OBSERVERS FOR COMPRESSIBLE NAVIER-STOKES EQUATION

AMIT APTE∗, DIDIER AUROUX† , AND MYTHILY RAMASWAMY‡§

Abstract. We consider a one dimensional model of a compressible fluid in a bounded interval.We want to estimate the density and velocity of the fluid based on the observations for only velocity.We build an observer exploring the symmetries of the fluid dynamics laws. Our main result is thatfor the linearised system with full observations of the velocity field, we can find an observer whichconverges to the true state of the system at any desired convergence rate for finitely many butarbitrarily large number of Fourier modes. Our numerical results corroborate the results for thelinearised, fully observed system, and also show similar convergence for the full nonlinear system andalso for the case when the velocity field is observed only over a subdomain.

Key words. Data assimilation; Observer; Navier-Stokes equation;

AMS subject classifications. 93C20; 93C95; 93B40

1. Introduction. Data assimilation is the problem of estimating the state ofa dynamical system described by an evolution equation, typically partial differentialequations (PDE), using noisy and partial observations of that system. This has beenwidely studied in the geophysical context, e.g. meteorology, oceanography, fluid flows,etc.[5, 17, 22]. One of the main tools for this state estimation problem is the analysisof observability, but both mathematical and applied aspects of observability havereceived only a little attention.[25, 23, 19] An even less studied aspect has been theconstruction of appropriate observers in the context of data assimilation.

Observers are essentially a modification of the original evolution equation forthe system, to incorporate the observations in a feedback term, with the aim thatthe solution of the observer converges to the solution of the original system beingobserved. Observers for finite dimensional systems have been well studied in theliterature, see, for example [30, 33]. But in many applications such as earth sciencesor engineering, the systems are modeled using PDE that are highly nonlinear and,in many instances, chaotic. In such cases of infinite dimensional systems governedby PDE, there are only a few examples available in the literature, mostly for linearsystems and very few for nonlinear systems.[20, 29, 32, 11, 13, 14, 16, 31, 24, 6, 28]

Some of the commonly used observers are Kalman Filters or Luenberger observersbut the main drawback of these is that they often may break intrinsic properties of themodel, e.g. symmetries and/or physical constraints such as balances in geophysicalmodels, see for example [12] and references therein. For nonlinear system possessingcertain symmetries, it is natural to seek a correction term which also preserves thosesymmetries. Such an invariance may make the correction term nonlocal but it mayhave other desirable properties, for example, independence from the change of coor-dinates. Recently, there have been attempts to construct, for a variety of systems,observers based on considerations of symmetry.[9, 10, 3]. The observers we constructare motivated by these recent works, as we will see in section 2. Another motivation

∗International Centre for Theoretical Sciences - TIFR, IISc Campus, Bangalore-560012, India.(apte@icts.res.in)†Laboratoire J.A. Dieudonne, UMR 7351, Universite Nice Sophia Antipolis, Parc Valrose, 06108

Nice cedex 2, France. (auroux@unice.fr)‡T.I.F.R Centre for Applicable Mathematics, Post Bag No. 6503, GKVK Post Office, Bangalore-

560065, India. (mythily@math.tifrbng.res.in)§The authors would like to thank the Indo-French Centre for Applied Mathematics (IFCAM) for

financial support under the “Observers and data assimilation” project.

1

2

for observer design is the computational cost: a full Kalman filter is usually too ex-pensive for real applications, due to the size of the gain matrices, while an observermight be much more affordable, without degrading the identification process.

The main aim of this work is to develop an observer for a class of PDE inspiredby earth system applications, all of which use some approximations of Navier-Stokesequations for fluid flow. In particular, we propose an observer for the nonlinear PDE[equations (2.1)] describing the evolution of a compressible, adiabatic fluid whosevelocity field is observed either fully or partially. Our main theoretical result (Propo-sition 1), supported by extensive numerical results (section 4), is that in the case ofcomplete observations of the velocity, an appropriate choice of parameters leads toconvergence of the observer to the true solution at any prescribed rate for arbitrarilylarge but finite number of Fourier modes.

There has been some work on developing observers for Navier-Stokes based sys-tems. The papers [31, 16, 27] work with finite dimensional approximations of the PDEinvolved whereas we work with the full PDE itself. A completely different approachbased on developing observers using appropriate continuous time limits of discretetime 3D-var or Kalman filter is developed in a series of papers [18, 21, 8] and alsoin [7, 2, 1, 26, 15, 4]. One of the main contributions of this paper is that we alsoderive the decay rates for the convergence of the observers and indeed find observersthat can decay arbitrarily fast. Since Navier-Stokes based PDE are commonly usedin practical data assimilation problems in earth sciences, our work has the potentialto be directly relevant to these applications, as we discuss in section 5.

The outline of the paper is as follows. The model and the observer are bothintroduced in the next section 2. In that section, we also state the linearized version ofthis problem. We analyze the convergence of this proposed observer for the linearizedPDE in section 3 and prove the main result (Proposition 1). We also briefly discussthe difficulties that arise in the theoretical analysis of the nonlinear equation or of thecases with either partial observations or unknown forcing for the linearized equations.In section 4, we present numerical results to substantiate the linear theory. We alsopresent the numerical results showing the efficacy of the observer in estimating thetrue solution for the partially observed linear system, the observer in the case ofunknown forcing term, and the fully nonlinear case. The last section 5 discusses somefuture directions of research.

2. Compressible Navier-Stokes equations and observers. In this paper,we study a model of compressible fluid in a bounded interval. The density ρ(t, x)and velocity v(t, x) of the fluid form the state vector of this model and they obey thefollowing compressible Navier-Stokes system :

ρt + (ρu)x = 0 , (ρu)t + (p+ ρu2)x = νuxx ,(2.1)

where the pressure p is given by the adiabatic equation of state:

(2.2) p = p(ρ) = ργ ,

with γ the adiabatic exponent taken to be 1.4 throughout this paper. We considersolutions over a finite time [0, T ] for the space domain [0, 1] with periodic boundaryconditions:

(2.3) ρ(t, x) = ρ(t, x+ 1) , u(t, x) = u(t, x+ 1) , ∀ t ∈ [0, T ] , and ∀ x ∈ [0, 1] ,

and similar periodic conditions for all derivatives of ρ and u.

3

We assume that the initial conditions ρ(0, x) = ρI(x) and u(0, x) = uI(x) areunknown, but we have some information on the solution u(t, x) of equations (2.1).The goal is to build an observer (ρ, u) for this system, in such a way that boththe density and the velocity of the observer (ρ, u) converges towards the solution(ρ, u). We will assume that we have observations of the velocity denoted by u(t, x)for t ∈ [0, T ] and x ∈ Ω ⊂ [0, 1] where Ω is a subset of [0, 1]. We will only considerthe case Ω = [0, L] with L ≤ 1.

2.1. Observers. We introduce the observer (ρ, u), based on the symmetries forthe system, satisfying the following set of equations:

ρt + (ρu)x = Fρ(ρ, u, u) ,

(ρu)t + (p+ ρu2)x = νuxx + Fu(ρ, u, u) , 0 < t < T , x ∈ [0, 1] .(2.4)

As there are no observations of the density, we first assume that the feedback termsdo not depend on ρ. Additionally, since these terms should be equal to 0 when u andu coincide, it is reasonable to consider the following classes of functions:

Fρ(ρ, u, u) = ϕρ ∗Dρ(u− u) , Fu(ρ, u, u) = ϕu ∗Du(u− u),(2.5)

where Dρ and Du are differential (or integral) operators (e.g. ∂x, ∂t, . . . ), and ϕρand ϕu are convolution kernels. We assume that these kernels are time independentand hence only functions of x, and we assume that they are isotropic (for 2D or 3Dapplications). With this choice, the observer preserves the symmetries of the system,since the correction terms are based on a convolution product with an isotropic kernel,making it invariant by rotation or translation.

Note that for the case of partial observation, i.e., when Ω 6= [0, 1], the feedbackterms are present only in Ω so that we essentially write the feedback term as:

Fρ(ρ, u, u) = ϕρ ∗Dρ [1Ω(u− u)] , Fu(ρ, u, u) = ϕu ∗Du [1Ω(u− u)] ,(2.6)

Probably, the most simple observer is defined by Luenberger observer (or asymp-totic observer): as only u is observed, then the feedback term is added only in thevelocity equation (Fρ = 0), and the feedback term is simply Fu(ρ, u, u) = ku(u− u),where ku > 0 is a constant. We will study more general observers, with the aim atcorrecting also the density equation with the velocity.

We would like to examine the convergence of the observer solution (ρ, u) to thesolution of the observed system (ρ, u) asymptotically in time. In particular, we areinterested in the rate of convergence of ‖ρ− ρ‖ and of ‖u−u‖ towards zero where wewill only consider the L2 norm in this paper.

In order to study the theoretical behavior of the observer, we first linearize thesystem around a steady state of the nonlinear system (2.1) and study the convergenceof the observer for the linearized system. We postpone the presentation of numericalresults for the nonlinear case to section 4.5.

2.2. Stationary solutions. In order to linearize equations (2.1), let us firstlook for the stationary solutions: ρt = 0 and ut = 0. We write the stationarysolution as ρ(t, x) = ρ0(x) and u(t, x) = u0(x). Then the first of equations (2.1) givesρ0(x) = C/u0(x) for a constant C determined by the initial conditions.

The second of equations (2.1) now reads:

(Cγu−γ0 + Cu0)x = ν0u0xx ⇔ −γCγu−γ−10 u0x + Cu0x = ν0u0xx.

4

Let us multiply this equation by u0 and integrate over the space domain [0; 1]:

−γCγ∫ 1

0

u−γ0 u0x dx+ C

∫ 1

0

u0u0x dx = ν

∫ 1

0

u0xxu0 dx.

The two first integrals are both equal to zero, by considering an integration by part,as the boundary conditions are periodic. Then

0 = −ν∫ 1

0

(u0x)2 dx,

meaning that u0x ≡ 0, and hence the only stationary solutions of 1D Navier Stokesequation are constants:

(2.7) ρ(t, x) = ρ0, u(t, x) = u0.

2.3. Linearization around an equilibrium state. We now consider an equi-librium state (ρ0, u0), and we linearize equations (2.1) around this state:

ρt + u0ρx + ρ0ux = 0,(2.8)

ρ0ut + u0ρt + γργ−10 ρx + u2

0ρx + 2ρ0u0ux = νuxx ,(2.9)

where only terms linear in (ρ, u) appear. Using (2.8), multiplied by u0, in (2.9), weget:

ρt + u0ρx + ρ0ux = 0 , ρ0ut + γργ−10 ρx + ρ0u0ux = νuxx .(2.10)

We can rewrite the solution along the characteristics of the equation. The transportcoefficient for both density and velocity is u0, and we define

(2.11) σ(t, x) = ρ(t, x+ u0t) and v(t, x) = u(t, x+ u0t).

Then σx(t, x) = ρx(t, x + u0t) and σt(t, x) = ρt(t, x + u0t) + u0ρx(t, x + u0t), andsimilarly for derivatives of v(t, x). Writing equations (2.10) at point (t, x + u0t), weget the following set of equations:

σt + ρ0vx = 0 , ρ0vt + γργ−10 σx = νvxx ,

which is nothing else than equations (2.10) with u0 = 0. So we can now assume thatu0 = 0 without any loss of generality, and for simplicity reasons, we set ρ0 = 1. Thusthe linearized Navier-Stokes system that we will work with is:

ρt + ux = 0 , ut + γρx = νuxx ,(2.12)

with initial conditions ρ(0, x) = ρI(x) and u(0, x) = uI(x), and periodic boundaryconditions as in equation (2.3).

As stated above, we assume that the initial conditions ρI(x) and uI(x) are un-known, but we have some information on the solution u(t, x) of equations (2.12). Thegoal is to build an observer (ρ, u) for this system, in such a way that the observer(ρ, u) converges towards the solution (ρ, u). We will use the same observer as intro-duced in equations (2.4)-(2.6), except that the left hand side is now linear, just as inequations (2.12):

ρt + ux = ϕρ ∗Dρ [1Ω(u− u)] , ut + γρx = νuxx + ϕu ∗Du [1Ω(u− u)] ,(2.13)

5

with periodic boundary conditions, and initial conditions different from the true initialconditions: ρ(0, x) = ρI(x) 6= ρI(x) and u(0, x) = uI(x) 6= uI(x)

We will now present the main theoretical result about this observer in the linearcase, stating that we can choose the feedback terms in order to guarantee any specifiedrate of convergence for arbitrarily large but finitely many number of Fourier modes.Since we are dealing with linear PDE, we will use Fourier series representation as ourmain tool.

3. Theoretical study of an observer for 1D linear Navier-Stokes system.For the case of velocity observations over the full domain x ∈ [0, 1], subtractingequations (2.13) from (2.12) (with Ω = [0, 1]), we get the following equations for theerrors r = ρ− ρ and w = u− u:

rt + wx = −ϕρ ∗Dρw , wt + γρx = νwxx − ϕu ∗Duw ,(3.1)

which is exactly identical to equations (2.13), with (r, w) replacing (ρ, u) and withu = 0. Hence in the rest of this section, we simply work with equations (2.13) withu = 0, with the understanding that the results really apply to the errors (r, w).

3.1. Damped wave equation formulation. We now eliminate the density inthe velocity equation, in order to get an equation for the velocity alone. Starting from(2.13), where feedbacks are given by (2.6), we take the space derivative of the densityequation, and the time derivative of the velocity equations:

ρtx + uxx = −ϕρ ∗Dρux,utt + γρtx = νutxx − ϕu ∗Duut .

We replace ρtx in the second equation by its expression given by the first equation,and we obtain:

(3.2) utt − γuxx = νutxx + ϕu ∗Duut − γϕρ ∗Dρux .

Equation (3.2) is a damped wave equation, with two forcing terms coming fromthe observer feedbacks. In this case, the goal is to make u converge towards 0. Notethat if there is a known forcing term in the linear system, it will also be added in theobserver equation, and then, by linearity, it will also disappear.

We now assume that the following differential operators are used:

(3.3) Dρ(f) = ∂xf, Du(f) = f,

which means that we want the velocity equation to be controlled by the velocity, andthe density equation by the space derivative of the velocity. Equation (3.2) now reads:

(3.4) utt − γuxx = νutxx − ϕu ∗ ut + γϕρ ∗ uxx.

Remark 1. If we consider the symmetric case, in which only the density ρ isobserved, and not the velocity u, then we obtain the following system:

ρt + ux = −ψρ ∗ Dρρ,ut + γρx = νuxx − ψu ∗ Duρ.

We now want to eliminate u in order to obtain an equation for the density alone.From the first equation, we can extract ux = −ρt−ψρ ∗ Dρρ and then we need to takethe space derivative of the second equation:

utx + γρxx = νuxxx − ψu ∗ Duρx.

6

We get then

−ρtt − ψρ ∗ Dρρt + γρxx = −νρtxx − νψρ ∗ Dρρxx − ψu ∗ Duρx,

or equivalently,

(3.5) ρtt − γρxx = νρtxx − ψρ ∗ Dρρt + νψρ ∗ Dρρxx + ψu ∗ Duρx .

By choosing Dρ equal to the identity operator, and Du = ∂x, we can then see that weobtain the same system as in (3.4) with ϕu = ψρ, and ϕρ = (νψρ + ψu)/γ. Then allfollowing results also hold in this case.

3.2. Fourier transform. As we are on a periodic space domain [0; 1], we canconsider the following Fourier decomposition of the velocity:

(3.6) u(t, x) =∑k 6=0

ak(t)ei2kπx,

where ak(t) are the time dependent Fourier coefficients.Note that if the mean value m of uI(x) is not equal to 0, then the solution

converges towards u(t, x) + m instead of u(t, x), as the equation defines the solutionup to a constant. Thus, we assume here that the mean value of u(x) is 0 which isequivalent to assuming that the mean values of the initial conditions of the observeru and the solution u are the same. This does not change the following computations,except for the k = 0 Fourier mode. We denote by ϕρk and ϕuk the Fourier coefficientsof the (time independent) functions ϕρ(x) and ϕu(x) respectively.

Substituting (3.6) in (3.2), we obtain the following equation for the kth mode:

(3.7) a′′k(t) + (ϕuk + 4νk2π2) a′k(t) + 4k2π2γ(1 + ϕρk) ak(t) = 0.

3.3. Spectral analysis. Equation (3.7) is a second order differential equationwith constant coefficients. The corresponding discriminant is

(3.8) ∆ = (ϕuk + 4νk2π2)2 − 16k2π2γ(1 + ϕρk),

and the two eigenvalues are

(3.9) λ± = −ϕuk + 4νk2π2

2±√

∆

2,

with the notation√

∆ = i√−∆ if the discriminant is negative.

Then the Fourier coefficient of u is given by:

if ∆ ≥ 0, ak(t) = Ck1 eλ+t + Ck2 e

λ−t;(3.10)

if ∆ < 0, ak(t) = Ck1 exp

(−ϕuk + 4νk2π2

2t

)cos

(√−∆

2t+ Ck2

),(3.11)

where Ck1 and Ck2 are constants given by the initial conditions.If the discriminant is positive, then the minimal decay rate of the solution is given

by the largest eigenvalue λ+. If the discriminant is negative, it is given by the realpart of the eigenvalues <(λ±). The imaginary part gives some information about theoscillations in time of the solution. From the equations (2.13), we can also see thatthe decay rate for ρ is the same as that for u.

7

3.4. Main result. We now give the main result in this framework: for anyspecified decay rate, we can find convolution kernels ϕρ and ϕu such that the observeru converges towards u at this specified rate up to any Fourier mode K. Indeed, forany mode k ≥ 1, we can choose appropriate Fourier coefficients of these kernels, butas we will see, their expression does not ensure the convergence of the Fourier series.

Proposition 3.1. For any d > 0, for any K > 0, one can find (ϕρk)k≥1 and(ϕuk)k≥1 such that the decay rate of the solution u(t, x) of equations (2.13) towardsu = 0 is at least d for any Fourier mode k ≤ K.

Proof. Let d > 0, K > 0, and let 1 ≤ k ≤ K. In order to ensure the positivity ofall Fourier coefficients of the convolution kernels, from (3.9), we set

(3.12) ϕuk = max(0; 2d− 4νk2π2).

For small modes, the diffusion process is not large enough to ensure the decay rate,so we need to add the feedback term. For larger modes, diffusion will be enough, andthere is no need to add the feedback (but one can still add a feedback term, and thedecay rate will become larger than the specified rate for such modes). Note that evenif we drop the max in (3.12), it still ensures that the decay rate will be at least d, but

ϕuk will become negative for k >√

2d4νπ2 .

Then, from (3.8), we set

(3.13) ϕρk = max

(0;

(ϕuk + 4νk2π2)2

16γk2π2− 1

).

We still consider the positive part, as for small modes and small values of ϕuk, thisexpression can lead to negative values. The positive part is also theoretically optional,as even for negative ϕρk coefficients, the following results hold.

The above choice of ϕρk ensures that ∆ ≤ 0, and the decay rate is then exactlyϕuk + 4νk2π2

2, which is always larger than (or equal to) d, because of the choice of

ϕuk in equation (3.12).The above choice from equations (3.12)-(3.13) can be used for any mode k ≤ K

which proves the proposition.Note that we could use the same definitions from equation (3.13) for k > K, but

then the Fourier series does not converge, as ϕρk = O(k2). So we need to truncatethe series, and set ϕρk = 0 for k > K.

For large modes, namely k >√

2d4νπ2 , ϕuk = 0, and then there is no issue in

considering the inverse Fourier transform and define a convolution kernel ϕu(x) withthese coefficients.

3.5. Remarks and comparison with the nudging feedback. For largemodes, both ϕρk and ϕuk are set equal to 0. In such modes, the observer is simplya solution of the damped wave equation without any forcing, and the correspondingdecay rate λ+ of equation (3.7) is equal to γ/ν asymptotically for k →∞.

We now compare the result presented in the previous section with what can bedone with the standard nudging observer, i.e. when ϕρ ≡ 0. Let us fix k and definea = 4νk2π2, b = 4kπ

√γ, and x = ϕuk.

• If b− a ≥ 0:– If x ≥ b − a, then the discriminant ∆ is positive, and then the max-

imal decay rate is −λ+ = −1

2

[−(x+ a) +

√(x+ a)2 − b2

]. This is a

8

decreasing function of x for positive x. So the optimal decay rate isobtained for x = b− a, and the optimal rate is then b

2 = 2kπ√γ.

– If x < b−a, then the discriminant is negative, and the decay rate is givenby −<(λ±) = x+a

2 . So the optimal decay rate is obtained for x = b− a,and the optimal rate is the same as in the previous case.

• If b − a < 0, then as x ≥ 0, the optimal rate is reached for x = 0, and it is

given by a−√a2−b22 .

So if k ≤√γ

νπ , then b − a ≥ 0 and the optimal rate is 2kπ√γ. If k >

√γ

νπ , the

optimal rate is 2νk2π2

(1−

√1− γ

ν2k2π2

).

The optimal rate increases then with k when k ≤√γ

νπ , and then decreases for

larger values of k. The maximum is reached when k =√γ

νπ , and it is 2γν . At the limit

when k goes to infinity, the optimal rate goes to γν , so there is no more effect of the

feedback, only the diffusion has some impact. In some sense, simple nudging feedbackcan only increase the decay rate of the very first modes, with an upper bound of2kπ√γ, and then, for increasing modes, it can only double the decay rate of diffusion,

and even, its action disappears as the mode goes to infinity. This shows that theobserver presented in the previous subsection is much better than simple nudgingterm, as the decay rate can be increased to any arbitrary value.

3.6. Unknown forcing term. We now assume that the original equation hasa forcing term in the velocity equation:

ρt + ux = 0 , ut + γρx = νuxx − f(x, t) .(3.14)

If this forcing term f(t, x) is known, then we also add it to the velocity equation of theobserver (2.13). Then, by considering the difference between the observer and originalequations, the forcing term disappears and we are still considering equation (3.4) forthe error.

So we now assume that the forcing term is unknown. In this case, we cannot addit inside the observer equation. So the observer equation remains unchanged, andthen, the difference between the reference velocity and the observer velocity satisfiesequation (3.4) with a forcing term:

(3.15) utt − γuxx = νutxx − ϕu ∗ ut + γϕρ ∗ uxx + ft.

We now adapt the spectral analysis. We assume that the mean of f is 0 (no biasin the forcing), and then:

f(t, x) =∑k 6=0

bk(t)ei2kπx.

Then from (3.7), we obtain the following new equation for the kth mode:

(3.16) a′′k(t) + (ϕuk + 4νk2π2) a′k(t) + 4k2π2γ(1 + ϕρk) ak(t) = b′k(t).

We just need to find a particular solution to this equation, and add it to the generalsolution that we found in section 3.3.

We consider a simple case, where the time dependence of the forcing is a sine (orcosine) function:

(3.17) bk(t) = ck sin(2ωkπt),

9

where ωk is the frequency of the forcing oscillation of mode k.Defining αk = ϕuk+4νk2π2, and βk = 4k2π2γ(1+ϕρk), equation (3.16) becomes:

(3.18) a′′k(t) + αka′k(t) + βkak(t) = 2ckωkπ cos(2ωkπt).

A particular solution is then given by

(3.19) ak(t) = Ak cos(2ωkπt) +Bk sin(2ωkπt).

The constants Ak and Bk are solution of the following linear system:

Ak(βk − 4ω2kπ

2) +Bk(2ωkπαk) = 2ckωkπ,Ak(−2ωkπαk) +Bk(βk − 4ω2

kπ2) = 0.

Then, we get:

Ak = 2ckωkπβk − 4ω2

kπ2

(βk − 4ω2kπ

2)2 + (2ωkπαk)2,(3.20)

Bk = 2ckωkπ2ωkπαk

(βk − 4ω2kπ

2)2 + (2ωkπαk)2.(3.21)

The amplitude of the particular solution is then given by

(3.22) Dk =√A2k +B2

k =2ckωkπ√

(βk − 4ω2kπ

2)2 + (2ωkπαk)2

Increasing ϕρk or ϕuk (or both) will make βk or αk (or both) increase, and thenDk will decrease. This means that we can make Dk become as small as we wantand the observer will converge towards the true state. But of course, the numericalperformance of the observer is severely degraded in comparison with previous cases,as it is usually not possible to consider extremely high values of feedback coefficientsfrom a numerical point of view.

Concerning the density, adapting the previous calculations on the decrease of ρ(knowing the decrease of u), we get:

(3.23) Ek =k

ωk(1 + ϕρk)Dk,

which means that one can adapt the limit amplitude of ρ by changing the values ofϕρk. Theoretically, choosing ϕρk = −1 has the effect of completely removing theinfluence of the forcing term on the density, but of course, it is not numerically stableor physically consistent to consider negative feedback coefficients.

3.7. Partial observations. In the case of partial observations, we define theobservation domain Ω = [0, L] with L < 1 In this case, in order to proceed with Fourieranalysis, we will need to find the Fourier transform of 1Ωu because the feedback termsϕ ∗ u will be replaced with ϕ ∗ (1Ωu). The Fourier series for 1Ω is

1Ω = L+

∞∑k=1

[sin 2πkL

ksin 2πkx+

1− cos 2πkL

kcos 2πkx

].

Thus we see that the Fourier series of 1Ωu will have Fourier components for all k evenin the case when u has just a single Fourier mode.

10

Such coupling of Fourier modes will numerically slightly degrade the performanceof the observer, as at any time, some energy will be transferred between differentFourier modes, as in the full nonlinear model.

If we rewrite equation (3.4), we get:

(3.24) utt − γuxx = νutxx − ϕu ∗ (1Ωut) + γϕρ ∗ (1Ωuxx).

Then, for simplicity reasons, assuming u only has one single Fourier mode k at agiven time, equation (3.7) rewrites:

(3.25) a′′k(t) + (Lϕuk + 4νk2π2) a′k(t) + 4k2π2γ(1 + Lϕρk) ak(t) = 0.

Indeed, only the 0th order term (L) in the Fourier decomposition of 1Ω will bekept through the convolution with u (or one of its derivatives). So, the decay ratebecomes

(3.26)ϕukL+ 4νk2π2

2.

Of course, this is an approximation, as even if u only has a single Fourier modeat time t = 0, the convolution with a characteristic function leads to mode mixing, asin a nonlinear situation. But we assume here that most of the energy is along modek (if only this mode is present at the initial time), which is confirmed by numericalexperiments.

4. Numerical experiments on the 1D compressible Navier-Stokes ob-server. In this section, we report some of the numerical investigations that illustratethe linear theory we discussed above. We also present results of using the same ob-server as in the linear case for two scenarios for which we have not presented anytheoretical results, namely, (i) the observer when the velocity is observed only over asubinterval of the domain in section 4.4, and (ii) in section 4.5, observer for the fullynonlinear equations.

4.1. Numerical configuration. The space domain is [0; 1] and is supposed tobe periodic. The discretization involves 102 grid points, with a step ∆x = 10−2.We consider a time step ∆t = 10−3. We also experimented with increasing thespatial resolution (and correspondingly decreasing the time step), but the results arealmost identical and not presented here. The adiabatic exponent is set to γ = 1.4,and the diffusion is set to ν = 5.10−2 (except in section 4.5). The numerical codeuses a conservation form of the incompressible Navier-Stokes system, with ρ and ρuas variables. A finite volume scheme is used, in which the inviscid flux is computedusing an approximate Riemann solver (e.g. VFRoe scheme). Time integration schemeis a third order explicit Runge-Kutta scheme, where the time step is chosen based ona CFL condition.

In order to reproduce a quasi-linear situation, we consider the true (observed)solution ρ(t, x) = 1 and u(t, x) = 0 while the initial conditions for the observer areset to:

ρI(x) = 1 + 5.10−2 sin(2kπkx), uI(x) = 5.10−2 sin(2kπkx),(4.1)

so that the mean values of ρ and u are ρI = 1 and uI = 0 respectively, and where kis a given mode, usually the first one (k = 1, unless differently specified).

11

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

err

or

time

k = 1; ϕρ = 0.0

ϕu = 00.0 density

ϕu = 00.0 velocity

ϕu = 10.0

ϕu = 15.0 1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0 0.5 1 1.5 2 2.5 3

err

or

time

k = 1; ϕu = 20.0

ϕρ = 00.5

ϕρ = 01.5

ϕρ = 10.0

Fig. 1. The L2 norm of the difference between the observer (ρ, u) and the solution (ρ, u) versustime. Solid and dotted lines are the errors in ρ and u, respectively. The left panel is for fixed ϕρ = 0with varying ϕu while the right panel is for fixed ϕu = 20 with varying ϕρ.

We first look at the solution without any feedback term. Figure 1 (solid anddashed curves, left panel) show the evolution (in log scale) of the L2 norm of thedifference between the observer (ρ, u) and the solution (ρ, u). As there is no feedback,all the Fourier coefficients ϕρk and ϕuk are equal to 0, and then from equation (3.8),the discriminant is ∆ ' −217.2, and the theoretical decay rate (only due to diffusion)is given by equation (3.9): dth = 0.987. Also the oscillation period can be computedfrom (3.11): ωth = 4π√

−∆' 0.85.

Numerically, the slope of the (red) curve in the left panel of figure 1 gives anumerical decay rate dnum = 0.980. Note that the figures show the errors to base10, hence the slope of the semilog plot is dnum/ log(10) = 0.426. The numericaloscillation period is approximately ωnum = 0.852. Note that one period correspondsto two oscillationson the figure for the norm of the cosine. This excellent agreementbetween theoretical and numerical values can be reproduced for other modes andother values of the parameters.

4.2. Simple nudging observer. We now suppose that there is some feedbackonly in the velocity equation. We then first let ϕρ = 0, and only modify the valuesof ϕu. This simulates the nudging, or asymptotic observer: as only the velocity ismeasured, only the velocity is corrected in the observer system. Table 1 shows thetheoretical and numerical decay rates and oscillation periods for several values of ϕu1.

The first remark is that the numerical results perfectly match the theoreticalresults, except for the particular value of ϕu1 = 12.895. In this case, the numericaldecay rate is slightly smaller than the theoretical one. Also, no oscillations can beseen on the results, which is reasonably in agreement with a theoretical period of 81which will be impossible to see with a final time of T = 5.

As ϕu1 increases, the decay rate increases, until ϕu1 reaches 4π√γ−4νπ2 ' 12.895

(see remark in section 3.5), for which the discriminant is equal to 0, and then positivefor increasing ϕu1. The corresponding optimal decay rate is dth = 2π

√γ ' 7.434 (see

section 3.5). We can see that the decay rate then decreases, as the discriminant takeslarger positive values, so that one of the two eigenvalues gets closer to 0.

One can see on figure 1 (left panel, blue and magenta curves) that the errordecreases much stronger than the case of no feedback (figure 1, left panel, red curve).We can also see that with increasing ϕu, the period of oscillations increases andeventually there are no oscillations (discriminant is positive). It confirms that the

12

ϕu1Theoreticaldecay rate

Numericaldecay rate

Theoreticaloscillation period

Numericaloscillation period

0 0.987 0.980(0.426) 0.85 0.860.1 1.037 1.032 0.85 0.850.5 1.237 1.237 0.86 0.851 1.487 1.486 0.86 0.875 3.487 3.485 0.97 0.9810 5.987 6.012(2.61) 1.42 1.38

12.895 7.434 6.590 81.0 −15 4.393 4.364(1.895) − −20 2.897 2.861 − −

Table 1Theoretical and numerical decay rates and oscillation periods for several values of ϕu (k = 1,

ϕρ = 0). The values in parenthesis give the numerical decay rates in base-10, in order to comparewith slopes of lines in figure 1

ϕρ1Theoreticaldecay rate

Numericaldecay rate

Theoreticaloscillation period

Numericaloscillation period

0 2.897 2.861 − −0.5 4.838 4.790(2.080) − −

1.184 10.94 9.741.5 10.98 11.03(4.791) 1.50 1.575 10.99 11.03 0.43 0.4310 10.99 11.06(4.802) 0.28 0.28

Table 2Theoretical and numerical decay rates and oscillation periods for several values of ϕρ with fixed

ϕu = 20 for the k = 1 mode. (The values in parenthesis are again decay rates in base-10 forcomparison with figure 1.)

decay rate decreases if ϕu1 is increased too much.

Similar results have been observed for other values of the diffusion ν, and for othermodes k, and in each case, these numerical results match well with the theoreticalpredictions.

4.3. Results on the full observer. We now use the full observer, with anadditional feedback term in the density equation. We first set ϕu1 = 20, for whichthe discriminant is positive, the largest eigenvalue gets closer to 0, and then the decayrate becomes non optimal.

Table 2 shows the theoretical and numerical decay rates (and oscillation periods)for several values of ϕρ1. As the discriminant is positive when ϕu1 = 20 and ϕρ1 = 0,equation (3.13) gives the theoretical value for which the discriminant comes back to

negative values: ϕρ1 = (ϕu1+4νπ2)2

16γπ2 − 1 ' 1.184.

We observe numerically that the decay rate is very similar to the theoretical rate,and we clearly observe the transition from positive to negative discriminant with thestabilization of the decay rate, and the apparition of oscillations. Increasing ϕρ1 toa much larger value than the optimum given by (3.13) is not necessary, as the decayrate does not increase, and the period of oscillations increases quite quickly. This isclearly seen from the plots in the right panel of figure 1.

As we have seen, by adding the derivative of the velocity as a feedback to the

13

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

err

or

in d

en

sity

Ω = [0,0.1]

Ω = [0,0.3]

Ω = [0,0.5]

Ω = [0,0.8]

Ω = [0,1.0]

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

err

or

in d

en

sity

Ω = [0,0.1]

Ω = [0,0.3]

Ω = [0,0.5]

Ω = [0,0.8]

Ω = [0,1.0]

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

err

or

in v

elo

city

Ω = [0,0.1]

Ω = [0,0.3]

Ω = [0,0.5]

Ω = [0,0.8]

Ω = [0,1.0]

0.975

0.98

0.985

0.99

0.995

1

1.005

1.01

1.015

1.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.06

-0.04

-0.02

0

0.02

0.04

De

nsity

Ve

locity

Density observer 1Density observer 2Velocity observer 1Velocity observer 2

Fig. 2. L2 norm of the difference between the observer ρ, u and the solution ρ, u, in the caseof partial observations over interval [0, L] for L = 0.1, 0.3, 0.5, 0.8, 1, all of them with strength offeedback terms being ϕρ = 0.5 and ϕu = 10. Left panels are for feedback from equation (2.6) whilethe top right panel is for feedback from equation (4.2). The error in velocity for the feedback fromequation (4.2) is almost identical to the bottom left panel and hence not shown. The bottom rightpanel shows the observer solutions at time t = 0.16. Note that “observer 1” refers to (2.6) while“observer 2” refers to (4.2).

density equation, we were able to significantly increase the decay rate of the error, incomparison with only a feedback in the velocity equation.

4.4. Observers with partial observations. In this section we present thenumerical results of using observer with feedback term in equation (2.6). The leftpanels of figure 2 shows the decay rate of the L2 error between the observer (ρ, u) andthe actual solution (ρ, u).

We see that, as expected, the rate of decay is smaller for smaller observationalintervals. We also see that the error in velocity decreases linearly (with oscillations)whereas the error in density saturates at a fairly high but constant value. This isbecause the solution of the observer equation converges to a solution with u = 0 butwith ρ = ρn where ρn 6= ρI - i.e. the observer density is shifted by an amount whichcompensates for the initial discrepancy between the mean of ρ and the mean of ρ.This is not surprising since the original equations themselves are invariant under theconstant shift in density.

In order to overcome this problem in the case of partial observations, we proposethe following modification of the feedback terms in equation (2.6):

Fρ(ρ, u, u) = ϕρ ∗Dρ [1Ω(u− u)− 〈u− u〉] ,(4.2)

where 〈f〉 indicates average of f over the interval [0, L]. This ensures that the averageof the feedback term is zero and hence the equilibrium solution of this equation alsohas mean zero. Note that in the case of L = 1, i.e., the case of full observations,

14

0.0 0.2 0.4 0.6 0.8 1.0Width of observed interval (L)

1

2

3

4

5

6

7

Slop

e of sem

ilog plot of observer e

rror

ϕρ =0.5,ϕu =10.0,k=1

Density error decay rateVelocity error decay rateBest fit line (density)Best fit line (velocity)(ϕu ·L+4νπ2 )/2

0.0 0.2 0.4 0.6 0.8 1.0Width of observed interval (L)

1

2

3

4

5

6

7

Slop

e of sem

ilog plot of observer e

rror

ϕρ =0.5,ϕu =10.0,k=1

Density error decay rateVelocity error decay rateBest fit line (density)Best fit line (velocity)(ϕu ·L+4νπ2 )/2

Fig. 3. The decay rates for the velocity and the density, as a function of the length of theobservation interval (quasi-linear case in the left panel and fully nonlinear case in the right panel).Note that the slope of the best fit line ≈ 4.5 whereas the slope of best fit line by taking only the pointsfor L = 0.1, 0.2, 0.3, 0.4, 0.9, 1.0 is ≈ 5.7.

this average is just zero and the feedback in equation (4.2) is identical to that inequation (2.6). The errors obtained by using this new observer are shown in theright upper panel of figure 2. We see clearly that the observer ρ now approaches thesolution ρ and the error decreases as expected.

In order to clearly see the effect of the observer, the lower right panel of figure 2shows the actual observer solutions for the case when Ω = [0, 0.3]. They clearlyshow the effect of incorporating the observations, and also the difference between theobserver with the incorrect mean and the one with correct mean. The effect is ofcourse more pronounced on the density than on the velocity.

In this case the discriminant of the decay rate is clearly negative (as evidenced byoscillations in the plot for errors) but we can calculate the decay rate. Figure 3 showsthe decay rate of the observer as a function of the length of the interval over whichthe velocity is observed, for the case when ϕρ = 0.5, ϕu = 10 for the k = 1 mode.We notice that this is pretty close to a straight line and the best fit line is given asfollows:

D = 4.99L+ 0.24 .(4.3)

Comparing this with the rate given in equation 3.26, we see that the last equality isa reasonable approximation. Thus even though we cannot calculate the exact decayrate (see section 3.7), a reasonable estimate can be obtained by using formula 3.26:

D ≈ ϕuL+ 4νπ2k2

2= 5L+ 0.987(4.4)

in experiments corresponding to figure 3 (ϕu = 10).This figure shows that the global slope of the numerical decay rates versus the

length of the observation interval is close to ϕu

2 = 5. There is also a good agreement be-tween numerical decay rates and the approximated theoretical ones (see equation 3.26)when L is either small (almost no observations) or large (almost all the domain is ob-served), while the numerical decay rates are degraded when only half the domain isobserved. When L ≈ 0.5, previous spectral studies show that mixing of Fourier modeshas a higher effect than when L ≈ 0 or 1. Convolution with the characteristic functionleads to non negligible transfers of energy from mode k to other modes, and then toa smaller decay rate.

15

10-16

10-12

10-8

10-4

100

0 1 2 3 4 5 6 7 8

err

or

in d

ensity

time

nonlinear, no feedback

nonlinear, with feedback

linear, with feedback

k = 1k = 3k = 6

0.6

0.8

1

1.2

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.96

0.98

1

1.02

1.04

De

nsity (

no

nlin

ea

r)

De

nsity (

line

ar)

Initial conditiont = 0.1 (nonlinear, with feedback)

t = 0.1 (nonlinear, no feedback)t = 0.1 (linear, with feedback)

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0 2 4 6 8 10 12 14 16 18 20

err

or

in d

en

sity

Ω = [0,0.1]

Ω = [0,0.3]

Ω = [0,0.5]

Ω = [0,0.8]

Ω = [0,1.0]

0.6

0.8

1

1.2

1.4

1.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.96

0.98

1

1.02

1.04

1.06

De

nsity (

no

nlin

ea

r)

De

nsity (

line

ar)

Initial conditiont = 0.15 (nonlinear, with feedback)

t = 0.15 (nonlinear, no feedback)t = 0.15 (linear, with feedback)

Fig. 4. The decay rates of the nonlinear observer compared with that of the linear observer(top left). The actual observer solutions are also shown (k = 1 in the top right, k = 3 in the bottomright). The decay rates for the nonlinear observer with partial observations is in the bottom left.

But figure 3 also shows that even with a small observed subdomain, our observeris still very efficient and we can still control the decay rate by increasing ϕu.

4.5. Observer for nonlinear Navier-Stokes system. In this section, we willreport the numerical results of using the observer for the nonlinear system of equations,i.e., from equations (2.4), with the feedback terms as in equations (2.6). Note that wedo not have theoretical estimates of the decay rates towards the equilibrium solution.The aim of this section is to understand the efficacy of the above nonlinear observer.

Firstly, we consider observations of the equilibrium solution, so that we are essen-tially studying the decay of the observer towards the equilibrium. The left panel of fig-ure 4 shows the decay rates of the observer solution with the following two initial con-ditions, with the feedback term set to zero, and for the case with (ϕρ, ϕu) = (0.2, 10):

ρI(x) = 1 + 5.10−1 sin(2kπkx), uI(x) = 5.10−1 sin(2kπkx),

Note that the perturbation strength is 10 times larger than the “quasi-linear” casewith the perturbation in equation (4.1). We consider two cases, k = 1 and k = 3.The cases with other values of k > 1 show behaviour very similar to the case k = 3and hence is not shown here explicitly. For reference, we have also plotted the decayrates of the linear case as well.

The right panel of that figure shows the actual observer solutions for the k = 1case. Note the different scales for the nonlinear (left axis) and linear (right axis)regimes. We clearly see that with perturbation strength of 5.10−1, even though theinitial condition only has k = 1 mode present, at time t = 0.1, higher modes are excited(dotted line, with no feedback). When the observer feedback is added, the solutionvery quickly decays to before higher modes are excited, bringing the perturbation to

16

the level where the linear theory is a good approximation and thus the decay rate isidentical to that of the linear case of perturbation strength 5.10−2.

The case for k = 3 (and indeed all higher modes with k > 1) is quite interesting.In this case, the initial condition of k = 3 excites the k = 1 mode. Thus even withthe observer feedback is added, the decay rate is not the same as the linear k = 3decay rate but rather it is closer to the linear k = 1 rate. The same behavior is seenfor other modes with k > 1. Figure 4 shows example of this decay for k = 3, 6, alongwith the actual observers. It is quite clear that in the quasi-linear case, modes otherthan the one contained in the initial condition are not excited while in the nonlinearcase, they are quite clearly excited.

Finally, we also performed numerical experiments with fully nonlinear observerwith partial observations over various domain sizes, as discussed in detail in sec-tion 3.7. Surprisingly, the behavior in the nonlinear case is very similar to the linearcase: the decay rate is close to being linear in the size of the domain - see the rightpanel of figure 3 and the bottom left panel of figure 4.

5. Conclusion. In this paper, we were interested in observer design for a viscousNavier-Stokes equation. Thanks to intrinsic properties of the system (symmetries), wewere able to design observers in order to reconstruct the full solution (both velocityand density) when only one variable (either velocity or density) is observed, evenpartially.

A spectral study showed that for a tangent linear system (linearized around anequilibrium state), we can prove the convergence of the observer towards the solution,and we can control the decay rate of the error, with explicit formulas for the feedbackcoefficients as functions of the desired decay rate.

Numerical experiments are in perfect agreement with theory in the linearizedsituation: we can obtain any decay rate by increasing the observer coefficients. Nu-merical experiments also show that our observer is still very efficient in the case ofpartial observations, and also in the full nonlinear case.

The application of this kind of observer in other specific cases such as two orthree dimensional Navier-Stokes will be a natural extension of this work, as will bethe case of full primitive equations of the ocean or the atmosphere. Observers for fluidequations coupled to other quantities such as temperature or salinity with observationsof these fields instead of velocity observations will also be an interesting extensionthat will be of great interest in practice since these types of measurements are morecommon. It will also be quite challenging and interesting for practical applicationsto consider observers in the case when observations are available discretely in timeand/or in space.

REFERENCES

[1] M. J. A. Farhat and E. Titi, Continuous data assimilation for the 2d Benard convectionthrough velocity measurements alone. arXiv:1410.1767.

[2] D. Albanez, H. N. Lopes, and E. Titi, Continuous data assimilation for the three dimensionalNavier-Stokes-α model. arXiv:1408.5470.

[3] D. Auroux and S. Bonnabel, Symmetry-based observers for some water-tank problems, IEEETrans. Automat. Control, 56 (2011), pp. 1046–1058.

[4] A. Azouani, E. Olson, and E. S. Titi, Continuous data assimilation using general interpolantobservables, Journal of Nonlinear Science, 24 (2014), pp. 277–304.

[5] A. F. Bennett, Inverse modeling of the ocean and atmosphere, Cambridge University Press,2002.

17

[6] A. Bensoussan, G. Da Prato, M. Delfour, and S. Mitter, Representation and Control ofInfinite Dimensional Systems, Birkhauser Basel, 2007.

[7] H. Bessaih, E. Olson, and E. Titi, Continuous assimilation of data with stochastic noise.arXiv:1406.153.

[8] D. Bloemker, K. Law, A. Stuart, and K. Zygalalkis, Accuracy and stability of thecontinuous-time 3dvar filter for the navier-stokes equation, Nonlinearity, 26 (2013),pp. 2193–2219.

[9] S. Bonnabel, P. Martin, and P. Rouchon, Symmetry-preserving observers, IEEE Trans.Automat. Control, 53 (2008), pp. 2514–2526.

[10] , Non-linear symmetry-preserving observers on Lie groups, IEEE Trans. Automat. Con-trol, 54 (2009), pp. 1709–1713.

[11] J. Deguenon, G. Sallet, and C. Z. Xu., A Kalman observer for infinite dimensional skew-symmetric systems with applications to an elastic beam, in 2nd Int. Symp. Communica-tions, Control, Signal Processing, 2006.

[12] G. A. Gottwald, Controlling balance in an ensemble kalman filter, Nonlinear Processes inGeophysics, 21 (2014), pp. 417–426.

[13] B. Z. Guo and Z. C. Shao, Stabilization of an abstract second order system with applicationto wave equations under non-collocated control and observations, Systems and ControlLetters, 58 (2009), pp. 334–341.

[14] B.-Z. Guo and C. Z. Xu., The stabilization of a one dimensional wave equation by boundaryfeedback with noncollocated observation, IEEE Trans. on Automatic Control, 52 (2007),pp. 371–377.

[15] K. Hayden, E. Olson, and E. S. Titi, Discrete data assimilation in the lorenz and 2d navier-stokes equations, Physica D: Nonlinear Phenomena, 240 (2011), pp. 1416 – 1425.

[16] T. John, M. Guay, N. Hariharan, and S. Naranayan, POD-based observer for estimationin NavierStokes flow, Computers and Chemical Engineering, 34 (2010), p. 965975.

[17] E. Kalnay, Atmospheric modeling, data assimilation and predictability, Cambridge UniversityPress, 2003.

[18] D. Kelly, K. Law, and A. Stuart, Well-posedness and accuracy of the ensemble kalmanfilter in discrete and continuous time, Nonlinearity, 27 (2014), pp. 2579–2603.

[19] A. J. Krener, Eulerian and Lagrangian observability of point vortex flows, Tellus A, 60 (2008),pp. 1089–1102.

[20] I. Lasiecka and R. Triggiani, Control theory for partial differential equations: continuousand approximation theories, Cambridge University Press, 2000.

[21] K. Law, A. Shukla, and A. Stuart, Analysis of the 3dvar filter for the partially observedlorenz ’63 model, Discrete and Continuous Dynamical Systems A, 34 (2014), pp. 1061–1078.

[22] J. M. Lewis, S. Lakshmivarahan, and S. Dhall, Dynamic Data Assimilation: A LeastSquares Approach, vol. 104 of Encyclopedia of Mathematics and its Applications, Cam-bridge University Press, 2006.

[23] C. Lupu and P. Gauthier, Observability of flow-dependent structure functions for use in dataassimilation, Monthly Weather Review, 139 (2010), pp. 713–725.

[24] G. Mohamed and M. Boutayeb, State observer design for non linear coupled partial differen-tial equations with application to radiative-conductive heat transfer systems, in 53rd IEEEConference on Decision and Control, 2014.

[25] N. Nichols, Mathematical concepts of data assimilation, in Data Assimilation, W. Lahoz,B. Khattatov, and R. Menard, eds., Springer Berlin Heidelberg, 2010, pp. 13–39.

[26] E. Olson and E. S. Titi, Determining modes for continuous data assimilation in 2d turbu-lence, Journal of Statistical Physics, 113 (2003), pp. 799–840.

[27] D. Park and M. Guay, Distributed estimation for a class of nonlinear systems with applicationto navier-stokes flow estimation, The Canadian Journal of Chemical Engineering, (2014),pp. n/a–n/a.

[28] K. Ramdani, M. Tucsnak, and G. Weiss, Recovering the initial state of an infinite-dimensional system using observers, Automatica, 46 (2010), pp. 1616–1625.

[29] A. Smyshlyaev and M. Krstic, Backstepping observers for a class of parabolic pdes, SystemsControl Letters, 54 (2005), pp. 613–625.

[30] E. Sontag, Mathematical Control Theory, vol. 6 of Texts in Applied Mathematics, Springer,1998.

[31] R. Vazquez, E. Schuster, and M. Krstic, A closed-form observer for the 3d inductionlessmhd and navier-stokes channel flow, in Decision and Control, 2006 45th IEEE Conferenceon, Dec 2006, pp. 739–746.

[32] C. Z. Xu, J. Deguenon, and G. Sallet, Infinite dimensional observers for vibrating systems,45th IEEE Conf. on Decision and Control, (2006), pp. 3979–3983.

18

[33] J. Zabczyk, Mathematical Control Theory, Birkhauser Basel, 2007.