Optimal Control for Linear Second-Order Hyperbolic Equations by BV-Functions in Time, KFU Graz 2019

International Research Training Group IGDK 1754
Optimal Control for Linear Second-Order Hyperbolic
Equations by BV-Functions in Time
Sebastian Engel
Rigorosum
01.03.2019
Sebastian Engel Optimal Control and BV-Functions Rigorosum 1

Optimal Control of the Wave Equation with BV-Functions
We focus on optimal control of hyperbolic partial dierential equations:
Figure: [Hor16]

Background - Related Work
§ Optimization with 2nd order linear hyperbolic equations:
§ Lions (Hilbert control).
and Kröner et al. (Distributed, Dirichlet, Neumann L2
-control).
Kunisch et al., and Trautmann et al.(Mp
; L2
p0;Tqq,
L2
w˚ p0;T; Mp
qq-control).
§ Thesis: 2nd order linear hyperbolic equations with BV(0,T)-controls.
§ FE with hyperbolic equations in optimal control:
§ Kröner et al. lin. cont. FE for state and control discretization.
Trautmann et al. lin. cont. state, variational discretization control.
§ Thesis: Variational discretization of u, lin. cont. FE for Dtu and state.
§ Hyperbolic equations and sparse controls
§ Kunisch et al., and Trautmann et al..
§ Thesis: Sparse BV-controls.
§ Optimization with BV-controls and other PDE constraints
§ Casas, Kunisch (semilinear elliptic, BVp
q).
Casas, Kruse, Kunisch (semilinear parabolic, BVp0;Tq-controls).
§ Thesis: 2nd order Hyperbolic equations with BVp0;Tq-controls

pPq
$
’’’’’’’’’’
’’’’’’’’’’%
min
uPBVp0YTqm
1
2
ż
ˆr0YTs
pyu ´ ydq2
dxdt `
mÿ
j“1
j
ż
r0YTs
d|Dtuj|ptq “: Jpuq
s.t.
$
’’’
’’’%
Btty ´ 4y “
mÿ
j“1
ujgj in p0YTq ˆ
y “ 0 on p0YTq ˆ B
pyYBtyq “ py0Yy1q in t0u ˆ
§ Ă Rn (n=1,2,3) open bounded, B Lipschitz, T P p0Y8q
§ yd P W1Y1
pr0YTs; L2
pqq, py0Yy1q P H1
0
pq ˆ L2
pq
§ pgjqm
j Ă L8
pqzt0u pairwise disjoint supports wj
This strictly convex problem has a unique solution.
Clason, Kunisch [CK11]
§ Penalization by the } ¨ }BV -norm corresponds to settings where the cost is
proportional to changes in the control.
- On/O structure, less blurring (H1
´control).

Equivalent Problem p˜Pq
Consider the following equivalent optimal control problem w.r.t. pPq:
p˜Pq
#
min
pvYcqPMp0YTqmˆRm
1
2
}SpvYcq ´ yd}2
L2pT q `
mÿ
j“1
j
ż T
0
|vj|dx “: JpvYcq
with uptq “
ˆtş
0
dvjpsq ` cj
˙m
i“1
resp.
ˆ
Dtu
up0q
˙
“
ˆ
v
c
˙ ˆ
Fundamental
theorem of calculus
˙
.
§ S is the ane control-to-state operator.
§ BVp0YTqm – Mp0YTqm ˆ Rm only possible in one dim.

Optimality Conditions
Adjoint Wave Equation
We dene by p˚
phq the solution of the adjoint wave equation:
$

%
Bttp˚
´ 4p˚
“ h in p0YTq ˆ
p˚
“ 0 on p0YTq ˆ B
pp˚
YBtp˚
q “ p0Y0q in tTu ˆ
Consider the following integrated adjoint functions for j=1,...,m:
p1YjpvYcqptq “
Tż
t
ż
wj
p˚
pSpvYcq ´ ydq gjdxdsY pvYcq P Mp0YTqm ˆ Rm

Theorem (Necessary and Sucient Condition)
pÝÑv YÝÑc q P Mp0YTqm ˆ Rm is the solution of p˜Pq if, for all j “ 1YXXXYm holds
´
ˆ
p1pÝÑv YÝÑc q
p1pÝÑv YÝÑc qp0q
˙
P
ˆ`
iB}ÝÑv i}Mp0YTq
˘m
i“1
0Rm
˙
Y
with p1pÝÑv YÝÑc q P C0pr0YTsq X C2
pr0YTsq.

Regularized Problem
Regularized Problem

Regularized Problem
For the numerical realization, we would now like to consider a problem that is
numerically easier to solve (Pieper [Pi15]):
p˜Pq
#
min
pvYcqPL2p0YTqmˆRm
JpvYcq `
2
˜
mÿ
j“1
}vj}2
L2p0YTq ` }c}2
Rm
¸
“: J1
pvYcq
Theorem
Denote by pÝÑv YÝÑc q the unique solutions of p˜Pq and by u their BVp0YTqm
representation. Then we have:
§ 0 ď J1
pÝÑv YÝÑc q ´ Jpuq “ Opq
§ u
Ñ0
ÝÝÝÑ u strictly in BVp0YTqm (Fundamental theorem of calculus)
§ p
1
H2
p0YTqm
ÝÝÝÝÝÝÑ p1 with p
1
ptq :“
Tş
t
ş
wj
p˚
pSpÝÑv YÝÑc q ´ ydqgjdxds

Regularized Problem - Optimality Conditions
Due to the additional L2
´regularization, we can use a Prox-Operator approach:
Theorem
ˆÝÑv
ÝÑc
˙
is optimal i
$
’
’%
ÝÑv “ Proxř
i
i }¨}L1p0;Tq
p´1
p
1
q
p
1
p0q ` ÝÑc “ 0Rm
,
/.
/-
with Proxř
i
i }¨}L1p0;Tq
p´1
p
1
q :“
¨
˝
max
´
0Y´1
p
1Yipsq ´ i

¯
`
` min
´
0Y´1
p
1Yipsq ` i

¯
˛
‚
m
i“1
§ Proxř
i
i }¨}L1p0;Tq
p´1
p
1
q is semismooth.

BV Path-Following Algorithm
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 p˜Pq we consider the Path-Following algorithm:
BV Path-Following Algorithm
Input: u0 P L2
p0YTqm ˆ Rm, 0 ą 0, TOL ą 0, TOLN ą 0, k “ 0 and
# P p0Y1q
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
A similar approach was used for a fully discretized control problem with
semi-linear parabolic constraints in [CKK17].

BV Path-Following Algorithm - Super Linearity
Theorem (Super Linearity of the Newton Method)
For each ą 0 there exists a ą 0 s.t. for all pÝÑv YÝÑc q P L2
p0YTqm ˆ Rm with
}pÝÑv YÝÑc q ´ pÝÑv YÝÑc q}L2p0YTqmˆRm ă Y
the semismooth Newton method algorithm converging superlinearly.

Numerical Example
Example: “ r´1Y1s2
, T “ 2, “ 0X005, and patch gpxq “ 1r´0X5Y0X5s2 pxq, i.e.
Desired state: yd :“ Spuq ´ pBtt ´ 4q9ptYxq with py0Yy1q “ p0Y0q for S and
9ptYxq :“ sinp3%tq sinp3%
2
tq
dź
i“1
cosp
%
2
xiq with “ 3%l
4
´
2
?
2
%
¯´2

Numerical Example
Dirac-example until “ 10´8
(Full Discretization: v P S( and y P S( b Sh).

Additional Control Problems
All presented results are extended for other linear dierential equations, i.e.
§ Hyperbolic constraints: Btty ` Ay “
mř
i“1
ui ¨ gi,
§ A as elliptic operator,
§ with Dirichlet, Neumann, or Robin B.C.
§ Linear parabolic constraints, which extends the results of [CKK17].

Error Rates for Variational Discretization
Error Rates for
Variational Discretization

Error Rates BV-Control Problems
§ i[Bar12]: Image reconstruction by TV-based optimal control
(no PDE constraints).
§ linear continuous uh : }u ´ uh}L2p
q ď ch
1
6 .
§ 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).

Standard Approach - Error Estimates
§ In case of smooth cost functions Jpuq, the standard approach uses
coercivity properties, e.g.
§ appropriate testing of the 1st
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.

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
(Yh q
$
’
’%
min
v P Mp0YTqm
c P Rm
1
2
}S(YhpvYcq ´ yd}2
L2pT q `
mÿ
j“1
j
ż T
0
d|vj| “: J(YhpvYcq
pÝÑv (YhYÝÑc (Yhq P Mp0YTqm ˆ Rm is the solution of p˜Psemi
(Yh q, if
´
ˆ
p1Yp(Yhq
p1Yp(Yhqp0q
˙
:“ ´
ˆ
p1Yp(YhqpÝÑv (YhYÝÑc (Yhq
p1Yp(YhqpÝÑv (YhYÝÑc (Yhqp0q
˙
P
ˆ`
iB}ÝÑv iY(Yh}Mp0YTq
˘m
i“1
0Rm
˙
§ }p1Y(Yh ´ p1}L8p0YTqm Ñ 0X

Convergence Results - Standard Approach
“ 2
3
: yd P C1
pI; L2
pqq and py0Yy1q P H1
0
pq ˆ L2
pq.
“ 1: yd P C1
pI; H1
0
pqq, g P pHp2q
qm, and py0Yy1q P Hp3q
ˆ Hp2q
.
State Error Rates
}SpÝÑv YÝÑc q ´ S(YhpÝÑv (YhYÝÑc (Yhq}L2pT q P Op( ` hq
Cost Error Rates
|Jpuq ´ J(Yhpu(Yhq| P Op( ` hq Y
with “ for “ 2
3
and “ 2 elseX
ˇ
ˇ
ˇ
ˇ}Dtu}Mp0YTqm ´ }Dtu(Yh}Mp0YTqm
ˇ
ˇ
ˇ
ˇ P O`
( ` h
˘
Y
with u(Yhptq “
ş
r0Yts dÝÑv (Yh ` ÝÑc (YhX

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.

Optimality Conditions - Consequences
For the optimal control pÝÑv YÝÑc q of p˜Pq holds:
$

%
supppÝÑv ˘
i q Ă tt P r0YTs | p1Yiptq “ ¯i u
}p1Yi}C0pIq ď i
We have analogous results for pÝÑv (YhYh(Yhq.
§ If D :“ tp1Yi “ ˘iu is a nite set, we nd that u is piecewise constant, i.e.
uiptq “
ÿ
aPD
'a ¨ 1raYTsptq ` ÝÑc
§ In practice, we often observe piecewise constant controls.
§ In general, we cannot expect
piecewise constant controls u.

Optimal Convergence Results - Preliminaries
Assumptions
A1: t t P p0YTq | |p1Yiptq| “ u “ tt1YiY¨ ¨ ¨ Ytmi Yiu
for mi P Ną0, with i “ 1Y¨ ¨ ¨ Ym and mi P N.
A2: Bttp1ptjYiq ‰ 0, for i “ 1Y¨ ¨ ¨ Ym and j “ 1Y¨ ¨ ¨ Ymi.
§ Assumption A1 implies:
ui “
miř
–“1
ci
– 1rt`;i YTs ` ciY
whereby ci
– can be 0, if ui has no jump in t–Yi.
[HiDe] for an elliptic problem without L2
-regularization: |tp˚
pxq “ 0u| “ 0
Bang-Bang
control ´ Variational Discretization ´ }u´uh}L1pq ď ch2
|lnphq|

Due to the structural assumptions A1, A2, we obtain:
Explicit Form
There exists a p(0Yh0q such that @p(Yhq “ 5 ď p(0Yh0q it holds
uptq “
¨
˚
˝
c1
...
cm
˛
‹
‚`
¨
˚
˚
˚
˚
˚
˝
m1ř
j“1
c1
j 1ptj;1YTsptq
...
mmř
j“1
cm
j 1ptj;mYTsptq
˛
‹
‹
‹
‹
‹
‚
Y u5ptq “
¨
˚
˝
c1Y5
...
cmY5
˛
‹
‚`
¨
˚
˚
˚
˚
˚
˝
m1ř
j“1
c1
jY51pt1
j;#YTsptq
...
mmř
j“1
cm
jY51ptm
j;#YTsptq
˛
‹
‹
‹
‹
‹
‚

This implies the following estimate for the optimal controls:
L1
´ Estimate
For all p(Yhq “ 5 ď p(0Yh0q holds
}u ´ u5}L1p0YTqm ď rc
ˆ mř
i“1
|ci ´ ciY5| `
miř
j“1
|ci
j | ¨ |tjYi ´ ti
jY5| ` |ci
j ´ ci
jY5|
˙

Optimal Convergence Results
We dene “ 2
3
for pydYÝÑg Yy0Yy1q P C1
pI; L2
pqq ˆ L2
pqm ˆ H1
0
pq ˆ L2
pq
We eand “ 1 for pydYÝÑg Yy0Yy1q P C1
pI; H1
0
pqq ˆ pHp2q
qm ˆ Hp3q
ˆ Hp2q
.
Amplitude |ci
j ´ ci
jY5|
Jump |tjYi ´ ti
jY5|
Constant |ci ´ ciY5|
,
////.
////-
“ Op( ` hq if “ 2
3
Y
ď c
`
(2
` h2
` }Spuq ´ S5pu5q}L2pT q
˘
, if “ 1X
Control ´ Error Rate:
This implies:
}u ´ u5}L1p0YTqm P Op( ` hq (suboptimal)X

Optimal Convergence Results
In case of pydYÝÑg Yy0Yy1q P C1
pI; H1
0
pqq ˆ pHp2q
qm ˆ Hp3q
ˆ Hp2q
we obtain
(by Young inequality and control rates for “ 1):
Optimal Control Error Rates
}u ´ u5}L1pIqm Y |ci ´ ciY5|Y
|tjYi ´ ti
jY5|Y |ci
j ´ ci
jY5|
,
.
-
“ Op(2
` h2
q
with i “ 1Y¨ ¨ ¨ Ym, j “ 1Y¨ ¨ ¨ Ymi,
Optimal State and Total Variation Error Rates
}Spuq ´ S5pu5q}L2pT q “ Op(2
` h2
q
ˇ
ˇ
ˇ
ˇ}Dtu}MpIq ´ }Dtu5}MpIq
ˇ
ˇ
ˇ
ˇ “ Op(2
` h2
q pBV-Strict Convergence!qX

Optimal Control of Hyperbolic Equations
Additional Results
and Outlook

Sound Propagation in an Urban Environment(rHhsrTtsrTtsrPpsrSss?|rFf srTtsrPpsq:TrueWave:swf (rHhsrTtsrTtsrPpsrSss?|rFf srTtsrPpsq:
TrueWave:swf
§ Hyperbolic Equation:
Btty ´ divpbpxqOyq “
řm
i“1
uiptq ¨ gipxq
§ Sound is emitted by acoustic speakers,
placed in gipxq

Sound Propagation in an Urban Environment
Data of the microphone recordings can be used to obtain a possible position
using optimal control theory:
Optimal Control - Example of Penalization Costs
§ Examples of penalization terms, used to nd an optimal control, e.g.
Moving Microphone:
Tş
0
´
´
ş
Br pptqq yptqdx ´ ydataptq
¯2
dt
§ The solution of the following minimization problem leads to a possible
sound source position:
min
uPX
1
2
ř
iYj
´
´
ş
Br pxj q yptiqdx ´ ydata
¯2
`
2
}u}X
where X “ BVp0YTqm or X “ H1
p0YTqm with
}u}BVp0YTqm “
mř
i“1
}Btui}Mp0YTq or }u}H1p0YTqm “
mř
i“1
}Btui}2
L2p0YTq ` }up0q}2
Rm .

Sound Propagation in an Urban Environment(rHhsrTtsrTtsrPpsrSss?|rFf srTtsrPpsq:Comparision:swf (rHhsrTtsrTtsrPpsrSss?|rFf srTtsrPpsq:
Comparision:swf
i “ 0X1 for BV and H1
,
bpxq “
$

%
10´10
Y inside
0X35Y outside
r10´10
Y0X35sY else

Bayesian Inversion with BV-Prior
Based on [EHMS19], we are able to consider a stochastic view on the sound
source identication problem before:
ydata “
´
´
ş
Br pxj q yuptiqdx
¯
i“1M
`
with noise „ Np0Y'q and stochastic control u “
kř
–“0
–1pt`YTs ` c „ 0
Bayesian Inversion
§ Well-posedness of the posterior solution
dypuq “ expp´1
2
}yu ´ ydata}2
¦qd0puq.
§ Convergence results for the SMC-Algorithm and FE-Discretization
§ Expectation, condence region, MAP-Estimator, ...
A similar Bayesian inversion problem is considered by [Yao16]
(TV-Gaussian Prior, Splitting pCN).

Summary
We analyzed control problems with second-order hyperbolic equations, with
various types of cost functionals, which are regularized by a non-smooth
semi-norm of BVp0YTq.
§ Optimality conditions + sparsity + explicit examples.

Summary
§ Regularized problem pPq (H1
´norm)
+ convergence results (control, costs, adjoint)

Summary
´norm)
§ Numerical analysis of a semismooth Newton algorithm for pPq
+ super-linear convergence.

Summary
´norm)
§ Error estimates for a variational discretization
+ optimal rates under specic assumptions (sparsity in pPq).

Summary
´norm)
§ Error estimates for a variational discretization
+ optimal rates under specic assumptions (sparsity in pPq).
§ A Bayesian perspective on pPq.

Literature
[Bar12], S. Bartels, Total variation minimization with nite elements:
convergence and iterative solution. SIAM J. Numer. Anal.,
50(3):1162-1180, 2012.
[Bog98], V. I. Bogachev, Gaussian measures, vol.62, Mathematical Surveys
and Monographs, American Mathematical Society, 1998.
[Cas17], E. Casas, A review on sparse solutions in optimal control of partial
dierential equations. SEMA Journal, 74, pp. 319-344, 2017.
[CK17], E. Casas, K. Kunisch, Analysis of optimal control problems of
semilinear elliptic equations by bv-functions. Set-Valued and Variational
Analysis, 1-25, 2017.

Literature
[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.
[CK11], C. Clason and K. Kunisch, A duality based approach to elliptic
control problems in non refelxive banach spaces. ESIAM Control Optim.
Calc. Var., 17, pp. 243-266, 2011.
[HiDe], K. Deckelnick, M. Hinze, A note on the approximation of ellliptic
control problems with bang-bang controls. Comput. Optim. Appl.,
51:931-939, 2010.
[EHMS19], S. Engel, D. Hafemeyer, C. Münch, and D. Schaden. An
application of sparse measure valued Bayesian inversion to acoustic sound
source identication. Inverse Problems, accepted.

Literature
[HMNV19], D. Hafemeyer, F. Mannel, I. Neitzel, B. Vexler, Finite element
error estimates for elliptic optimal control by BV functions, arxiv, 2019.
[Hor16], M. Hornikx, Ten questions concerning computational urban
acoustics. Building and Environment 106 : 409-421, 2016.
[KTV16], K. Kunisch, P. Trautmann, B. Vexler, Optimal control of the
undamped linear wave equation with measure valued controls. SIAM
Journal on Control and Optimization, 54(3), 1212-1244, 2016.
[KKV10], A. Kröner, K. Kunisch, B. Vexler, Semismooth Newton Methods
for an Optimal Boundary Control Problem of Wave Equations, in: Recent
Advances in Optimization and its Applications in Engineering, M. Diehl, F.
Glineur, E. Jarlebring, W. Michiels (eds.), Springer, 389-398, 2010.

Literature
[KKV11], A. Kröner, K. Kunisch, B. Vexler, Semismooth Newton Methods
for Optimal Control of the Wave Equation with Control Constraints, SIAM
J.Control Optim. 49: 830-858, 2011.
[Lion], J. L. Lions: Optimal Control of Systems Governed by Partial
Dierential Equations. Springer-Verlag, Berlin, 1971.
[MR04], B. S. Mordukhovich, J. P. Raymond, Neumann boundary control
of hyperbolic equations with pointwise state constraints. SIAM journal on
control and optimization, 43(4), 1354-1372, 2004.
[Pi15], K. Pieper, Finite element discretization and ecient numerical
solution of elliptic and parabolic sparse control problems. PhD thesis,
Technical University Munich, 2015.

Literature
[Sta09], G. Stadler, Elliptic optimal control problems with L1
-control cost
and applications for the placement of control devices. Comput. Optim.
Appl., 44(2):159-181, 2009.
[TVZ17], P. Trautmann, B. Vexler, A. Zlotnik, Finite element error analysis
for measure-valued optimal control problems governed by a 1D wave
equation with variable coecients, arXiv , 2017.
[Yao16], Z. Yao, Z. Hu, and J. Li. A tv-gaussian prior for
innite-dimensional bayesian inverse problems and its numerical
implementations. IOPscience, 2016.
[Zlo94], A. A. Zlotnik, Convergence rate estimates of nite-element
methods for second-order hyperbolic equations. numerical methods and
applications, p.153 et.seq. Guri I. Marchuk, CRC Press, 1994.

Appendix
Variational Discretization

Consider the following semi-discretized optimal control problem:
pPsemi
(Yh q
$
’’’’
’’’’%
min
uPBVp0YTqm
1
2
}yuY(Yh ´ yd}2
L2pT q `
mÿ
j“1
j
ż
r0YTs
d|Dtuj|ptq “: J(Yhpuq
with }y ´ y(Yh}Cp0YT;L2pqq P Op(2
3 ` h
2
3 qY for pFEM Zlotnikq
f “forcing y0 “displacement y1 “velocity
3
C1
pr0YTs; H1
0
pqq or
BVp0YT; H2
q
H3
H2
2 C1
pr0YTs; L2
pqq H2
H1
0
pq
1 L2
pTq H1
0
pq L2
pq
§ H “

w P L2
pq
ˇ
ˇ
ˇ
ˇ
ř
kě1
!
k xwYky2
L2pq ă 8
*
with eigenvalues and
eigenfunctions p!kYkqkě1 of ´4 with homogeneous Dirichlet boundary
conditions.

Due to the structural assumptions A1 and A2, we obtain:
Lemma
There exists a ą 0, and (0Yh0 ą 0 such that
i
@p(Yhq “ 5 ď p(0Yh0q holds:
ÝÑv 5Yi “
miř
l“1
ci
lY5ti
l;#
with ti
jY5 P BptjYiq
where pBptjYiqqmi
j“1
are pairwise disjoint for a xed i “ 1Y¨ ¨ ¨ Ym with 0 ă 5 Ñ 0

Optimal Convergence Result
Optimal Rates:
1st-Step, State Dependence:
}u ´ u5}L1pIqm ď ˆcp(2
` h2
` }Spuq ´ S5pu5q}L2pT qq
2nd-Step, Scaled Young Inequality:
}Spuq ´ S5pu5q}L2pT q
ďljhn
FOOC
#
c}Spuq ´ S5puq}L2pT q
`cpÝÑg q}u5 ´ u}
1
2
L1pIqm }L˚
pSpuq ´ ydq ´ L˚
5pSpuq ´ ydq}
1
2
L8p0YT;L2pqq
+
ďljhn
Zlotnik
Young Ineq.Y ą 0
#
cp(2
` h2
q ` ˆc}Spuq ´ S5puq}L2pT q
`cp
ÝÑg q
4 }L˚
pSpuq ´ ydq ´ L˚
5pSpuq ´ ydq}L8p0YT;L2pqq
+

Optimal Convergence Results - Consequences
Let the assumptions A1 and A2 hold:
Properties of u5:
Consider an amplitude ci
jY5 of an optimal control of pPsemi
5 q, with i “ 1Y¨ ¨ ¨ Ym,
and j “ 1Y¨ ¨ ¨ Ymi:
a) Assume that |ci
j | ą 0. The optimal control of pPsemi
5 q has a jump
(|ci
jY5| ą 0) inside BptjYiq for all 0 ă 5 ă 50 and 50 small enough.
b) Assume that ci
j “ 0. The optimal control of pPsemi
5 q can have a jump in
BptjYiq for all 0 ă 5 ă 50, with 50 small enough, but the jump height has
to decrease with some specic rate pci
jY5 P Op( ` hqq.
c) Let tt P I||p1Yiptq| “ u “ H for i “ 1Y¨ ¨ ¨ Ym, and m ě 1. Then the
optimal control sub function u5Yi of pPsemi
5 q has no jumps pu5Yi “ const.q for
all 0 ă 5 ď 50.

Appendix
More General Costs

Control Problem
pP¥q
$
’’’’’’’’’
’’’’’’’’’%
min
uPBVp0YTqm
3ř
j“1
j
2
} r¥jpyuq ´ zj}2
Oj
`
mř
–“1
–}Dtu–}Mp0YTq “: J¥puq
s.t.
$
’’’
’’’%
lyu :“ Bttyu ´ 4yu “
mÿ
j“1
ujgj in p0YTq ˆ
y “ 0 on p0YTq ˆ B
pyp0qYBtyp0qq “ py0Yy1q P H1
0
pq ˆ L2
pq
§ zj P Oj sep. Hilbert spaces and pg–qm
– Ă L8
pqzt0u with pairwise disjoint
supports.
r¥2pyuq :“ ¥2pyupt–qqr
–“1
Y
r¥3pyuq :“ ¥3pBtyupt–qqr
–“1
Y
0 “ t0 ă t1 ă ¨ ¨ ¨ ă tr “ TY
Lin. indep. p r¥j pyg`qqm
`“1
for one j ą0
with
$

%
r¥1 P Lp L2
pTq Y O1qY
¥2 P Lp L2
pqr Y O2qY
¥3 P Lp H´1
pqr Y O3qX
This convex problem has a solution.

Equivalent Problem and Optimality Conditions
prP¥q
$
’’’
’’’%
min
pvYcqPMp0YTqmˆRm
J¥puq :“ rJ¥pvYcq
with uptq “
ˆtş
0
dvjpsq ` cj
˙m
i“1
resp. pDtuYup0qq “ pvYcqX
The rst-order optimality condition has the following form:
ˆ
p¥
1
ptq
p¥
1
p0q
˙
“
¨
˝´
Tş
t
ş

ˆppÝÑv YÝÑc qÝÑg
´p¥
1
p0q
˛
‚P
ˆ
piB}pÝÑv i}Mqqi
0
˙
text
with the jumping
wave function
ˆppÝÑv YÝÑc q :“
řr
i“1
pi1Ii
with Ii :“ pti´1Ytiq
l pi “ gi in Ii ˆ Y piptiq “ pi`1ptiq ` hiY Btpiptiq “ Btpi`1ptiq ` ki
gi “ 1¥˚
1
p¥1pyuq ´ z1q|Ii ˆ P L2
pIi ˆ qY
hi “ 3p´4q´1
”
¥˚
3
p r¥3pyuq ´ z3q
ı
i
P H1
0
pqY
ki “ ´2
”
¥˚
2
p r¥2pyuq ´ z2q
ı
i
P L2
pqX

Regularized Control Problem
prP¥
q
#
min
pvYcqPL2p0YTqmˆRm
rJ¥pvYcq `
2
˜
mř
j“1
}vj}2
L2p0YTq ` }c}Rm
¸
For the BV-representation u of the optimal control pvYcq holds u
strict
ÝÝÝÑ
BV
u.

Regularized Control Problem - Jumping Wave Example
Let 1 “ 3 “ 0, and consider –, – “ 1Y¨ ¨ ¨ Yk, as pairwise disj. balls inside :
r¥2pyuq :“
ˆ´
1
|`|
ş
`
yuptiqdx
¯k
–“1
˙r
i“1
P Rk¨r p„ Microphones)
One experimental result with ÝÑ
t “ p0X098Y0X199Y0X297Y0X347Y0X398Y1q:

Appendix
Bayesian Inversion

Bayesian Inversion and pP¥q - Specic Example
Consider the following stochastic problem:
pI¥q z2 “ r¥2pyuq ` Y „ Np0Y 1
2
¨ idRk¨r qY
where r¥2 is dened as in (Microphones) and u K .

2
¨ idRk¨r qY
§ Without any knowledge of z2, we assume that u „ 0 (Prior)
Ñ How should 0 be dened? Gaussian?

2
¨ idRk¨r qY
§ We consider ˜u „ Np0Y 1
Iq “ with:
$

%
I P LpL2
pIqq trace class
rgpI1{2
q “ H1
p0YTqY
(see [Bog98,EX 2.3.4]).

2
¨ idRk¨r qY
§ We consider ˜u „ Np0Y 1
Iq “ with:
$

%
I P LpL2
pIqq trace class
rgpI1{2
q “ H1
p0YTqY
(see [Bog98,EX 2.3.4]).
§ We weight by f pwq “ 1
£0
expp´}Btw}Mq
Ñ concentration to functions with small total variation, i.e.
0p¨q “
ş
¨
f pwqdpwq (see [Yao18]).

§ We have u|z2 „ z, with zp¨q “ 1
£z
ş
¨
expp´2
2
}Ă¥2pyuq ´ z2}2
Rk¨r qd0puq
(Posterior).

£z
ş
¨
expp´2
2
}Ă¥2pyuq ´ z2}2
Rk¨r qd0puq
(Posterior).
§ Maximum a Posteriori Estimator:
min
uPH1p0YTq
2
2
} r¥2pyuq ´ z2}2
Rk¨m ` }Btu}M ` p}Btu}2
L2 ` }u}2
L2 q

£z
ş
¨
expp´2
2
}Ă¥2pyuq ´ z2}2
Rk¨r qd0puq
(Posterior).
min
uPH1p0YTq
2
2
} r¥2pyuq ´ z2}2
L2 ` }u}2
L2 q
§ Sampling and Empirical MAP (Splitting pCN with 100,000 sample steps):
We used ˜uptq “
8ř
–“0
Np0Y!2
– q ¨ 9–ptq „ Np0Yp´1
40q´1
q,
with MAP:
min
uPH1
0
p0YTq
2
2
} r¥2pyuq ´ z2}2
Rk¨m
`}Btu}M ` }Btu}2
L2

£z
ş
¨
expp´2
2
}Ă¥2pyuq ´ z2}2
Rk¨r qd0puq
(Posterior).
min
uPH1p0YTq
2
2
} r¥2pyuq ´ z2}2
L2 ` }u}2
L2 q
§ Sampling and Empirical MAP (Splitting pCN with 100,000 sample steps):
We used ˜uptq “
8ř
–“0
Np0Y!2
– q ¨ 9–ptq „ Np0Yp´1
40q´1
q,
with MAP:
min
uPH1
0
p0YTq
2
2
} r¥2pyuq ´ z2}2
Rk¨m
`}Btu}M ` }Btu}2
L2
§ Disadvantages (true function is sparse): Computationally expensive (KLE),
high number of samples needed (lack of sparsity), good initial value needed.

BV-Prior
§ We now assume that u „
kř
–“0
–1pt`YTsptq ` c („ 0 BV-Prior).

BV-Prior
kř
–“0
§ The set of all realizations of u is dense in BVp0YTq, in the strict BV-topo.

BV-Prior
kř
–“0
§
Sampling + Empirical MAP +
Optimal Control pP¥q
(SMC 5,000 samples):

BV-Prior
kř
–“0
§
Sampling + Empirical MAP +
Optimal Control pP¥q
(SMC 5,000 samples):
§ Advantage: Faster algorithms by sparsity.

Gaussian Measure
[Bog98, Theorem 3.6.1]
Let be a Radon Gaussian measure on a locally convex space X. Then the
topological support of coincides with the ane subspace Epq ` H
X
, where
H
X
stands for the closure in X.

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

Optimal Control for Linear Second-Order Hyperbolic Equations by BV-Functions in Time, KFU Graz 2019

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