M(I)-D_tu_ϑ_M(I)| =O(
     τ^2+h^2).

 Furthermore, u_ϑ converges strictly in BV(0,T) to u for ϑ→ 0 with the convergence rate O(τ^2+h^2).


The statements are a consequence of (<ref>), (<ref>), and Theorem <ref>.


	Based on the same techniques we have used so far, the following convergence rates can be shown for less regular data (g_i)_i=1^m⊂ L^∞(Ω) and (y_d,y_0,y_1)∈ C^1(I̅;L^2(Ω))× H^1_0(Ω) × L^2(Ω) as assumed in the problem (P) at the beginning:
	
		
  a) y̅ - y̅_ϑ_L^2(Ω_T)= O(τ^2/3+h^2/3),
		
  b) | J(v̅,c̅)-J_ϑ(v̅_ϑ,c̅_ϑ)| = O(τ^2/3+h^2/3),
		
  c) u̅ - u̅_ϑ_L^1(I)^m+∑_i=1^m| D_t u̅_i_M(I) - D_t u̅_ϑ,i_M(I)|= O(τ^2/3+h^2/3).
	
In particular, statements of Theorem <ref> has to be extended by <cit.> with respect to lower regular data chosen for (g_i)_i=1^m and (y_d,y_0,y_1).



§ NUMERICAL EXPERIMENTS

In order to numerically verify the previously presented optimal error rates, an appropriate algorithm is of particular importance due to the variational discretization of problem (P). Similarly as in <cit.>, we solve the BV-control problem using Algorithm <ref>, which is a modified version of the primal dual active point (PDAP) algorithm introduced in <cit.>. This method is based on a conditional gradient method, see <cit.>. The algorithm calculates the derivative v̅∈ M(I)^m and the offset c̅∈ℝ^m of the optimal control u̅. The iterates for v are given by a linear combination of Dirac measures (∑_l=1^m_iλ_l,iδ_t_l,i)_i=1^m. In every iteration the positions of the m global maxima of

    p_1,i^k t↦|∫_t^T ∫_Ω L^*_ϑ(S_ϑ(v_k,c_k)-y_d)g_i  dx ds|,   i=1,…,m

are found. Then new Dirac measures are added at these positions. Finally a non-smooth optimization problem in terms of the magnitudes λ and the constants c is solved. The L^1-norm in the corresponding cost functional enhances sparsity in the vector of the magnitudes. If a magnitude is set to zero, the corresponding Dirac measure is erased from the current iterate.
For a convenient notation we define the map 𝒰_𝒜(λ):=∑_ℓ∈𝒜λ_ℓδ_ℓ for any finite set 𝒜⊂ I and λ∈ℝ^|𝒜|. 

In the pure measure-valued case (B(v,c)=v) it is proven that this algorithm converges with a sublinear rate in terms of the cost functional. Under additional assumptions on the problem a linear rate is proven, see <cit.>.
We consider a specific configuration of the input data for (P), such that the solution is known explicitly and is a piecewise constant BV-function with finitely many jumps. We use a construction procedure for the solution which can be found in <cit.>. We fix the following scenario:


  * Ω:=(-1,1)^2 and T=2

  * α=6· 10^-3

  * g(x):=cos(π/2 x_1)cos(π/2 x_2), thus g∈ℍ^2

  * (y_0,y_1)=(0,0)

  * y_d=S(v̅,c̅)-(∂_tt-Δ)φ∈ C^1(I̅,H^1_0(Ω)) with
φ(t,x):=3πα/2sin(2π t)sin(π t)cos(π/2x_1)cos(π/2x_2).

Thus the data has the required regularity which we assumed in the proof of the quadratic convergence rates. Based on this input data the derivative v̅ of the optimal control u̅ has the form

    v=δ_1/3-δ_1+δ_5/3

and c̅=0. Moreover this example fulfills Assumption <ref> since p_1,1 has the form
p_1,1(t) =αsin(3π t/2)^3.
The function y_d contains the optimal state S(v̅,c̅), which cannot be obtained exactly. Thus we approximate it by its finite element solution on the finest grid level. More precisely, we set for the reference state S(v̅,c̅) the mesh fineness in time and space to τ=2^-9 and h=2·√(2)· 2^-8, respectively. All spatial integrals used for the numerical computation of S(v̅,c̅) are calculated by a seven-point Gaussian quadrature rule different to the numerical calculations used for the approximated optimal state with respect to the BV-PDAP algorithm, where we considered a three-point Gauss quadrature rule. The function p_1,1^k is computed exactly. Since they are piecewise quadratic in time, their global maxima can be attained at arbitrary points in I̅, not necessarily at grid points. Thus B(v^k,c^k) can have jumps outside of grid points. However, the integrals ∫_0^TB(v^k,c^k)e_i dt with a hat function e_i in the discrete state equation are calculated exactly. Moreover, the candidates for the global maxima of p_1,1^k are also computed explicitly using the fact that ∂_t p_1,1^k is piecewise linear. An additional L^2-regularization is added to the non-smooth optimization problem for the magnitudes λ and the constants c. Then it is solved by a continuation strategy and a semi smooth Newton method, see also <cit.>.


In Figure <ref> we observe that the error in state variable measured in the L^2(Ω_T)-norm, in the control variable measured in the L^1(I)-norm as well as in the cost functional converge like O(τ^2+h^2). The same is true for the error in the jump positions t_j,1, the magnitudes of the jumps c_j^1 and the offset c_1. Thus the predicted error rates are verified.



§ ACKNOWLEDGMENTS

Sebastian Engel and Philip Trautmann were supported by the
International Research Training Group IGDK, funded by the German Science Foundation (DFG) and the Austrian Science Fund (FWF).

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