Stability of Observations of Partial Differential Equations under Uncertain Perturbations

0.23 MB·English

by  Martin Lazar

STABILITY OF OBSERVATIONS OF PARTIAL DIFFERENTIAL EQUATIONS
UNDER UNCERTAIN PERTURBATIONS
MARTINLAZAR
5
1
0 Abstract. We analyse stability of observability estimates for solutions to wave and Scr¨odinger equations
2 subjectedtoadditive perturbations. Thepaper generalisestherecent averagedobservability/control result
byallowingforsystemsconsistingofoperatorsofdifferenttypes. Themethodalsoappliestothesimultaneous
n
observability problem by which one tries to estimate the energy of each component of a system under
a
consideration. The analysis relies on microlocal defect tools; in particular on standard H-measures, when
J
themaindynamicofthesystemisgovernedbythewaveoperator,whileparabolicH-measuresareexplored
0 inthecaseoftheSchro¨dingeroperator.
3

]
P
A
1. Introduction
.
h
t A notion of averaged control has been recently introduced in [15, 10], both for parameter dependent
a
ODEs, as well as for systems of PDEs with variable coefficients. Its goal is to control the average (or more
m
generally a suitable linear combination) of system components by a single control. The problem is relevant
[
in practice where the control has to be chosen independently of the coefficient value.

1 The notionis equivalentto the averagedobservability,by whichthe energy of the systemis recordedby
v observing the average of solutions on a suitable subdomain.
3
Inthe paperweinvestigateamoregeneralproblembasedonasystemwhosefirstcomponentrepresents
6
6 its main dynamic, while the other ones correspondto perturbations. Assuming that the main component is
0 observable, we explore conditions by which that property remains stable under additive perturbations.
0 In general, the operators entering the system are not assumed to be of the same type. In a special case
.
2 ofasystemconsistingofasametypeoperators,theresultcorrespondstotheaveragedcontrolofthesystem,
0 thus incorporating results obtained in [10].
5
Themethodsareappliedtothesimultaneousobservabilityproblemaswell,bywhichonetriestoestimate
1
energy of all system components by observing their average. The corresponds dual problem consists of
:
v controlling each individual component of the adjoint system by means of a same control.
i
X Thestudyoftheproblemexploresmicrolocalanalysistools,inparticularH-measuresandtheirvariants.
H-measures, introduced independently in [7, 14], are kind of defect tools, measuring deflection of the weak
r
a from strong convergence of L2 sequences. Since their introduction, they have been successfully applied
in many mathematical fields - let us just mention generalisation of compensated compactness results to
equations with variable coefficients [7, 14] and applications in the control theory [4, 5, 10]. Most of these
applications apply the so called localisation principle providing constraints on the support of H-measures
(e.g. [14]), and the proofs of this paper rely on it as well.
The paper is organised as follows. In the next section we provide an averagedobservability result for a
system whose main dynamic is governed by the wave operator. The finite system is analysed first, followed
by generalisations to an infinite discrete setting. Application of the approach to simultaneous observability
is provided in the subsection 2.3. The third section is devoted to observation of the Schr¨odinger equation
under perturbations determined either by a hyperbolic or by a parabolic type operator. In the latter case,
parabolicH-measures(generalisationoforiginalones to a parabolicsetting)haveto be explored. The paper
is closed with concluding remarks, and by pointing toward some open and related problems.
1991 Mathematics Subject Classification. 93B05,93B07,93C20, 93D09.
Keywords and phrases. averagedcontrol,robustobservability,parabolicH-measures.
1
2 MARTINLAZAR
2. Observation of the wave equation under uncertain perturbations
2.1. Averaged observability. We analyse the problem of recovering the energy of the wave equation by
observinganadditiveperturbationofthesolution. Theperturbationisdeterminedbyadifferentialoperator
P , in general different from the wave one.
2
More precisely, we consider the following system of equations:
P u =∂ u div(A (t,x) u )=0, (t,x) R+ Ω
1 1 tt 1 1 1
− ∇ ∈ ×
P u =0, (t,x) R+ Ω
2 2
∈ ×
u =0, (t,x) R+ ∂Ω (2.1)
1
∈ ×
u (0, )=β L2(Ω)
1 0
· ∈
∂ u (0, )=β H−1(Ω),
t 1 1
· ∈
where Ω is an open, bounded set in Rd, A is a bounded, positive definite matrix field, while P is some,
1 2
almost arbitrary, differential operator (precise conditions on it will be given below). In the sequel we shall
also use the notation = div(A ) for the elliptic part of P .
1 1 1
A − ∇
For the moment, we specify neither initial nor boundary conditions for the second equation, we just
assume that corresponding problem is well posed and that it admits an L2 solution. For the coefficients of
both the operators we assume that are merely bounded and continuous.
Proposition 2.1. Suppose that there is a constant C˜, time T and an open subdomain ω such that for any
choice of initial conditions β ,β the solution u of (2.1) satisfies
0 1 1
T
E (0):= β 2 + β 2 C˜ u 2dxdt. (2.2)
1 k 0kL2 k 1kH−1 ≤ | 1|
Z0 Zω
In addition, we assume that characteristic sets p (t,x,τ,ξ) = 0 ,i = 1,2 have no intersection for (t,x)
i
{ } ∈
0,T ω,(τ,ξ) Sd, where p stands for the principal symbol of the operator P .
i i
h i× ∈
Then for any θ ,θ R,θ =0 there exists a constant C˜ such that the observability inequality
1 2 ∈ 1 6 θ1
T
E (0) C˜ θ u +θ u 2dxdt+ β 2 + β 2 (2.3)
1 ≤ θ1 Z0 Zω| 1 1 2 2| k 0kH−1 k 1kH−2!
holds for any pair of solutions (u ,u ) to (2.1).
1 2
Proof: Ofcourse,thecaseθ =0holdstrivially,andforsimplicityisexcludedfromthefurtheranalysis.
2
We argueby contradiction. Assuming the contrary,there exists a sequence ofsolutions un,un suchthat
1 2
T
En(0)>n θ un+θ un 2dxdt+ βn 2 + βn 2 . (2.4)
1 Z0 Zω| 1 1 2 2| k 0kH−1 k 1kH−2!
As the considered problem is linear, without loosing generality we can assume that En(0) = 1. Thus (2.4)
implies that βn 2 + βn 2 0, resulting in the weak convergence (βn,βn) ⇀ (0,0) in L2(Ω)
k 0kH−1 k 1kH−2 → 0 1 − ×
H−1(Ω). Therefore the solutions (un) converge weakly to zero in L2(Ω 0,T ) as well. In order to obtain
1 ×h i
a contradiction, we have to show that the last convergence is strong, at least on the observability region.
From the contradictory assumption (2.4) we have that the H-measure ν associated to a subsequence of
(θ un+θ un) vanishes on 0,T ω. Furthermore, it is of the form
1 1 2 2 h i×
ν =θ2µ +θ2µ +θ θ 2 µ ,
1 1 2 2 1 2 ℜ 12
where on the right hand side the elements of the matrix measure associated to the vector subsequence of
(un,un) arelisted, with µ denoting the off-diagonalelement. Note that (un) is bounded in L2( 0,T ω),
1 2 12 2 h i×
since thatis the casefor (un) (by boundedness ofinitialdata), andforthe linearcombination(θ un+θ un)
1 1 1 2 2
(by contradictory assumption (2.4)), which enables one to associate an H-measure to it.
According to the localisation property for H-measures, each µ is supported within the corresponding
j
characteristic set p (t,x,τ,ξ)=0 , i=1,2, which, by assumption, are disjoint on the observability region.
i
{ }
On the other hand, from the very definition of matrix H-measures it follows that off-diagonal entries are
STABILITY OF OBSERVATIONS OF PDE-S UNDER UNCERTAIN PERTURBATIONS 3
dominated by the corresponding diagonal elements. More precisely, it holds that suppµ suppµ
12 1
⊆ ∩
suppµ , implying that µ =0 on the observability region.
2 12
Thus we get that
ν =θ2µ +θ2µ =0 on 0,T ω.
1 1 2 2 h i×
As µ and µ are positive measures and θ > 0, it follows that µ vanishes on 0,T ω as well. Thus
1 2 1 1
h i×
we get strong convergence of (un) in L2( 0,T ω), which together with the assumption of the constant,
non-zero initial energy contradic1ts the obshervaibi×lity estimate (2.2). (cid:3)
Remark 1. The last result provides surprising stability of the observability estimate (2.2) under uncertain
perturbations,uptocompactreminders. Essentially, theonlyrequirementfortheperturbationisseparationof
the characteristic sets. This implies that the wave component can be observed robustly when adding unknown
perturbations, up to a finite number of low frequencies.
In the next step we would like to obtain the strong observability inequality for initial energy E (0) by
1
removing compact terms in (2.3). To this effect, we have to specify some additional constraints on the
problem for the perturbation u .
2
We take P to be an evolution operator of the form
2
P =(∂ )k+c (x) , k N, (2.5)
2 t 2 1
A ∈
where is an elliptic part of the wave operator P , while c is a bounded and continuous function.
1 1 2
A
Theorem 2.2. As above, we assume that the coefficients of the operator P are bounded and continuous,
1
and that the corresponding solution u satisfies the observability inequality (2.2).
1
In addition we assume that the perturbation operator P is of the form (2.5). In the case k =2 (i.e. P
2 2
being a wave operator) the separation of coefficients c (x) 1=0 is supposed on ω.
2
− 6
For the initial values of solutions u ,i=1,2 we supposed these are related by a linear operator such that
i
whenever ((θ u (0)+θ u (0)) =0) then u (0) =u (0) =0 , and the analogous implication holds for
1 1 2 2 ω 1 ω 2 |ω
| | |
the initial first order time derivatives.
(cid:0) (cid:1)
Then there is a positive constant C such that the strong observability inequality holds:
θ1
T
E (0) C θ u +θ u 2dxdt. (2.6)
1 ≤ θ1 | 1 1 2 2|
Z0 Zω
Remark 2. Notethat theabove assumptions directly imply that characteristic sets ofP andP aredisjoint.
1 2
Indeed, for P being an evolution operator of order k = 2, its principal symbol equals 0 only in poles ξ = 0
2
6
(case k = 1), or on the equator τ = 0 (case k > 2) of the unit sphere in the dual space, where p =
1
τ2 A (x)ξ2 differs from zero.
1
−
In the case k=2 separation of the characteristic sets is provided by the assumption c (x)=1 on ω.
2
6
Proof: As in the proof of Proposition 2.1, let us suppose the contrary. Then there exists a sequence of
solutions un,un to (2.1) such that En(0)=1 and
1 2 1
T
θ un+θ un 2dxdt 0. (2.7)
| 1 1 2 2| −→
Z0 Zω
Thus the corresponding weak limits satisfy both the equation P u =0, as well as the relation
i i
θ u +θ u =0 (2.8)
1 1 2 2
on the observability region. As functions u ,∂ u are continuous with respect to time, the bounds on initial
i t i
data imply β =β =0 on ω.
0 1
Assumptions on the operatorsP ,i=1,2 ensure that corresponding characteristicsets do not intersect.
i
By applying localisationproperty of H-measures as in the proof of Proposition2.1 we get that un converges
1
to u strongly on 0,T ω.
1
h i×
It remains to show that the limit u vanishes on the observability region, which, together with the
1
assumption of the constant non-zero initial energy, will contradict the observability assumption (2.2).
We split the rest of the proof into several cases.
4 MARTINLAZAR
a) (k=2) Due to the relation (2.8) it follows that
(c 1)∂ u =0, (t,x) 0,T ω.
2 tt 1
− ∈h i×
As c 1 >0, and the initial data are 0 on ω, it implies u =0 on the observability region.
2 1
| − |
b) (k=1) Relation (2.8) implies
∂ u =c (x)∂ u ,
t 1 2 tt 1
which together with u (0, )=∂ u (0, )=0 on ω provides the claim.
1 t 1
· ·
c) (k>2) Similar as above we obtain
∂k−2(∂ u )=c (x)∂ u .
t tt 1 2 tt 1
As ∂ u (0, ) = u (0, ) = 0 on ω, and similarly for the higher order derivatives, the claim
tt 1 1 1
· −A ·
follows.
(cid:3)
Remark 3. Several remarks are in order.
The observability assumption (2.2) on a solution of the wave equation is equivalent to the Geometric
•
Control Condition (GCC, [3]), stating that projection of each bicharacteristic ray on a physical space
has to enter the observability region in a finite time.
The last theorem also holds if, instead of initial data of two components being linked by an operator,
•
we assume a cone condition u (0) c u (0) u (0) c u (0) , with a
k 2 kL2(ω) ≤ k 1 kL2(ω) k 2 kL2(ω) ≥ k 1 kL2(ω)
constant c < θ1/θ2(c>θ1/θ2). The latter condition is sta(cid:16)ble under passing to a limit,(cid:17)and also
ensures the implication ((θ u (0)+θ u (0)) =0) = u (0) =u (0) =0 , which suffices for
1 1 2 2 ω 1 ω 2 |ω
| ⇒ | |
the proof.
(cid:0) (cid:1)
The result (2.6) can be generalise to a more general perturbation operator P by assuming that
2
•
coefficients of both the operators are analytic. In that case the separation of characteristic sets
implies the separation of corresponding analytic wave front sets ([9, Theorem 9.5.1]). Together with
(2.8) it provides that u is analytic on the observability region. Constraints on initial data and finite
1
velocity of propagation imply u = 0 on an open set near t = 0, and as the solution is analytic it
1
vanishes on the whole observability region which contradicts the observability assumption (2.2).
If P is a wave operator the strong observability inequality (2.6) is equivalent to the controllability of
2
•
a suitable linear combination (determined by the operator linking the initial data) of solutions to the
adjoint system under a single control (cf. [10]).
Meanwhile, theweakobservabilityresult (2.3)inthatcasecorrespondstotheaveragecontrollability
of the adjoint system up to a finite number of low frequencies.
The last theorem generalises the results of [10] by allowing for a general evolution operator P which
2
•
does not have to be the wave one.
In addition, it allows for an arbitrary linear combination of system components, while in [10]
just their (weighted) average is explored. Specially, if the difference u u is considered, the result
1 2
−
corresponds to the synchronisation problem (e.g. [12]) in which all the components are driven to the
same state by applying the null controllability of their differences.
Furthermore, unlike in [10], the proof of the relaxed observability inequality (2.3) does not rely on
the propagation property of H-measures, which allows for system’s coefficients to be merely continu-
ous. On the other hand, such approach avoid technical issues related to the reflection of H-measures
on the domain boundary.
The theorem to some extent also generalises the results of [16] in which a similar result is provided
•
for the system (2.1) consisting of a wave and a heat operator with constant coefficients (or more
generally with a common elliptic part).
However, although allowing for a more general perturbation operator, it requires initial data of
two components of the system (2.1) to be related, while in [16] no assumptions on initial data for the
second component is assumed.
The constant C can be taken to be uniform for θ θ and θ >0.
• θ1 1 ≥ ∗ ∗
STABILITY OF OBSERVATIONS OF PDE-S UNDER UNCERTAIN PERTURBATIONS 5
The weak observability result (2.3) is easily generalised to a system with a finite number of compo-
•
nents, under assumption that the characteristic set of the leading operator P is separated from the
1
characteristic sets of all the other operators, while the latter ones can be arbitrary related.
However, the generalisation of the result to an infinite dimensional setting is not straightforward.
It requires study of the localisation property for H-measures determined by a sequence of function
series, and is the subject of the next subsection.
On the other side, the generalisation of the strong estimate result (2.6) to a system consisting of
more than two components has still not been obtained, and is a subject of the current investigations.
The result (2.3) also holds if the observability region is not of a cylindrical type, but a more general
•
set satisfying the GCC for the first component of the system. Such generalisation corresponds to a
moving control (cf. [11]). However, possible derivation of the corresponding stronger observability
resultremainsopen, duetothelastpart oftheproofofTheorem 2.2,inwhich thespecial (cylindrical)
shape of the observability region is used.
Note that the observability result (2.6) is weaker than the one required in the simultaneous control,
•
where one has to estimate initial energy of all components entering the system. Of course, the
assumptions in the latter case are stronger, as one has to assume that the observability set satisfies
the GCC for the second component as well.
2.2. Infinite discrete setting. Inthis subsectionwe wanttoanalysethe stability ofthe observabilityesti-
matesforthewaveequationwhentheperturbationisgivenasasuperpositionofinfinitelymanycomponents,
eachdeterminedbya differentialoperatorP , which,ingeneral,doesnothavetobe the waveone. Thus the
i
system of interest reads as:
P u =∂ u div(A (t,x) u )=0, (t,x) R+ Ω
1 1 tt 1 1 1
− ∇ ∈ ×
P u =0, (t,x) R+ Ω,i 2
i i
∈ × ≥
u =0, (t,x) R+ ∂Ω (2.9)
1
∈ ×
u (0, )=β L2(Ω)
1 0
· ∈
∂ u (0, )=β H−1(Ω),
t 1 1
· ∈
with the same assumptions on the domain Ω and the operator P being assumed for the system (2.1). For
1
the other equations, neither initial nor boundary conditions are specified. For the moment, we just assume
the correspondingcoefficients areboundedandcontinuous,andthe problemsarewelldefinedwith solutions
in L2(Ω).
In this setting, the same microlocal analysis tool as in the finite case, in particular the localisation
property of H-measures, is applied in the study of the stability of the observability estimates. However, as
perturbations are determined by a superposition of infinitely many solutions, this requires analysis of the
mentioned property for a sequence of function series.
Namely, it is well known that an H-measure associated to a linear combination of two sequences is
supported within the union of supports of measures determined by eachcomponent, and the same property
holdsforanyfinitelinearcombination. However,ingeneralitfailswhenconsideringsuperpositionofinfinite
many sequences, as shown by the next example.
Example 1. Let (un) and (fn) be L2(Rd) sequences, whose corresponding H-measures µ and µ have
u f
disjoint supports, and let (θ ) be a sequence of nonnegative numbers summing in 1.
i
Define the following sequences
θ un i=n
vn = i 6
i fi i=n.
(cid:26)
Thus for each i an H-measure ν associated to vn equals θ2µ .
i i i u
On the other side we have that vn =(1 θ )un+fn, and the corresponding measure equals µ +µ .
i i − n u f
Thus in order to constrain suppPort of an H-measure by supports of corresponding components we have
to impose additional assumptions on constituting sequences. More precisely, the following result holds.
6 MARTINLAZAR
Lemma 2.3. Let (θ ) be an averaging sequence of positive numbers summing to 1, and let (un) ,i N be
i i n ∈
a family of uniformly bounded L2 sequences, i.e. we assume there exists a constant C such that un
u k ikL2 ≤
C ,i,n N.
u
∈
Define the linear combination v = θ un, and denote by µ and ν H-measures associated to (sub)se-
n i i i i
quences (of) (un) and (v ), respectively. Then
i n n P
suppν Cl suppµ . (2.10)
i i
⊆ ∪
(cid:16) (cid:17)
Proof: Takeanarbitrarypseudodifferentialoperatoroforderzero,P Ψ0,withasymbolp(x,ξ)being
∈ c
compactly supported within the complement of the closure of suppµ .
i i
∪
By the definition of H-measures we have
∞ ∞
ν,p =lim P θ un (x) θ un (x)dx. (2.11)
h i n ZRd (cid:16)X1 i i(cid:17) (cid:16)X1 j j(cid:17)
As P is a continuous operator on L2(Rd) it follows that
∞ ∞ ∞ ∞
lim P θ un (x) θ un (x)dx limsupC θ un θ un
n (cid:12)(cid:12)ZRd (cid:16)Xk i i(cid:17) (cid:16)X1 j j(cid:17) (cid:12)(cid:12)≤ n P(cid:13)Xk i i(cid:13)L2(cid:13)X1 i i(cid:13)L2 (2.12)
(cid:12) (cid:12) (cid:13) ∞ (cid:13) (cid:13) (cid:13)
(cid:12) (cid:12) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:12) (cid:12) limsupC C2 θ k 0,
≤ n P u i!−→
k
X
where C is the L2 bound of the operator P. The last sum is a remainder of a convergent series, and the
P
above limit converges to zero uniformly with respect to n.
Similarly, one shows the same property holds for lim P ∞θnun (x) ∞θnun (x)dx . Thus
n Rd 1 i i l j j
we can exchange limits in (2.11), getting (cid:12) (cid:16) (cid:17) (cid:16) (cid:17) (cid:12)
(cid:12)R P P (cid:12)
(cid:12) (cid:12)
∞ ∞
ν,p = lim P (θ un)(x)θ un(x)dx
h i i=1j=1 n ZRd i i j j
XX (2.13)
∞ ∞
= θ θ µ ,p =0,
i j ij
h i
i=1j=1
XX
where µ are H-measures determined by sequences (un) and (un), supported within the closure given in
ij i j
(2.10), outside which p is supported. (cid:3)
As a consequence of the last lemma, in order to apply the localisation property within the analysis of
observability estimates for solutions to (2.9), some kind of uniform boundedness on the solutions has to be
assumed.
Proposition 2.4. Suppose the observability inequality (2.2) holds for a solution u to (2.9).
1
As for the system (2.9), suppose that L2 norm of all the solutions u is dominated (up to a multi-
i
plicative constant, independent of a choice of initial data) by the energy norm of u . In addition as-
1
sume that characteristic set p (t,x,τ,ξ) = 0 has no intersection with Cl p (t,x,τ,ξ) = 0 for
1 i≥2 i
{ } ∪ { }
(t,x) 0,T ω,(τ,ξ) Sd, where p stands for the principal symbol of the(cid:16)operator P . (cid:17)
i i
∈h i× ∈
Then for any averaging sequence (θ ) of positive numbers summing to 1, with θ > 0, there exists a
i 1
constant C˜ such that the observability inequality
θ
T
E (0) C˜ θ u 2dxdt+ β 2 + β 2 (2.14)
1 ≤ θ Z0 Zω| i i| k 0kH−1 k 1kH−2!
X
holds for any family of solutions (u ) to (2.9).
i
STABILITY OF OBSERVATIONS OF PDE-S UNDER UNCERTAIN PERTURBATIONS 7
Proof: Assume the contrary. Then there exist sequences of initial conditions (βn),(βn), and of associ-
0 1
ated solutions (un), such that
i
T ∞
1=En(0):= βn 2 + βn 2 >n θ un 2dxdt+ βn 2 + βn 2 . (2.15)
1 k 0kL2 k 1kH−1 Z0 Zω|i=1 i i| k 0kH−1 k 1kH−2!
X
Let ν be an H-measure associated to a (sub)sequence of ∞ θnun. Due to the inequality (2.15), it equals
i=1 i i
zero on 0,T ω.
We hsplitith×e last sum into two parts θ un+ ∞ θ uPn, and we rewrite ν in the form
1 1 i=2 i i
ν =ν +ν +2 ν ,
P1 2 12
ℜ
whereν andν areH-measuresassociatedto(sub)sequences(of)(θnun)and( ∞θnun),respectively,while
1 2 1 1 2 i i
ν is a measure corresponding to their product. In addition, ν = θ2µ , where by µ we denote a measure
12 1 1 1 P i
associated to a (sub)sequence (of) the i th component un.
− i
From here the statement of the theorem is obtained easily (following the lines of the proof in finite
discrete case, Proposition 2.1), once we show that ν and ν have disjoint supports.
1 2
BythelocalisationpropertyforH-measures,eachmeasureµ issupportedwithintheset p (t,x,τ,ξ)=
i i
{
0 .
}
The assumption on the domination of solutions to (2.9) by an energy norm of u , together with the
1
constantinitial energyEn(0)implies uniformbound onsolutionsun,both withrespectto i andn. Thuswe
1 i
can apply Lemma 2.3 to conclude that ν is supported within the set
2
Cl p (t,x,τ,ξ)=0 ,
i≥2 i
∪ { }
which, due to the assumption on separa(cid:16)tion of the characteristic(cid:17)s set, does not intersect the support of
ν = θ2µ . As θ is strictly positive, we get that un converges to 0 strongly in L2( 0,T ω), which
co1ntrad1ict1s the obs1ervability estimate (2.2). 1 h i× (cid:3)
Remark 4.
The assumption of the last proposition requiring solutions u of (2.9) to be dominated by the energy
i
•
norm of u occurs, for example, in a case of a system consisting of the operators of the same form,
1
P = τ2 A (t,x)ξ ξ, with uniformly bounded (both from below and above) coefficients and initial
i i
− ·
energies.
The assumption on separation of characteristics sets in that case can be stated as
A (t,x)ξ ξ >(<) sup(inf)A (t,x)ξ ξ, (t,x) 0,T ω, ξ =0,
1 i
· i≥2 i≥2 · ∈h i× 6
i.e. the fastest (or the slowest) velocity is strictly separated from all the others.
In that case the weak observability (2.14) result is equivalent to the averaged controllability of the
adjoint system up to a finite number of low frequencies.
Of course, one can construct more general systems, including operators of different types as well,
that satisfy the required boundedness assumption.
As already mentioned in previous subsection, obtaining corresponding strong observability result in
•
this setting remains an open problem.
The constantC appearing in (2.14)can betaken uniformly for afamily of averaging sequences, each
θ
•
satisfying
(i) θ θ ,
1 ∗
(ii) ∞≥θ ǫ ,
k i ≤ k
where θ 0,1] and (ε ) is a null sequence, both independent of a choice of a particular sequence
∗ k
P ∈ h
(θ ).
i
2.3. Simultaneous observability. The subsection deals with a problem of recovering energy of a system
by observing an average of solutions on a suitable subdomain. For this purpose one has to estimate initial
energies of all system components, unlike the case of the average observability where this was required just
for the first one. A two component system is analysed firstly, while generalisations to a more dimensional
case is discussed at the end.
8 MARTINLAZAR
We reconsider the system (2.1) assuming that P is an evolution operator of the form
2
P =(∂ )k+ , k N, (2.16)
2 t 2
A ∈
where is an (uniformly) elliptic operator (in general different from ), and the problem for the pertur-
2 1
A A
bation u is accompanied by a series of initial conditions
2
(∂ )ju (0)=γ H−j(Ω), j =0,...,k 1.
t 2 j
∈ −
Its initial energy is denoted by(cid:0) (cid:1)
k−1
E (0)= γ .
2 k jkH−j(Ω)
0
X
As in the previous subsection, we start with a weak observability result.
Proposition 2.5. Suppose that there is a constant C˜, time T and an open subdomain ω such that for any
choice of initial conditions the solutions to (2.1) satisfy
T
E (0) C˜ u 2dxdt, i=1,2. (2.17)
i i
≤ | |
Z0 Zω
In addition assume that characteristic sets p (t,x,τ,ξ) = 0 ,i = 1,2 have no intersection for (t,x)
i
{ } ∈
0,T ω,(τ,ξ) Sd, where p stands for the principal symbol of the operator P .
i i
h i× ∈
Then for any θ ,θ R 0 there exists a constant C˜ such that the observability inequality
1 2 θ
∈ \{ }
T
E (0)+E (0) C˜ θ u +θ u 2dxdt+ β 2 + β 2 + γ 2 +...+ γ 2 (2.18)
1 2 ≤ θ Z0 Zω| 1 1 2 2| k 0kH−1 k 1kH−2 k 0kH−1 k k−1kH−k!
holds for any pair of solutions (u ,u ) to (2.1).
1 2
The result is obtained easily by following the steps of the proof presented above in the averaged ob-
servability setting. Assuming the contrary and implying microlocal analysis tools, one shows that both
components un converge to 0 strongly on the observability region, thus obtaining the contradiction.
i
However,adifferentapproachisrequiredinordertoobtainthe strongobservabilityinequalityforinitial
energy by removing compact terms in (2.18). It is based on a standard compactness-uniqueness procedure
of reducing the observability for low frequencies to an elliptic unique continuation result [3, 5].
We introduce a subspace N(T) of H = L2(Ω) H−1(Ω) L2(Ω) H1−k(Ω), consisting of initial
× × ×···×
data for which the averageof solutions to (2.1) vanishes on the observability region
N(T):= (β ,β ,γ ,...,γ ) H θ u +θ u =0on 0,T ω .
0 1 0 1−k 1 1 2 2
{ ∈ | h i× }
Based on the relaxed observability inequality (2.18) it follows that N(T) is a finite dimensional space.
Furthermore, the following characterisationholds.
Lemma 2.6. We assume one of the following statements holds:
a) The order k of time derivative in (2.16) is odd. Coefficients of both the operators P and P are time
1 2
independent and of class C1,1,
b) The time derivative order k is even, and k/2 (or ( k/2 )) is an uniformly elliptic
A1 −A2 − A1 − A2
operator. Coefficients of both the operators P and P are analytic.
1 2
Then N(T)= 0 .
{ }
Proof: One first shows that N(T) is an A-invariant, where A is an unbounded operator on H:
0 1
− 0
0
A= A1  , (2.19)
0 I
0 − k−1
 0 
 A2 
 
with the domain D(A)=H1(Ω) L2(Ω) H1(Ω) H2−k(Ω).
0 × × 0 ×···×
STABILITY OF OBSERVATIONS OF PDE-S UNDER UNCERTAIN PERTURBATIONS 9
Being A-invariant and finite-dimensional, it contains an eigenfunction of A. Thus there is a λ C and
∈
(β ,β ,γ ,...,γ ) N(T) such that
0 1 0 1−k
∈
β = λ2β
1 0 0
A −
γ =( 1)k−1λkγ
2 0 0
A − (2.20)
β = λβ
1 0
−
γ =( 1)jλjγ , j =1,...,k 1.
j 0
− −
By the definition of N(T) it follows θ u +θ u =0 on 0,T ω, and specially
1 1 2 2
h i×
θ β +θ γ =0 on ω. (2.21)
1 0 2 0
At this level, we want to show that each assumption of the lemma implies β =γ =0.
0 0
a) As ,i=1,2arepositiveoperatorsandkisodd,from(2.20)itfollowsthatoneoffunctionsβ ,γ is
i 0 0
A
trivial. Byrelation(2.21)itfollowsthatthe otheronealsoequalszeroonω. Being aneigenfunction
of an elliptic operator, the unique continuation argument (e.g. [8, Theorem 3]) implies it is zero
everywhere.
b) Analyticity of coefficients implies analyticity of eigenfunctions. Specially it follows θ β +θ γ =0
1 0 2 0
everywhere, and relations (2.20) imply
( k/2 )β =0.
A1 −A2 0
Assumptions on the operator k/2 imply β =0 on Ω.
A1 −A2 0
(cid:3)
Remark 5. In a special case = div(c (x) ) one easily proves that the last Lemma holds with analytic
2 2
A − ∇
coefficients c ,c being separated just on an arbitrary non-empty open set, and not on the entire Ω.
1 2
Theorem 2.7. Under the assumptions of Proposition 2.1 and Lemma 2.6 there is a positive constant C
θ
such that the strong observability inequality holds:
T
E (0)+E (0) C θ u +θ u 2dxdt. (2.22)
1 2 θ 1 1 2 2
≤ | |
Z0 Zω
Proof: As in the proof of Proposition 2.1, let us suppose the contrary. Then there exists a sequence of
solutions un,un to (2.1) such that En(0)=1 and
1 2 1
T
θ un+θ un 2dxdt 0.
| 1 1 2 2| −→
Z0 Zω
Thus for weak limits (u ,u ) of solutions on the observability region we have θ u +θ u = 0, implying
1 2 1 1 2 2
u (0),∂ u (0),u (0),..., (∂ )k−1u (0) N(T). By means of the above lemma and taking into account
1 t 1 2 t 2
∈
the relaxed observability inequality, it follows
(cid:0) (cid:0) (cid:1) (cid:1)
T
1 C˜ θ un+θ un 2dxdt+ βn 2 + βn 2 + γn 2 +...+ γn 2 0,
≤ θ Z0 Zω| 1 1 2 2| k 0kH−1 k 1kH−2 k 0kH−1 k k−1kH−k!−→
thus obtaining a contradiction. (cid:3)
We close this subsection by the following remarks.
Remark 6.
If P is a second order evolution operator the strong observability inequality (2.22) is equivalent to
2
•
the simultaneous controllability of the adjoint system, also studied in [10], by which one controls each
component individually (and not just their average).
The notion of simultaneous observability is stronger than the average one, as it estimates energy of
•
all system components, whose initial data, in this case, are not related. Consequently, it requires
stronger assumption of GCC being satisfied by each component.
10 MARTINLAZAR
The application of the compactness-uniqueness procedure in the passage from the weak to the strong
•
observability estimate allows the perturbation P to bean evolution operator with an arbitrary elliptic
2
part. However, such approach is not possible in the averaged observability setting. Namely, in order
for subspace N(T) to be finite dimensional one has to relate the initial data of two components by a
bounded linear operator. But such constrain would not be preserved under action of the operator A
given by (2.19), and as a consequence N(T) would not be A-invariant.
The weak observability result (2.18) is easily generalised to a system with a finite number of compo-
•
nents, under assumption that the characteristic sets of all operators are mutually disjoint.
As in the averaged observability case, the generalisation of strong estimate result (2.22) to a system
•
consisting of more than two components has still not been obtained, and is a subject of the current
investigations.
3. Observation of the Schro¨dinger equation under uncertain perturbations
In this section we consider a system in which the first component, the one whose energy is observed,
satisfies the Schr¨odinger equation, while the second one, corresponding to a perturbation, is governedby an
evolution operator P :
2
P u =i∂ u +div(A (x) u )=0, (t,x) R+ Ω
1 1 t 1 1 1
∇ ∈ ×
P u =0, (t,x) R+ Ω
2 2
∈ × (3.1)
u =0, (t,x) R+ ∂Ω
1
∈ ×
u (0, )=β L2(Ω).
1 0
· ∈
As in the study of perturbations of the wave dynamics in Section 2, we specify no initial or boundary
conditions for the second operators, we just assume that corresponding problem is well posed and that it
admitsanL2solution. Asforthesystemcoefficients,asbeforeweimposemerelyboundednessandcontinuous
assumptions, and suppose that A is a positive definite matrix field.
1
3.1. Averaged observability under non-parabolic perturbations. For the reasons explained below,
in this subsectionwe restrictthe analysisto evolutionoperatorsP oforderstrictly largerthanone. In that
2
case the stability of the Schr¨odinger observability estimate is given by the next theorem.
Theorem 3.1. Suppose that there is a constant C˜, time T and an open subdomain ω such that for any
choice of initial datum β the solution u of (2.1) satisfies
0 1
T
E (0):= β 2 C˜ u 2dxdt. (3.2)
1 k 0kL2 ≤ | 1|
Z0 Zω
In addition, for the system (3.1) we assume the following:
a) The perturbation operator is an evolution operator of the form (2.5) and of the order k >1.
b) The initial values of solutions u ,i = 1,2 are related by a linear operator such that whenever
i
((θ u (0)+θ u (0)) =0) then u (0) =u (0) =0 .
1 1 2 2 ω 1 ω 2 |ω
| | |
Then, for any θ 0,1] there exists a constant C such that the observability inequality
1 (cid:0) θ (cid:1)
∈h
T
E (0) C θ u +θ u 2dxdt (3.3)
1 θ 1 1 2 2
≤ | |
Z0 Zω
holds for any pair of solutions (u ,u ) to (2.1).
1 2
The proof goes similarly as for the observations of the wave equation. Required conditions a), b) are
necessary for obtaining the strong observability inequality, without a compact term.
On the other hand, in order to obtain a relaxed inequality with a compact term, no assumption is
required at all. Namely, the assumption on separation of characteristic sets p (t,x,τ,ξ) = 0 required
i
{ }
in Proposition 2.1 becomes superfluous in this setting, as being directly satisfied by an arbitrary evolution
operator P of order k strictly larger than 1. Namely, its characteristic set does not contain the poles ξ =0
2
of the unit sphere in the dual space, which constitute the characteristic set of the Schr¨odinger operator P .
1