This document discusses optimal error estimates for a semi-discrete optimal control problem of the wave equation with time-dependent bounded variation controls. It presents an optimal control problem involving the wave equation with BV controls. It then discusses variational discretization of the problem and derives convergence results. Standard error estimates for the state, cost, and BV norm of controls are provided. The discussion focuses on obtaining optimal error rates for the controls in the strict BV topology as well as the state and cost.

1. International Research Training Group IGDK 1754
Optimal Error Estimates for a Semi-Discrete
Optimal Control Problem of the Wave Equation with
Time-Depending Bounded Variation Controls
Sebastian Engel (TUM), P. Trautmann (KFU), B. Vexler (TUM)
MAFELAP 2019
18 21 June, 2019
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 1

2. International Research Training Group IGDK 1754
Topics of Discussion
§ Control problem pPq
§ Error rates in BV-control problems with PDEs - related works
§ Variational discretization of pPq and convergence results
§ Numerical approach and experimental results
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 2

3. International Research Training Group IGDK 1754
Motivation - Optimal Control with BV functions
§ The focus on BV functions for control problems is motivated by structure
of the optimal solutions which favor controls with few jumps, i.e. little
switching.
§ H1 regularized optimal control problems behave continuously. BV functions
allow jumps.
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 3

4. International Research Training Group IGDK 1754
Optimal Control of the Wave Equation with BV-Functions
Consider the following optimal control problem:
pPq
$
’’’’’’’’’’
’’’’’’’’’’%
min
uPBVp0;Tqm
1
2 }pyu ´ ydq}2
L2
p
T q `
mÿ
j“1
j
ż
r0;Ts
d|Dtuj|ptq “: Jpuq
s.t.
$
’’’
’’’%
Bttyu ´ 4yu “
mÿ
j“1
ujgj in p0;Tq ˆ
yu “ 0 on p0;Tq ˆ B
pyu;Btyuq “ py0;y1q in t0u ˆ
§
Ă Rn (n=1,2,3) polygonal or polyhedral, T P p0;8q.
§ yd P L2pp0;Tq ˆ
qq, py0;y1q P H1
0 p
q ˆ L2p
q.
§ pgjqm
j Ă L8
p
qzt0u pairwise disjoint supports wj.
This strictly convex problem has a unique solution.
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 4

5. International Research Training Group IGDK 1754
Optimal Control of the Wave Equation with BV-Functions
Consider the following optimal control problem:
pPq
$
’’’’’’’’’’’
’’’’’’’’’’’%
min
uPBVp0;Tq¡em
1
2 }pyu ´ ydq}2
L2
p
T q `
£
£
££g
g
gg
mÿ
j“1
j
ż
r0;Ts
d|Dtuj|ptq “: Jpuq
s.t.
$
’’’
’’’%
Bttyu ´ 4yu “
£
£
££g
g
gg
mÿ
j“1
ujgj in p0;Tq ˆ
yu “ 0 on p0;Tq ˆ B
pyu;Btyuq “ py0;y1q in t0u ˆ
§
Ă Rn (n=1,2,3) polygonal or polyhedral, T P p0;8q.
§ yd P L2pp0;Tq ˆ
qq, py0;y1q P H1
0 p
q ˆ L2p
q.
§ pgjqm
j Ă L8
p
qzt0u pairwise disjoint supports wj.
This strictly convex problem has a unique solution.
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 4

6. International Research Training Group IGDK 1754
Optimal Control of the Wave Equation with BV-Functions
Consider the following optimal control problem:
pPq
$
’’’’’’’
’’’’’’’%
min
uPBVp0;Tq
1
2 }pyu ´ ydq}2
L2
p
T q `
ż
r0;Ts
d|Dtu|ptq “: Jpuq
s.t.
$
%
Bttyu ´ 4yu “ u ¨ g in p0;Tq ˆ
yu “ 0 on p0;Tq ˆ B
pyu;Btyuq “ py0;y1q in t0u ˆ
§
Ă Rn (n=1,2,3) polygonal or polyhedral, T P p0;8q.
§ yd P L2pp0;Tq ˆ
qq, py0;y1q P H1
0 p
q ˆ L2p
q.
§ g Ă L8
p
qzt0u.
This strictly convex problem has a unique solution.
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 4

7. International Research Training Group IGDK 1754
Equivalent Problem p˜Pq
Consider the following equivalent optimal control problem w.r.t. pPq:
p˜Pq
min
pv;cqPMp0;TqˆR
1
2 }Spv;cq ´ yd}2
L2
p
T q `
şT
0 |v|dx “: Jpv;cq
with uptq “
tş
0
dvpsq ` c resp.
ˆ
Dtu
up0q
˙
“
ˆ
v
c
˙ ˆ
Fundamental
theorem of calculus
˙
.
§ S is the ane control-to-state operator.
§ BVp0;Tq – Mp0;Tq ˆ R only possible in one dim.
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 5

8. International Research Training Group IGDK 1754
Error Rates BV-Control Problems
§ i[CKK17]: Optimal control of a semi-linear parabolic equation.
§ One dimensional BV-controls, u;h cellwise constant.
}y ´ y;h}L2p
q ` |Jpuq ´ J;hpu;hq| ď cp
?
` hq
§ i[HMNV19]: Optimal control of one dimensional elliptic equations.
§ Variational discretization:
Optimal convergence results for state, adjoint, and control
(piecewise constant assumption).
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 6

9. International Research Training Group IGDK 1754
Standard Approach - Error Estimates
§ In case of smooth cost functions Jpuq, the standard approach uses
coercivity properties, e.g.
§ appropriate testing of the 1
st
order optimality conditions, to derive error
estimates for the control.
§ Error rates for the optimal states, costs and TV-semi-norm of the optimal
controls can be obtained in a direct manner, which are sub-optimal.
§ For optimal error rates of the controls in the strict BV-topology, state, and
costs, we need several assumptions on the adjoint function p1.
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 7

10. International Research Training Group IGDK 1754
Variational Discretization of p˜Pq
In the following we discretize the state equation by linear continuous FE in time
and space (S b Sh), where controls will not be changed:
p˜Psemi
;h q
$
%
min
v P Mp0;Tq
c P R
1
2 }S;hpv;cq ´ yd}2
L2
p
T q `
şT
0 d|vj| “: J;hpv;cq
Optimality Conditions
pv;h;c;hq P Mp0;Tq ˆ R is the solution of p˜Psemi
;h q, if
´
ˆ
p1;p;hq
p1;p;hqp0q
˙
:“ ´
ˆ
p1;p;hqpv;h;c;hq
p1;p;hqpv;h;c;hqp0q
˙
P
ˆ
B}v;h}Mp0;Tq
0
˙
p1pv;cqptq “
Tż
t
ż
p˚
pSpv;cq ´ ydq g
$
%
Bttp˚
´ 4p˚
“ h in p0;Tq ˆ
p˚
“ 0 on p0;Tq ˆ B
pp˚
;Btp˚
q “ p0;0q in tTu ˆ
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 8

11. International Research Training Group IGDK 1754
Variational Discretization of p˜Pq
In the following we discretize the state equation by linear continuous FE in time
and space (S b Sh), where controls will not be changed:
p˜Psemi
;h q
$
%
min
v P Mp0;Tq
c P R
1
2 }S;hpv;cq ´ yd}2
L2
p
T q `
şT
0 d|vj| “: J;hpv;cq
Optimality Conditions
pv;h;c;hq P Mp0;Tq ˆ R is the solution of p˜Psemi
;h q, if
´
ˆ
p1;p;hq
p1;p;hqp0q
˙
:“ ´
ˆ
p1;p;hqpv;h;c;hq
p1;p;hqpv;h;c;hqp0q
˙
P
ˆ
B}v;h}Mp0;Tq
0
˙
p1pv;cqptq “
Tż
t
ż
p˚
pSpv;cq ´ ydq g
$
%
Bttp˚
´ 4p˚
“ h in p0;Tq ˆ
p˚
“ 0 on p0;Tq ˆ B
pp˚
;Btp˚
q “ p0;0q in tTu ˆ
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 8

12. International Research Training Group IGDK 1754
Convergence Results - Standard Approach

13. “ 1: yd P C1pI; H1
0 p
qq, g P H2 X H1
0 , and py0;y1q P Hp3q
ˆ H2 X H1
0 .
State Error Rates
}Spv;cq ´ S;hpv;h;c;hq}L2
p
T q P Op

14. ` h

15. q
Cost Error Rates
|Jpuq ´ J;hpu;hq| P O`
2 ` h2
˘
;
ˇ
ˇ
ˇ
ˇ}Dtu}Mp0;Tq ´ }Dtu;h}Mp0;Tq
ˇ
ˇ
ˇ
ˇ P O`

16. ` h

17. ˘
;
with u;hptq “
ş
r0;ts dv;h ` c;h:
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 9

18. International Research Training Group IGDK 1754
Optimal Control with PDE constraints and Sparse Controls
§ What are sparse controls?
Sparsity
[Cas17]: BV-controls are sparse, if their distributional derivative is singular with
respect to the Lebesgue measure.
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 10

19. International Research Training Group IGDK 1754
Optimality Conditions - Consequences
For the optimal control pv;cq of p˜Pq holds:
$
%
supppv˘
q Ă tt P r0;Ts | p1ptq “ ¯ u
}p1}C0pIq ď
We have analogous results for pv;h;c;hq.
§ If D :“ tp1 “ ˘u is a nite set, we nd that u is piecewise constant, i.e.
uptq “
ÿ
aPD
a ¨ 1ra;Tsptq ` c
§ In practice, we often observe piecewise constant controls.
§ In general, we cannot expect
piecewise constant controls u.
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 11

20. International Research Training Group IGDK 1754
Optimal Convergence Results - Preliminaries
Assumptions
A1: t t P p0;Tq | |p1ptq| “ u “ tt1;¨ ¨ ¨ ;tmu with m P N.
A2: Bttp1ptjq ‰ 0, for j “ 1;¨ ¨ ¨ ;m.
Assumption A1 implies:
u “
mř
`“1
c` 1rt`;Ts ` ¯c;
whereby c` can be 0, if u has no jump in t`.
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 12

21. International Research Training Group IGDK 1754
Optimal Convergence Results - Preliminaries
Due to the structural assumptions A1, A2, we obtain:
Explicit Form
There exists a p0;h0q such that @p;hq “ # ď p0;h0q it holds
uptq “
mř
j“1
cj1ptj;Tsptq ` ¯c; u#ptq “
mř
j“1
cj;#1ptj;#;Tsptq ` ¯c#
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 13

22. International Research Training Group IGDK 1754
Optimal Convergence Results - Preliminaries
This implies the following estimate for the optimal controls:
L1 ´ Estimate
For all p;hq “ # ď p0;h0q holds
}u ´ u#}L1
p0;Tqm ď rc
ˆ
|¯c ´ ¯c#| `
mř
j“1
|cj| ¨ |tj ´ tj;#| ` |cj ´ cj;#|
˙
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 14

23. International Research Training Group IGDK 1754
Optimal Convergence Results
We dene “ 1 for pyd;g;y0;y1q P C1pI; H1
0 p
qq ˆ H2 X H1
0 ˆ Hp3q
ˆ H2 X H1
0 .
Amplitude |cj ´ cj;#|
Jump |tj ´ tj;#|
Constant |¯c ´ ¯c#|
,
////.
////-
“ ď c
`
2 ` h2 ` }Spuq ´ S#pu#q}L2
p
T q
˘
:
Control ´ Error Rate:
This implies:
}u ´ u#}L1
p0;Tq P Op ` hq (suboptimal):
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 15

24. International Research Training Group IGDK 1754
Optimal Convergence Result
Optimal Rates:
1st-Step, State Dependence:
}u ´ u#}L1
pIq ď cp2 ` h2 ` }Spuq ´ S#pu#q}L2
p
T qq
2nd-Step, Scaled Young Inequality:
}Spuq ´ S#pu#q}L2
p
T q
ďljhn
FOOC
#
c}Spuq ´ S#puq}L2
p
T q
`cpgq}u# ´ u}
1
2
L1
pIq}p˚
pSpuq ´ ydq ´ p˚
#pSpuq ´ ydq}
1
2
L8p0;T;L2
p
qq
+
ďljhn
Zlotnik
Young Ineq.; ą 0
#
cp2 ` h2q ` ˆc}Spuq ´ S#puq}L2
p
T q
`cp
ÝÑg q
4 }p˚
pSpuq ´ ydq ´ p˚
#pSpuq ´ ydq}L8p0;T;L2
p
qq
+
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 16

25. International Research Training Group IGDK 1754
Optimal Convergence Results
In case of pyd;g;y0;y1q P C1pI; H1
0 p
qq ˆ H2 X H1
0 ˆ Hp3q
ˆ H2 X H1
0 we obtain:
Optimal Control Error Rates
}u ´ u#}L1
pIq; |¯c ´ ¯c#|;
|tj ´ tj;#|; |cj ´ cj;#|
,
.
-
“ Op2 ` h2q
with j “ 1;¨ ¨ ¨ ;m,
Optimal State and Total Variation Error Rates
}Spuq ´ S#pu#q}L2
p
T q “ Op2 ` h2q
ˇ
ˇ
ˇ
ˇ}Dtu}MpIq ´ }Dtu#}MpIq
ˇ
ˇ
ˇ
ˇ “ Op2 ` h2q pBV-Strict Convergence!q:
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 17

26. International Research Training Group IGDK 1754
Numerical Experiment - Optimal Rates
§ Variational discretization of p˜Pq Ñ BV-control is not discretised.
§ A discretization of BV-functions leads to mesh dependence which prevents
optimal rates (e.g. [HMNV19]).
§ PDAP Algorithm used for measure-valued control problems, see [PW19].
BV-PDAP
1. Set pv0;c0q “
`řm0
i“1 c0;it0;i ;c0
˘
, k=0;
2. Calculate t˚
“ arg maxtPp0;Tq |p1;;hpvk;ckqptq|.
3. Calculate p¯; ¯q “ arg min; J;h
`řmk
i“1 itk;i ` mk`1t˚ ;
˘
.
4. Set pvk`1;ck`1q “
`řmk
i“1
¯k;itk;i ` ¯0;mk`1t˚ ; ¯
˘
, k “ k ` 1, and go to 2.
Similar algorithm is used in [HMNV19].
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 18

27. International Research Training Group IGDK 1754
Numerical Experiment - Optimal Rates
Analytically solvable control problem:
§
“ r´1;1s2, T “ 2, “ 2:3 ¨ 10´4, and patch
gpxq “ cospx1{2q ˚ cospx2{2q
§ Dene u :“ 1r0:5;Tq ´ 1r1:5;Tq
§ Desired state: yd :“ Spuq ´ pBtt ´ 4q'pt;xq with py0;y1q “ p0;0q for S
and 'pt;xq :“

28. sinp3tq sinp3
2 tq
dź
i“1
cosp
2
xiq with

29. “ 3l
4
´
2
?
2
¯´2
.
Assumption A1 and A2 are fullled!
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 19

30. International Research Training Group IGDK 1754
Numerical Experiment - Optimal Rates
Analytically solvable control problem:
§
“ r´1;1s2, T “ 2, “ 2:3 ¨ 10´4, and patch
gpxq “ cospx1{2q ˚ cospx2{2q
§ Dene u :“ 1r0:5;Tq ´ 1r1:5;Tq
§ Desired state: yd :“ Spuq ´ pBtt ´ 4q'pt;xq with py0;y1q “ p0;0q for S
and 'pt;xq :“

31. sinp3tq sinp3
2 tq
dź
i“1
cosp
2
xiq with

32. “ 3l
4
´
2
?
2
¯´2
.
Assumption A1 and A2 are fullled!
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 19

33. International Research Training Group IGDK 1754
Numerical Experiment - Optimal Rates
10-2
10-1
10-8
10-6
10-4
10-2
100
O( 2
+h2
)
O( +h)
State Error
Jump Error
Amplitude Error
Constant Error
Control L1
-Error
J-Cost Error
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 20

34. International Research Training Group IGDK 1754
Literature
[Cas17], E. Casas, A review on sparse solutions in optimal control of partial
dierential equations. SEMA Journal, 74, pp. 319-344, 2017.
[CKK17], E. Casas, K. Kunisch, and F. Kruse. Optimal control of
semilinear parabolic equations by bv-functions. SIAM Journal on Control
and Optimization, 55:1752-1788, 2017.
[HMNV19], D. Hafemeyer, F. Mannel, I. Neitzel, B. Vexler, Finite element
error estimates for elliptic optimal control by BV functions, arxiv, 2019.
[HiDe10], K. Deckelnick, M. Hinze, A note on the approximation of ellliptic
control problems with bang-bang controls. Comput. Optim. Appl.,
51:931-939, 2010.
[PW19], K. Pieper, D. Walter, Linear convergence of accelerated
conditional gradient algorithms in spaces of measures, arxiv, 2019.
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 21

35. International Research Training Group IGDK 1754
Appendix
More detailed results
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 22

36. International Research Training Group IGDK 1754
Optimal Convergence Results - Consequences
Let the assumptions A1 and A2 hold:
Properties of u#:
Consider an amplitude cj;# of an optimal control of pPsemi
# q, with j “ 1;¨ ¨ ¨ ;m:
a) Assume that |cj| ą 0. The optimal control of pPsemi
# q has a jump
(|cj;#| ą 0) inside Bptjq for all 0 ă # ă #0 and #0 small enough.
b) Assume that cj “ 0. The optimal control of pPsemi
# q can have a jump in
Bptjq for all 0 ă # ă #0, with #0 small enough, but the jump height has to
decrease with some specic rate pcj;# P Op ` hqq.
c) Let tt P I||p1ptq| “ u “ H. Then the optimal control sub function u# of
pPsemi
# q has no jumps pu# “ const.q for all 0 ă # ď #0.
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 23

37. International Research Training Group IGDK 1754
Optimal Control of the Wave Equation with BV-Functions
Wave Solution - Error Rates:
}y ´ y;h}Cp0;T;L2
p
qq P Op2
3 ` h2
3 q
f “forcing y0 “displacement y1 “velocity
3
C1pr0;Ts; H1
0 p
qq or
BVp0;T; H2q
H3 H2
2 C1pr0;Ts; L2p
qq H2 H1
0 p
q
1 L2p
Tq H1
0 p
q L2p
q
§ H “
w P L2p
q
ˇ
ˇ
ˇ
ˇ
ř
kě1
k xw;ky2
L2
p
q ă 8
*
with eigenvalues and
eigenfunctions pk;kqkě1 of ´4 with homogeneous Dirichlet boundary
conditions.
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 24

38. International Research Training Group IGDK 1754
Error Rates - Data with Lower Regularity

39. “ 2
3 : yd P C1pI; L2p
qq and py0;y1q P H1
0 p
q ˆ L2p
q.
State Error Rates
}Spv;cq ´ S;hpv;h;c;hq}L2
p
T q P Op

40. ` h

41. q
Cost Error Rates
|Jpuq ´ J;hpu;hq| P O`

42. ` h

43. ˘
;
ˇ
ˇ
ˇ
ˇ}Dtu}Mp0;Tq ´ }Dtu;h}Mp0;Tq
ˇ
ˇ
ˇ
ˇ P O`

44. ` h

45. ˘
;
Control Error Rates
}u ´ u#}L1
pIq P Op

46. ` h

47. q
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 25

48. International Research Training Group IGDK 1754
Thank you for your attention
Supported by the DFG through the International Research Training Group IGDK
1754 Optimization and Numerical Analysis for Partial Dierential Equations
with Nonsmooth Structures
Sebastian Engel Optimal Control of the Wave Equation with BV-Functions MAFELAP 2019 26