optimal placement of sensors, actuators and dampers for...

Optimal placement of sensors, actuators and dampersfor waves

Enrique ZuazuaJoint work with Y. Privat and E. Trelat (CNRS/LJLL-Paris VI/IUF)

IKERBASQUE & BCAMBilbao, Basque Country, Spain

[email protected]://www.bcamath.org/zuazua/

Supported by ERC-Advanced Grant NUMERIWAVES & Mairie de Paris “Recherche a Paris”

LJLL/UMPC, February 2014

Enrique Zuazua (IKERBASQUE & BCAM) Waves, Control and Design LJLL/UMPC, February 2014 1 / 65

http://www.bcamath.org/zuazua/

Motivation

Table of Contents

1 Motivation

2 The control of wavesAn example: noise reductionA toy model

3 Pointwise observations

4 Damped wavesGeometric conditions for exponential decayComplexity of the decay rate

5 Optimal placement of sensors and actuatorsProblem formulationThe spectral problemMain results for the spectral criterium

6 The heat equation

7 Conclusions and Perspectives


Motivation

What is the best shape and placement ofsensors?

- Reduce the cost of instruments.- Maximize the efficiency of reconstructionand estimations.


Motivation

The mathematical theory needed to understand these issues combines:

Hyperbolic PDEs (possibly on Networks)

Control Theory

Optimal Design

Optimization

Spectral analysis

Microlocal analysis

Numerical analysis

. . .

In this talk we aim to present some toy models and problems, togetherwith some key results and research perspectives.


The control of waves

Table of Contents

1 Motivation





6 The heat equation



The control of waves An example: noise reduction

Outline

1 Motivation





6 The heat equation



The control of waves An example: noise reduction

Acoustic noise reductionActive versus passive controllers.


The control of waves A toy model

Outline

1 Motivation





6 The heat equation




Control of 1− d vibrations of a string

The 1-d wave equation, with Dirichlet boundary conditions, describing thevibrations of a flexible string, with control on one end:

ytt − yxx = 0, 0 < x < 1, 0 < t < Ty(0, t) = 0; y(1, t) =v(t), 0 < t < Ty(x , 0) = y 0(x), yt(x , 0) = y 1(x), 0 < x < 1

y = y(x , t) is the state and v = v(t) is the control.The goal is to stop the vibrations, i.e. to drive the solution to equilibriumin a given time T : Given initial data y 0(x), y 1(x) to find a controlv = v(t) such that

y(x ,T ) = yt(x ,T ) = 0, 0 < x < 1.



The dual observation problem

The control problem above is equivalent to the following one, on theadjoint wave equation:

ϕtt − ϕxx = 0, 0 < x < 1, 0 < t < Tϕ(0, t) = ϕ(1, t) = 0, 0 < t < Tϕ(x , 0) = ϕ0(x), ϕt(x , 0) = ϕ1(x), 0 < x < 1.

The energy of solutions is conserved in time, i.e.

E (t) =1

2

∫ 1

0

[|ϕx(x , t)|2 + |ϕt(x , t)|2

]dx = E (0), ∀0 ≤ t ≤ T .

The question is then reduced to analyze whether the folllowing inequalityis true. This is the so called observability inequality:

E (0) ≤ C (T )

∫ T

0|ϕx(1, t)|2 dt.



The answer to this question is easy to gues: The observability inequalityholds if and only if T ≥ 2.

E(0) ≤ T

0|ϕx(1, t)|2dt

Wave localized at t = 0 near the extreme x = 1 propagating with velocityone to the left, bounces on the boundary point x = 0 and reaches thepoint of observation x = 1 in a time of the order of 2.



Construction of the Control

Following J.L. Lions’ HUM (Hilbert Uniqueness Method)1 , the control is

v(t) = ϕx(1, t),

where ϕ is the solution of the adjoint system corresponding to initial data(ϕ0, ϕ1) ∈ H1

0 (0, 1)× L2(0, 1) minimizing the functional

J(ϕ0, ϕ1) =1

2

∫ T

0|ϕx(1, t)|2dt+

∫ 1

0y 0ϕ1dx− < y 1, ϕ0 >H−1×H1

0,

in the space H10 (0, 1)× L2(0, 1). J is convex, continuous (because of the

fact that ϕx(1, t) ∈ L2(0,T ) (hidden regularity)). Moreover,

COERCIVITY OF J = OBSERVABILITY INEQUALITY. 2

1J. -L. Lions, Controlabilite Exacte, Stabilisation et Perturbations deSystemes Distribues. Tome 1. Controlabilite Exacte, Masson, Paris, RMA 8,1988.

2Norbert Wiener (1894–1964) defined Cybernetics as the science of controland communication in animals and machines


Pointwise observations

Table of Contents

1 Motivation





6 The heat equation




Interior pointwise measurements

Take x0 ∈ (0, 1). How much energy we can recover from measurementsdone on x0?

ϕ(x0, t) =∑

k∈Z

ake ikπtsin(kπx0).

Furthermore, if T > 2,

∫ T

0

∣∣∣∑

ake ikπtsin(kπx0)∣∣∣2dt ∼

∑sin2(kπx0)|ak |2.

Obviously, two cases:

The case: x0 ∈ Q, some of the weights sin2(kπx0) vanish an thequadratic term is not a norm.

The case x0 /∈ Q: sin2(kπx0) 6= 0 for all k and the quantity underconsideration is a norm, i.e. it provides information on all the Fouriercomponents of the solutions.

But, even if, sin2(kπx0) 6= 0 for all k, the norm under consideration is notthe L2-one we expect!!!!



Can we explain this in terms of rays, and the propagation of waves (andantiwaves)?

If x0 is rational we can build a finite number of rays and anti-rays thatalways intersect in x0 for the time interval (0, 2) of periodicity of solutions.



The case x0 irrational.

Note that the following is impossible!!!∣∣∣ sin(kπx0)

∣∣∣ ≥ α > 0, ∀k?

Indeed, this would mean that∣∣∣kπx0 −mπ| ≥ β

for all k ,m ∈ Z. And this is obviously false.For suitable irrational numbers x0 we can get the optimal lower bound

∣∣∣kπx0 −mπ| ≥ β/k.

In this case we get an observation inequality but with a loss of onederivative.But for some other irrational numbers (Liouville ones, for instance) thedegeneracy may be arbitrary fast. 3

3K. Ammari, A. Henrot and M.Tucsnak,Optimal location of the actuator forthe pointwise stabilization of a string, C. R. Acad. Sci. Paris, t. 330, Serie I, p.275–280, 2000.



1− d conclusion

D’Alembert ∼ Fourier

Ray propagation ∼ Spectrum


Damped waves

Table of Contents

1 Motivation





6 The heat equation



Damped waves Geometric conditions for exponential decay

Outline

1 Motivation





6 The heat equation




Internal stabilization of waves:

Let ω be an open subset of Ω. Consider:

ytt −∆y =−yt1ω in Q = Ω× (0,∞)y = 0 on Σ = Γ× (0,∞)y(x , 0) = y 0(x), yt(x , 0) = y 1(x) in Ω,

where 1ω stands for the characteristic function of the subset ω.The energy dissipation law is then

dE (t)

dt= −

∫

ω|yt |2dx .

Question: Do they exist C > 0 and γ > 0 such that

E (t) ≤ Ce−γtE (0), ∀t ≥ 0,

for all solution of the dissipative system?



This is equivalent to an observability property4: There exists C > 0 andT > 0 such that

E (0) ≤ C

∫ T

0

∫

ω|yt |2dxdt.

This estimate, together with the energy dissipation law, shows that

E (T ) ≤ σE (0)

with 0 < σ < 1. Accordingly the semigroup map S(T ) is a strictcontraction. By the semigroup property one deduces immediately theexponential decay rate.

4A. Haraux, Une remarque sur la stabilisation de certains systemes dudeuxieme ordre en temps. Portugaliae mathematica, 46 (3), 1989, 245-258.



The observability inequality and, accordingly, the exponential decayproperty holds if and only if the support of the dissipativemechanism, Γ0 or ω, satisfies the so called the Geometric ControlCondition (GCC) (Ralston, Rauch-Taylor, Bardos-Lebeau-Rauch5 ,...)

Rays propagating inside the domain Ω following straight lines that arereflected on the boundary according to the laws of Geometric Optics. Thecontrol region is the red subset of the boundary. The GCC is satisfied inthis case. The proof requires tools from Microlocal Analysis.

5C. Bardos, G. Lebeau, and J. Rauch, “Sharp sufficient conditions for theobservation, control and stabilization of waves from the boundary”, SIAM J.Cont. Optim., 30 (1992), 1024–1065.



Qualitative change from 1− d to multi-d

Trapped ray

Trapped

ray

ω

Ω

A trapped ray scaping the damping region ω makes it impossible the decayrate to be exponential. Each trajectory tends to zero as t →∞ but the

decay can be arbitrarily slow.


Damped waves Complexity of the decay rate

Outline

1 Motivation





6 The heat equation




The optimal shape design problem

Optimize ω (or Γ0) and the damping profile a = a(x) to enhance theexponential decay rate:

ytt −∆y =−a(x)yt1ω in Q = Ω× (0,∞)y = 0 on Σ = Γ× (0,∞)y(x , 0) = y 0(x), yt(x , 0) = y 1(x) in Ω.

Within the class of ω and a > 0 such the exponential decay property holds:

E (t) ≤ Ce−γa,ωtE (0), ∀t ≥ 0

MAXIMIZE : (ω, a)→ γa,ω.



Overdamping!

Obviously, the decay rate γa depends on the damping potential a.But, against the very first intuition, this map is not monotonic withrespect to the size of the damping. A 1− d spectral computation forconstant coefficients yields:



Some known results: 1− d

1− d : The exponential decay rate coincides with the spectral abscissawithin the class of BV damping potentials. For large eigenvaluesRe(λ) ∼ −

∫ω a(x)dx/2. 6 Thus:

γa ≤∫

ωa(x)dx .

k k/2 0

0

400

Re( )

Imag

()

Real (blue) and imaginary (red) eigenvalues of the damped wave equation utt uxx+kut=0, x in (0,1), k=100*

6S. Cox & E. Z., The rate at which the energy decays in a damped string.Comm. P.D.E. 19 (1&2). 213–243. 1994.



The singular potential

a(x) =2

x

produces an arbitrarily fast decay rate. 7

Connections with:

Transparent boundary conditions.

Perfectly matching layers.

7C. Castro and S. Cox, Achieving arbitrarily large decay in the damped waveequation. SIAM J. Cont. Optim., 39 (6), 1748–1755, 2001.



Multi−d

In the multidimensional case the situation is even more complex. Thedecay rate is determined as the minimum of two quantities8:

The spectral abscissa;

The minimum asymptotic average (as T →∞) of the dampingpotential along rays of Geometric Optics.The later is in agreement with our intuition of waves traveling alongrays of Geometric Optics.

8G. Lebeau, Equation des nodes amorties, Algebraic and geometric methodsin mathematical physics (Kaciveli), 1993, 73-109, Math. Phys. Stud., 19,Kluwer Acad. Publ., Dordrecht, 1996.



Geometric configuration in which the spectral abscissa does not suffice tocapture the decay rate. The decay rate vanishes due to a trapped ray, butthe spectrum is uniformly shifted in the left complex half space.



Truth = Spectrum + Rays

Truth = Fourier ∪ D’Alembert



Moving ray control: “Oceans Twelve”, 2014, Laser dance.


Optimal placement of sensors and actuators

Table of Contents

1 Motivation





6 The heat equation



Optimal placement of sensors and actuators Problem formulation

Outline

1 Motivation





6 The heat equation




Consider the conservative wave equation:

ztt −∆z = 0 in Q = Ω× (0,T )z = 0 for x ∈ ∂Ω; t ∈ (0,T )z(x , 0) = z0(x), zt(x , 0) = z1(x) in (0, π).

Optimal placement problems are then of variational nature!

It corresponds to the analysis of the behavior of the damped systeminfinitesimally small damping.

Observability:

||z0||2L2(Ω) + ||z1||2H−1(Ω) ≤ C (ω,T )

∫ T

0

∫

ωz2dxdt.

Inspired in previous works by, among others: P. Hebrard & A. Henrot andA. Munch, P. Pedregal & F. Periago.



Fourier series shows that, in general, Fourier modes are mixed in quite ancomplicated manner thus making the understanding of these issuescomplex:

z(t) =∑

zke i√λk tφk(x).

Thus,

∫ T

0

∫

ω|z |2dxdt =

∑∑zk zj

∫

ωφk(x)φj(x)dx

∫ T

0e [i√λk−i√λj ]t .



Earlier numerical experiments

Optimal design of the billiard: Hebrard-Humbert, 20039

9P. Hebrard and E. Humbert. The geometrical quantity in damped waveequations on a square. ESAIM:COCV, 12(4), 2006.



Optimal location of the sensor for the first eigenfunction by a level setapproach: A. Munch, 2005.10

10A. Munch, P. Pedregal and F. Periago. A variational approach to a shapedesign problem for the wave equation. CRAS, Paris, 343 (5) (2006)371-376.



0 0.5 1 1.5 2 2.5 3 3.5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Simulations performed using AMPL + IPOPT



Main results with fixed initial data

For initial data that are analytic (exponential decay of Fouriercoefficients), there is a unique minimizer with a finite number ofconnected components. 11

The optimal set always exists but it can be a Cantor set even forC∞ smooth data.

α1 α2 α3 α4 π0

ϕ∗γ ,T(Lπ) ϕ∗

γ ,T(Lπ)

Lπ π0

11Szolem Mandelbrojt, Sur un probleme concernant les series de Fourier,Boulletin de la SMF, 62 (1934), 143–150.


Optimal placement of sensors and actuators The spectral problem

Outline

1 Motivation





6 The heat equation




Reduction to a spectral problem

The problem becomes much simpler in several cases:

The case T =∞. We then look at the

limT→∞

1

T

∫ T

−T

∫

ω|z |2dxdt.

Randomizing initial data and considering the expected observabilityconstant (Zygmund lemma, recent works by N. Burq et al.12)

In 1-d, Ω = (0, π) in which case solutions are 2π-time periodic.

Cross terms vanish and we are led to the following observability problem:

∑|zk |2 ≤ C (ω)

∑|zk |2

∫

ωφ2k(x)dx .

12N. Burq, N. Tzvetkov, Random data Cauchy theory for supercritical waveequations. I. Local theory, Invent. Math. 173 (2008), no. 3, 449–475.



These issues can be considered, as mentioned above, in two differentcases:

Fixed initial data, and therefore fixed weights |zk |2 in `1.All possible initial data of finite energy. Then, the problem becomesthat of finding ω so that the following minimum is maximized:

J(ω) = infk

∫

ωφ2k(x)dx .

I = sup|ω|=L

J(ω).

Warning!

This spectral criterium, is not sufficient to fully characterize theobservability constant since a second microlocal one is also required.We are ignoring the rays!!!

Spectral criterium = Truth /2

We are ignoring the ray contribution...

Are we?Enrique Zuazua (IKERBASQUE & BCAM) Waves, Control and Design LJLL/UMPC, February 2014 50 / 65

Optimal placement of sensors and actuators Main results for the spectral criterium

Outline

1 Motivation





6 The heat equation




Spectral criterium: 1− d

Relaxation occurs (Hebrart-Henrot, 2005)13: the optimum is achievedby a density function ρ(x) so that

∫ π0 ρ(x)dx = L and not by a

measurable set with bang-bang densities (except for L = π/2). Theconstant density is optimal but is not the unique one.

Spillover occurs (1− d): The optimal design for the first N Fouriermodes is the worst choice for the N + 1-th one.

No gap! The infimum over measurable sets and over densitiescoincides. The functional is not lower semicontinuous!!!

13P. Hebrard, A. Henrot, A spillover phenomenon in the optimal location ofactuators, SIAM J. Control Optim. 44 (2005), 349–366.



The spillover phenomenon



Spectral criterium: multi−d

The truncation of the optimal design problem to the first N Fourier modesleads to an efficient approximation algorithm.

JN(ω) = infk=1,...,N

∫

ωφ2k(x)dx .

IN = sup|ω|=L

JN(ω).

JN(ωN)→ I .

Note this is true despite the lack of weak lower semicontinuity of thefunctional J and compatible with spillover!



Spectral criterium: multi−d

The constant density is optimal and the non-gap result holds under theadditional assumption that

the eigenfunctions are uniformly bounded in L∞.

Note that this applies to the case where Ω is a rectangle, for the classicalbasis of eigenfunctions in separated variables: sin(kx) sin(my).

Our original proof used an added technical assumption recently removedby Lior Silberman employing the well known convergence of Cesaro means:

1

N

N∑

j=1

φ2j 1/|Ω|, as N →∞, vaguely.

The L∞ bound allows showing that Cesaro means weakly converge in L1

which allows testing against any L∞ density.



Some references:

A. Shnirelman, Ergodic properties of eigenfunctions, Usp. Math.Nauk. 29, 181-182 (1974).

Y. Colin de Verdiere, Ergodicite et fonctions propres du Laplacien,Comm. Math. Phys. 102 (1985), 497–502.

P. Gerard, E. Leichtnam, Ergodic properties of eigenfunctions for theDirichlet problem, Duke Math. J. 71 (1993), 559–607.

Note that the existing results guarantee the converge to the uniformmeasure for a density-one subsequence of eigenfunctions (SchnirelmanThm).



But in the multi-dimensional case very rarely eigenfunctions are uniformlybounded in L∞!

Despite of this an in depth use of the structure of the eigenfunctions in thedisk allows shows that the constant density is optimal!



We put in evidence the intimate relations between optimal designproblems, the high frequency behavior of the spectrum and the (quantum)ergodicity properties of the dynamical system associated to the billiard ofthe domain.

Domain conjectured to satisfy the QUE in A.H. Barnett, Asymptotic rateof quantum ergodicity in chaotic Euclidean billiards, Comm. Pure Appl.

Math. 59 (2006), 1457–1488.



Optimal designs in 2− d for the square with volume fraction 1/2 in thespectral criterium. This is an exceptional case where classical optimal

domains exist.


The heat equation

Table of Contents

1 Motivation





6 The heat equation



The heat equation

Similar questions can be formulated for parabolic equations, and inparticular for the linear heat equation.Once more a randomized observability criterium leads to a spectralcriterium.Due to the intrinsic damping in the heat equation the high frequencycomponents are then penalized:

CT ,rand(χω) = infj∈N∗

e2λjT − 1

2λj

∫

ωφj(x)2 dx , (1)

One is then able to show that the optimal design is determined by a finitenumber of eigenfunctions, the relevant number of them diminishing as Tincreases. The proof uses recent results on the lack of concentration ofeigenfunctions on measurable sets. Namely, that14

∫

ω|φj(x)|2dx ≥ c1 exp(−c2

√λj).

uniformly over sets ω of a given measure.14J. Apraiz, L. Escauriaza, G. Wang, C. Zhang, Observability inequalities and

measurable sets, J. Europ. Math. Soc., to appear.Enrique Zuazua (IKERBASQUE & BCAM) Waves, Control and Design LJLL/UMPC, February 2014 62 / 65

The heat equation

for all j 2 IN and k 2 IN. According to [42, Theorem 2.1], there exists C > 0 such that

limj!+1

Z

Djk

r dr =

Z 1

0

r dr and

Z

Djk

Rjk(r)2r dr 6 Cj2/32j1/3

3 .

The first estimate ensures that, for a given k 2 IN and for j large enough, ! is contained in Djk.Using the inequalities j2↵ < jk = z2↵

jk < (2j)2↵ for all j 2 IN and k 2 IN, we easily infer that

limj!+1

jk(T )

Z 1

0

a(r)Rjk(r)2r dr = 0,

for every k 2 IN, which raises a contradiction. The conclusion follows.

2.6 Several numerical simulations

We provide hereafter several numerical simulations. The truncated problem of order N is obtainedby considering all couples (j, k) such that j 6 N and k 6 N . The simulations are made with aprimal-dual approach combined with an interior point line search filter method6.

On Figure 1 (resp., on Figure 2), we compute the optimal domain !N for the operator A0 = 4,the Dirichlet-Laplacian (resp., the Neumann-Laplacian on the domain defined with zero average)on the square = (0,)2. We can observe the expected stationarity property of the sequence ofoptimal domains !N from N = 4 on (i.e., 16 eigenmodes).

Figure 1: On this figure, = (0,)2, L = 0.2, T = 0.05, and A0 is the Dirichlet Laplacian. Row1, from left to right: optimal domain !N (in green) for N = 1, 2, 3. Row 2, from left to right:optimal domain !N (in green) for N = 4, 5, 6.

6More precisely, we used the optimization routine IPOPT (see [59]) combined with the modeling language AMPL(see [15]) on a standard desktop machine.

26

1515G. Allaire, A. Munch, F. Periago, Long time behavior of a two-phase

optimal design for the heat equation, SIAM J. Control Optim. 48 (2010), no.8, 5333–5356.


Conclusions and Perspectives

Table of Contents

1 Motivation





6 The heat equation



Conclusions and Perspectives

Conclusions

There is a rich (and difficult!) field to be explored at the intersectionof the following topics:

NetworksDamped wave equationsControl theoryOptimal design and OptimizationSpectral theoryErgodicity of billiardsMicrolocal analysisFine numerics

At the continuous level there is a huge need of better spectralunderstanding.

Also of gaining comprehension of complex dynamics when spectralanalysis does not suffice: What about the optimal placement ofsensors for the original time-evolution problem?

Lots to be done on the numerics of these problems.


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