An optimal control problem associated to the undamped linear wave equation with time depending controls of bounded variations (BV), multiplied by fixed spatial shape functions with pairwise disjoint supports is considered. More precisely the problem under consideration is given by the least squares distance of a desired state to the controlled wave equation in the space-time L2-Norm and an additional total variation term of the derivative of the BV-function controls. Using the total variation of a BV-function in the mentioned cost functional causes sparsity in the derivative of the optimal control (Dirac measures) enhances locally constant controls with jumps at the corresponding Diracs. This sparsity property can be partially represented with the necessary and sufficient first order optimality condition. Numerically we employ a L2 regularization of the weak derivative of the controls times a constant gamma, which we later take to 0. As a consequence the optimal controls live in the Sobolev space H1. One is then able to show that the BV optimal controls can be approximated in the BV weak* topology with the unique optimal H1 controls for gamma going to 0. The main purpose of this regularization is to use the semi-smooth Newton algorithm. In the full discretized problem, we consider a three level finite element method for the weak formulation of the wave equation with linear continuous finite elements in time and space. The control "u" can be identified with their unique decomposition into a L2 function "dt u" and a constant "u(0)", representing the derivative of the control and the initial value at time 0. In the full discretized problem we discretize "dt u" by linear continuous finite elements. Finally, we apply the semi-smooth Newton algorithm to approximate our H1 controls and a BV-path following algorithm to get an approximation of the BV optimal controls with respect to the time-space discretization refinement level.

1. International Research Training Group IGDK 1754
Optimal Control of the Wave Equation with
BV-Functions
Sebastian Engel (KFU), K. Kunisch (KFU), P. Trautmann (KFU)
Optimization 2017
September 6 8, 2017
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Contents
1. Introduction of Problem (P)
2. Equivalent Problem p˜Pq
3. First Order Optimality Condition of p˜Pq
Consequences
4. BV Path Following Algorithm
Reduced Problem and Regularized Problem pP
q
First Order Optimality Condition of pP
q
Semismooth Newton Method
BV Path Following Algorithm for pP
q
5. Numerical Example: Exact Solution
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Introduction - Problem
Consider the following optimal control problem:
pPq
$
’’’’’’’’’’
’’’’’’’’’’%
min
uPBVp0;Tqm
1
2
ż
ˆr0;Ts
pyu ´ ydq2
dxdt `
mÿ
j“1
j
ż
r0;Ts
d|Dtuj|ptq
s.t.
$
’’’
’’’%
Btty ´ 4y “
mÿ
j“1
ujgj in p0;Tq ˆ
y “ 0 on p0;Tq ˆ B
py;Btyq “ py0;y1q in t0u ˆ
§
Ă Rn (n=1,2,3) open bounded, B
Lipschitz, T P p0;8q
§ yd P W1;1pr0;Ts; L2p
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.
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Motivation - Optimal Control with BV functions
§ H1 regularized optimal control problems do behave continuously. BV
functions allow jumps.
§ BV regularized optimal control problems exhibit sparsity in Dtuj.
§ Practical applications want simpler functions, e.g. piecewise constant
functions where jump locations are imposed.
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Contents
1. Introduction of Problem (P)
2. Equivalent Problem p˜Pq
3. First Order Optimality Condition of p˜Pq
Consequences
4. BV Path Following Algorithm
Reduced Problem and Regularized Problem pP
q
First Order Optimality Condition of pP
q
Semismooth Newton Method
BV Path Following Algorithm for pP
q
5. Numerical Example: Exact Solution
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Equivalent Problem p˜Pq
Consider the following equivalent optimal control problem w.r.t. pPq:
p˜Pq
$
’’’’’’
’’’’’’%
min
pv;cqPMp0;TqmˆRm
1
2
ż
ˆr0;Ts
pyu ´ ydq2
dxdt `
mÿ
j“1
j
ż
r0;Ts
d|vj|ptq
with uptq “
ˆtş
0
dvjpsq ` cj
˙m
i“1
resp. pDtu;up0qq “ pv;cq:
§ This approach is only possible in one dim.
We dene with pphq the solution of the adjoint wave equation:
$
%
Bttp ´ 4p “ h in p0;Tq ˆ
p “ 0 on p0;Tq ˆ B
pp;Btpq “ p0;0q in tTu ˆ
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Contents
1. Introduction of Problem (P)
2. Equivalent Problem p˜Pq
3. First Order Optimality Condition of p˜Pq
Consequences
4. BV Path Following Algorithm
Reduced Problem and Regularized Problem pP
q
First Order Optimality Condition of pP
q
Semismooth Newton Method
BV Path Following Algorithm for pP
q
5. Numerical Example: Exact Solution
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First Order Optimality Condition
Consider the following integrated adjoint functions for j=1,...,m:
p1;jptq “
Tż
t
ż
wj
ppypuq ´ ydqgjdxds
Theorem (Necessary and Sucient Condition)
pv;cq P Mp0;Tqm ˆ Rm is the solution of p˜Pq i for all j “ 1;:::;m holds
´
ˆ
p1
p1p0q
˙
P
ˆ`
iB}vi}MpIq
˘m
i“1
0Rm
˙
P C0p0;Tqm ˆ Rm
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First Order Optimality Condition - Consequences
Corollary
For the optimal control pv;cq holds:
supppv`
i q Ă tt P r0;Ts | p1;iptq “ ´i u
supppv´
i q Ă tt P r0;Ts | p1;iptq “ i u
where vi “ v`
i ´ v´
i is the
Jordan decomposition of the measure vi.
Furthermore we have:
A
´p1;i
i
;vi
E
C0pIq;MpIq
“ }vi}MpIq (Polar Decomposition)
Remark (Sparsity)
If D :“ tp1;i “ ˘iu is nite we get that u is local constant, i.e.
uiptq “
ÿ
aPD
a ¨ 1ra;Tsptq ` c
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Contents
1. Introduction of Problem (P)
2. Equivalent Problem p˜Pq
3. First Order Optimality Condition of p˜Pq
Consequences
4. BV Path Following Algorithm
Reduced Problem and Regularized Problem pP
q
First Order Optimality Condition of pP
q
Semismooth Newton Method
BV Path Following Algorithm for pP
q
5. Numerical Example: Exact Solution
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Reduced Problem
Reduced problem:
p˜Pq
#
min
pv;cqPMp0;TqmˆRm
1
2 }Spv;cq ´ yd}2
L2
p
T q `
mÿ
j“1
j
ż T
0
|vj|dx “: Jpv;cq
with the ane linear control-to-state operator S.
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Regularized Problem
The following approach can also be found in the doctoral thesis of K. Pieper or
P. Trautmann:
pP
q
#
min
pv;cqPL2
pIqmˆRm
Jpv;cq `
2
˜
mÿ
j“1
}vj}2
L2
p0;Tq ` }c}2
Rm
¸
“: J1
pv;cq
Theorem
Denote by pÝÑv
;ÝÑc
q the unique solutions of pP
q and by ÝÑu
their BVpIqm
representation. Then we have:
ÝÑu
w˚; BVpIqm
ÝÝÝÝÝÝÝÑ u
0 ď J1
pÝÑv
;ÝÑc
q ´ Jpuq “ Op
q
Both statements imply that ÝÑu
Ñ0
ÝÝÝÑ u strictly in BVpIqm.
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First Order Optimality Condition of pP
q
Theorem (Necessary and Sucient Condition)
ÝÑu
“ pÝÑv
;ÝÑc
q P L2p0;Tqm ˆ Rm is an optimal control of pP
q i
´
¨
˝p
1 psq :“
ş
Tş
s
ppSpÝÑu
q ´ ydqÝÑg
p
1 p0q
˛
‚´
¨
˝
ÝÑv
ÝÑc
˛
‚P
¨
˝
`
iB}v
;i}L1
p0;Tq
˘m
i“1
0Rm
˛
‚
Corollary
For the optimal control pv;cq P MpIqm ˆ Rm of p˜Pq and function p1 holds:
1) p
1
H2
pIqm
ÝÝÝÝÑ p1
2) We have for i “ 1;:::;m:
şT
0 ´
p
1;i
i
dv
;ipsq
Ñ0
ÝÝÝÑ
A
´p1;i
i
;vi
E
C0pIq;MpIq
“ }vi}MpIq
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Equivalent First Order Optimality Condition for pP
q
Using a Prox-Operator approach one is able to show the following:
Corollary (Rewritten: Necessary and Sucient Condition of pP
q)
p1q ÝÑv
“
¨
˝
max
´
0;´1
p
1;ipsq ´ i
¯
`
` min
´
0;´1
p
1;ipsq ` i
¯
˛
‚
m
i“1
P L2p0;Tqm
p2q p
1 p0q `
ÝÑc
“ 0Rm
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Equivalent First Order Optimality Condition for pP
q -
Consequences
Corollary (Sparsity)
For a.a. s P r0;Ts and i “ 1;:::;m holds:
v
;ipsq “
$
’’’’
’’’’%
0 ; |p
1;ipsq| ă i
´1
p
1;ipsq ` i
; p
1;ipsq ě i
´1
p
1;ipsq ´ i
; p
1;ipsq ď ´i
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BV Path Following Algorithm for pP
q
Dene the following function: F
pÝÑv ; ÝÑc q :“
˜ÝÑv ´ Prox
ř
i
i }¨}L1p0;Tq
p´ 1
p
1 q
ÝÑc ` 1
p1p0q
¸
Since we approximate p˜Pq by pP
q we consider the path following algorithm:
BV Path Following Algorithm
Input: u0 P L2p0;Tqm ˆ Rm,
0 ą 0, TOL
ą 0, TOLN ą 0, k “ 0 and
P p0;1q
while
k ą TOL
do
Set i “ 0, ui
k “ uk
while }F
k pui
kq}L2p0;TqmˆRm ą TOLN do
Solve DF
k pukqpuq “ ´F
k pukq, set ui`1
k`1 “ ui
k ` u; i “ i ` 1.
end
Dene uk`1 “ ui
k, and
k`1 “
k; set k “ k ` 1.
end
This approach was also used by E. Casas, K. Kunisch, and F. Kruse.
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BV Path Following Algorithm for pP
q - Super Linearity
Theorem (Super Linearity of the Newton Method)
For each
ą 0 there exists a ą 0 s. t for all pÝÑv ;ÝÑc q P L2pIqm ˆ Rm with
}pÝÑv ;ÝÑc q ´ pÝÑv
;ÝÑc
q}L2
pIqmˆRm ă ;
the semismooth Newton method algorithm converges superlinearly to pÝÑv
;ÝÑc
q.
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Contents
1. Introduction of Problem (P)
2. Equivalent Problem p˜Pq
3. First Order Optimality Condition of p˜Pq
Consequences
4. BV Path Following Algorithm
Reduced Problem and Regularized Problem pP
q
First Order Optimality Condition of pP
q
Semismooth Newton Method
BV Path Following Algorithm for pP
q
5. Numerical Example: Exact Solution
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Numerical Example: Exact Solution
§ Let
“ r´1;1s2, r0;T “ 2s and consider the space patch
gpxq “ 1r´0:5;0:5s2 pxq, and uptq “
2ÿ
n“0
p´1qn1r 1`2n
3
;2sptq , i.e.
and consider the adjoint wave 'pt;xq :“

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

21. “ 3l
4
´
2
?
2
¯´2
.
§ It holds: p1ptq “
2ż
t
ż
'pt;xqgpxqdxdt “ ´sin
ˆ
3
2
t
˙3
ñ }p1}C0pIq “ .
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Numerical Example: Exact Solution
§ Consider the desired state yd :“ Spuq ´ pBtt ´ 4q'pt;xq with
py0;y1q “ p0;0q for S.
§ Furthermore consider “ 0:005.
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Numerical Example: Exact Solution
Then we have for:
§ Time grid “ 65 (d.o.f) and triangulation 22N`1 “ 128 triangles, N “ 3:
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Numerical Example: Exact Solution
Then we have for:
§ Time grid “ 129 (d.o.f) and triangulation 22N`1 “ 512 triangles, N “ 4:
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Numerical Example: Exact Solution
Then we have for:
§ Time grid “ 257 (d.o.f) and triangulation 22N`1 “ 2048 triangles, N “ 5.
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Numerical Example: Exact Solution
Then we have for:
§ Time grid “ 513 (d.o.f) and triangulation 22N`1 “ 8192 triangles, N “ 6:
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Literature
1 K. Pieper, Finite element discretization and ecient numerical solution of
elliptic and parabolic sparse control problems, Dissertation, Technical
University of Munich.
2 E. Casas, F. Kruse, K. Kunisch, Optimal control of semilinear parabolic
equations by BV-functions, (accepted.).
3 K. Kunisch, P. Trautmann, B. Vexler, Optimal control of the undamped
linear wave equation with measure valued controls, SIAM J. Control
Optim., 54(3) 2016, 1212-1244.
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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
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