Int. J. Appl. Math. Comput. Sci., 2002, Vol.12, No.2, 143–151
REGIONAL BOUNDARY OBSERVABILITY: A NUMERICAL APPROACH
E
LH
ASSANEZERRIK
∗, H
AMIDBOURRAY
∗, A
LIBOUTOULOUT
∗∗
MACS Group. AFACS UFR, Moulay Ismail University Sciences Faculty, Meknes, Morocco
e-mail:
zerrik@fsmek.ac.maIn this paper we review the concept of regional boundary observability, developed in (Michelitti, 1976), by means of sensor structures. This leads to the so-called boundary strategic sensors. A characterization of such sensors which guarantees regional boundary observability is given. The results obtained are applied to a two-dimensional system, and various cases of sensors are considered. We also describe an approach which leads to the estimation of the initial boundary state, which is illustrated by simulations.
Keywords: observability, regional observability, regional boundary observability, boundary strategic sensors
1. Introduction
For a given distributed parameter system defined on a spatial domain Ω, we are interested in the knowledge of system states on the whole domain (Gilliam and Martin, 1988; Kobayashi, 1980). The regional observability con- cept, introduced by (El Jai et al., 1995), is focused on state observation on a given part ω of Ω. This concept was extended in (Zerrik et al., 1999) to the case where ω is located on the boundary of Ω. The approach, based on appropriate optimization techniques, shows how to con- struct the initial state on a part of the boundary, but the procedure can be adapted, in time, to observe the current boundary state on the same portion of the boundary.
The introduction of this concept is motivated by real situations. This is the case, e.g., in the energy exchange problem, where the aim is to determine the energy ex- changed in a casting plasma on a plane target which is perpendicular to the direction of the flow from measure- ments carried out by thermocouples (Fig. 1). It can also be of great help for a system which is not observable on the whole boundary ∂Ω of Ω, but observable on a part Γ ⊂ ∂Ω. For example, consider the system defined on Ω =]0, 1[×]0, 1[ by
∂h
∂t (x
1, x
2, t) = ∂
2h
∂x
21(x
1, x
2, t) + ∂
2h
∂x
22(x
1, x
2, t) in Ω×]0, T [,
∂h
∂ν (ξ, η, t) = 0 on ∂Ω×]0, T [, h(x
1, x
2, 0) = h
0(x
1, x
2) in Ω.
(1)
The measurements are given by the output function z(t) =
Z
Γ0
h(ξ, η, t)f (ξ, η) dξ dη, (2) where the boundary sensor (Γ
0, f ) is located in the sub- domain Γ
0= {0} × [0, 1] and f (ξ, η) = cos πη is the spatial distribution of the sensing measurements.
(1): torch of plasma (2): probe (of steel) (3): insulator Γ: face of exchange b1, b2: sensor locations
b1 b2
(3)
(2)
(1) •
Fig. 1. The problem of estimating the energy exchanged on Γ.
The state h
0(x
1, x
2) = cos(πx
1) cos(2πx
2) is not observable on ∂Ω but it is observable on Γ = [0, 1] × {0}. This shows that the regional boundary case is more general.
A regional boundary observability analysis has been
made from a purely theoretical viewpoint (Zerrik et al.,
1999), but the study may also become concrete, in some
sense, by using the structure of sensors, which form an
important link between the system and its environment,
have a passive role and allow the system evolution to be
measured. Their structure depends on the geometry, the
location of the support and the spatial distribution of the
sensing measurements.
Our interest is mainly focused on regional boundary observability linked with sensor structures, their location and number. Thus we give some fundamental results re- lated to sensors so that regional boundary observability can be achieved. This leads to the so-called boundary strategic sensors. The achieved results are also applied to two-dimensional diffusion systems. Secondly, we give a reconstruction method of the initial boundary conditions.
This is the purpose of this paper, which is organized as follows.
The second section is devoted to the presentation of the system under consideration, a brief recall of the re- gional boundary observability concept and its characteri- zation in terms of sensor structures. In Section 3 an appli- cation to a two-dimensional diffusion process is consid- ered. Examples of various situations are also given and specific results are summarized in tabular form. In Sec- tion 4 we develop a technical approach which leads to a state reconstruction algorithm. In the last section the pro- posed approach is successfully tested through computer simulations.
2. Regional Boundary Observability and Sensors
2.1. Problem Statement
Let Ω be a regular bounded open set of R
n(n ≥ 2) with boundary ∂Ω and Γ be a nonempty subset of ∂Ω, with positive Lebesgue measure. For a given T > 0, set Q = Ω×]0, T [ and Σ = ∂Ω×]0, T [. The system considered is described by the equation
(S)
∂y
∂t (x, t) = Ay(x, t) in Q,
∂y
∂ν
A(ξ, t) = 0 on Σ, y(x, 0) = y
0(x) in Ω,
(3)
where A is a second-order linear differential opera- tor with compact resolvent, which generates a strongly continuous semi-group (S(t))
t≥0on the state space X = H
1(Ω). A
∗indicates the adjoint operator of A and ∂y/∂ν
Adenotes the co-normal derivative associated with A.
Here y
0is supposed to be in H
1(Ω) and unknown, and the measurements are obtained through the output function
z(t) = Cy(x, t), (4)
C: H
1(Ω) −→ R
pbeing a bounded linear operator de- pending on the structure of the sensors considered. The observation space is O = L
2(0, T ; R
p).
We define the operator:
X −→ O, K : h −→ CS(·)h,
which is, in the zonal case, linear and bounded with an adjoint K
∗: O −→ X given by
K
∗z
∗= Z
T0
S
∗(s)C
∗z
∗(s) ds.
We also consider the trace operator of order zero γ
0: H
1(Ω) → H
12(∂Ω), which is linear, surjective and con- tinuous. γ
0∗denotes its adjoint and χ: H
12(∂Ω) −→
H
12(Γ) is the operator restriction to Γ when its adjoint is denoted by χ
∗.
Let y
01= χγ
0y
0be the restriction of the trace of y
0to Γ. Boundary regional observability explores the state reconstruction in the case where the subregion ω is a subset of the boundary ∂Ω. More precisely, we only have to reconstruct the component y
01of the unknown initial state. In this case we have to pay particular attention to space functions and the operators considered.
Definition 1. The system (3) together with the output (4) is said to be exactly (resp. approximately) regionally boundary observable on Γ if
Im(χγ
0K
∗) = H
12(Γ) (resp. Ker(Kγ
0∗χ
∗) = {0}). (5) Conventionally, we shall say that the system is exactly (resp. approximately) B-observable (B for the boundary) on Γ.
This is a natural definition of regional boundary ob- servability extending those given in (Curtain and Zwart, 1995; El Jai and Pritchard, 1988) to the case where we re- strict the state reconstruction to the boundary subregion Γ.
Obviously, we have the following assertions:
1. If the system (3) is exactly B-observable, then it is approximately B-observable.
2. If the system (4) is exactly (resp. approximately) B- observable on Γ, then it is exactly (resp. approxi- mately) B-observable on every subset Γ
1of Γ.
2.2. Boundary Strategic Sensor
The aim of this section is to give a characterization of sen- sors (number and location) in order for a system to be re- gionally approximately boundary observable.
Consider the system (3) and assume that the mea-
surements are made by p sensors (D
i, f
i)
1≤i≤p. The
output function is then given by z(t) = (z
1(t), . . . , z
p(t))
Regional boundary observability: A numerical approach 145
with z
i(t) = y(b
i, t), b
i∈ Ω for 1 ≤ i ≤ p in the point- wise case, and z
i(t) =
Z
Di
y(x, t)f
i(x) dx, D
i⊂ Ω for 1 ≤ i ≤ p in the zonal case.
Definition 2. A sequence of sensors (D
i, f
i)
1≤i≤pis said to be boundary strategic on Γ if the correspond- ing observed system is approximately B-observable on Γ. (In what follows we shall say that such sensors are Γ-strategic.)
Assume that there exists a complete set of eigen- functions (ϕ
mj)
m∈I; j=1,...,rmof A in H
1(Ω) associ- ated with the eigenvalues λ
mof multiplicities r
mand r = sup
m∈Ir
mis finite. For x = (x
1, . . . , x
n) ∈ Ω and m = (m
1, . . . , m
n) ∈ I, let x = (x
1, . . . , x
n−1) and m = (m
1, . . . , m
n−1). Suppose that the functions ψ
mj(x) = χγ
0ϕ
mj(x), m ∈ I, form a complete set in H
12(Γ).
Proposition 1. The sequence of sensors (D
i, f
i)
1≤i≤pis Γ-strategic if and only if
1. p ≥ r and 2. rank G
m= r
m, where
(G
m)
ij=
hϕ
mj, f
ii
L2(Di)in the zonal case, ϕ
mj(b
i) in the pointwise case, for 1 ≤ i ≤ p, 1 ≤ j ≤ r
m.
Proof. For brevity, the proof is limited to the case of zonal sensors. The techniques used constitute extensions of those given in (Fattorini, 1968) and are based on alge- braic manipulations.
For z
∗∈ H
12(Γ) we have Kγ
∗0χ
∗z
∗(t)
=
X
m∈I
e
λmtrm
X
j=1
hϕ
mj, f
ii
L2(Di)hϕ
mj, γ
∗0χ
∗z
∗i
H1(Ω)i=1,p
=
X
m∈I
e
λmtrm
X
j=1
hϕ
mj, f
ii
L2(Di)hχγ
0ϕ
mj, z
∗i
H12(Γ)
i=1,p
=
X
m∈I
e
λmtrm
X
j=1
hϕ
mj, f
ii
L2(Di)hψ
mj, z
∗i
H12(Γ)
i=1,p
.
If the system (3) together with the output (4) is not approximately B-observable on Γ, there exists z
∗6= 0 such that Kγ
0∗χ
∗z
∗= 0, which gives
rm
X
j=1
hψ
mj, z
∗i
H12(Γ)
hϕ
mj, f
ii
L2(Di)= 0,
∀ m ∈ I, 1 ≤ i ≤ p.
Consider
z
m=
hψ
m1, z
∗i
H12(Γ)
.. . hψ
mrm, z
∗i
H12(Γ)
.
Since G
mz
m= 0, it follows that rank G
m6= r
mfor any m.
Conversely, if rank G
m6= r
m, then there exists m ∈ I such that
z
m=
z
m1.. . z
mrm
6= 0
and G
mz
m= 0.
Let z
∗∈ H
12(Γ) be such that hψ
jk
, z
∗i
H12(Γ)
= 0 for j 6= m and
hψ
mk, z
∗i
H12(Γ)
= z
mkfor 1 ≤ k ≤ r
m. Therefore we have
rj
X
k=1
hf
i, ϕ
jki
Dihψ
jk
, z
∗i
H12(Γ)
= 0,
∀ j 6= m, 1 ≤ i ≤ p and
rm
X
k=1
hf
i, ϕ
mki
Dihψ
mk, z
∗i
H12(Γ)
= 0, 1 ≤ i ≤ p.
Thus there exists z
∗6= 0 (belonging to H
12(Γ)) such that Kγ
0∗χ
∗z
∗= 0, i.e. the system (3) together with the output (4) is not approximately B- observable on Γ.
Note that the proposition implies that the required number of sensors is at least equal to the highest multi- plicity of the eigenvalues.
Remark 1.
1. The above result remains also true for boundary sen- sors (pointwise or zonal cases).
2. By infinitesimally deforming the domain, the mul-
tiplicity of the eigenvalues can be reduced to one
(Michelitti, 1976). Consequently, the B-obser-
vability can be achieved by using only one sensor.
E. Zerrik et al.
146
3. Applications to Sensor Location
In this section, we present an application of the above results to a two-dimensional system defined on Ω = ]0, a[ × ]0, d[ by
∂y
∂t (x
1, x
2, t) = ∂
2y
∂x
21(x
1, x
2, t)+ ∂
2y
∂x
22(x
1, x
2, t) in Q,
∂y
∂ν (ξ
1, ξ
2, t) = 0 on Σ,
y(x
1, x
2, 0) = y
0(x
1, x
2) in Ω.
(6)
The output function is given by (4) and the sensors are considered to be pointwise or zonal, and located in the interior of the system domain Ω or on its boundary.
The eigenfunctions associated with the system (6) are of the form
ϕ
ij(x
1, x
2) = 2a
ij√
ad cos iπ x
1a
cos
jπ x
2d
with a
ij= (1 − λ
ij)
−12. They correspond to the eigen- values
λ
ij= − i
2a
2+ j
2d
2π
2of multiplicity one if a
2/d
26∈ Q. In this case the system (6) can be B-observable only by one sensor.
Let Γ =]0, a[×{0}, be the subregion target. In this case I = N
2, (i, j) = i, (x
1, x
2) = x
1, and the functions ψ
i(x
1) = p2/a cos(iπ(x
1/a)), i ∈ N, form a complete set in H
12(Γ).
The following results give information on the loca- tion of internal pointwise or zonal Γ-strategic sensors.
The case of boundary sensors is given in Tables 1 and 2.
3.1. Internal Pointwise Sensor
Consider the system (6) with the output function z(t) = y(b, t) where the pointwise sensor (b, δ
b) is located in- side the domain at a point b = (α, β) ∈ Ω (Fig. 2(a)).
(a) (b) (c)
Fig. 2. Location of a pointwise sensor.
Corollary 1. The sensor is not Γ-strategic if there exists k, l ∈ IN
∗such that 2kα/a or 2lβ/d is odd.
(a) (b) (c)
Fig. 3. Location of a zone sensor.
This shows that the regional boundary observability depends on the location of the sensor. We note that in real applications a sensor is considered as pointwise if the support area of the measure distribution is very small com- pared with the system domain.
3.2. Internal Zone Sensor
Here we consider the system (6) with the output func- tion z(t) = R
D
y(x
1, x
2, t)f (x
1, x
2) dx
1dx
2. Suppose that the sensor is located inside the domain Ω over D = ]α
1, α
2[×]β
1, β
2[ (cf. Fig. 3(a)), and f ∈ L
2(D) defines the spatial distribution of the sensing measurements on D.
For 0 < α
1< α
2< a and 0 < β
1< β
2< d, set
η
1= α
1+ α
22 , η
2= β
1+ β
22 ,
µ
1= α
2− α
12 , µ
2= β
2− β
12 .
Corollary 2.
1. If f is uniformly distributed on D, then the sensor is not Γ-strategic if one of the following properties is satisfied: µ
1/a ∈ Q or µ
2/d ∈ Q, or there exists k, l ∈ N
∗such that 2kη
1/a or 2lη
2/d is odd.
2. If f is symmetric with respect to the point (η
1, η
2), then the sensor is not Γ-strategic if η
1/a ∈ Q or η
2/d ∈ Q.
3. If f is symmetric with respect to the axis x = η
1(or with respect to the axis y = η
2), then the sensor is not Γ-strategic if there exists k ∈ N
∗such that 2kη
1/a is odd (resp. there exists l ∈ N
∗such that 2lη
2/d is odd).
Note that the regional boundary observability de-
pends on the geometry of the sensor support and the mea-
surement function. For the case where the sensor is lo-
cated on the boundary, we obtain analogous results (see
Tables 1 and 2).
Regional boundary observability: A numerical approach 147
Remark 2.
1. Note that the sensor support given above corresponds to a real geometry of a sensor in the diffusion system.
The hypothesis of symmetry and uniform distribu- tion are physically realistic. For example, this would be the case if the sensor was evenly distributed over its support (f = δχ
D, where χ is the characteristic function and D ⊂ Ω is the zone in which the mea- surements are carried out).
2. From a practical point of view, the distributed system is most often approximated by a finite-dimensional system. Then the conditions of the B-observability on Γ can be also verified for the finite-dimensional system. For instance, in the pointwise sensor case, if the system is approximated by a tree- dimensional system, then the condition of the non B-observability on Γ is α/a ∈ I
3where I
3= {1/6, 1/4, 1/2, 3/4, 5/6} for all β ∈]0, d[.
3.3. Recapitulating Tables
In this subsection we present the established results of the previous subsections in tabular form. The cases of bound- ary sensors (pointwise and zonal) are also considered.
Table 1. Pointwise sensor.
Sensor location Non -strategic cases α ∈]0, a[ and β = 0 or β = d ∃ k ∈ N
∗| 2kα/a is odd β ∈]0, d[ and α = 0 or α = a ∃ l ∈ N
∗| 2lβ/d is odd
α ∈]0, a[ and β ∈]0, d[ ∃ k ∈ N
∗| 2kα/a is odd or
∃ l ∈ N
∗| 2lβ/d is odd
For the pointwise case, the regional boundary ob- servability depends on the location of the sensor. For the zonal case, it depends on the form of the sensor support and its location (location of the support centre), as well as on the measurement function.
4. Reconstruction Method
In this section, we present an approach which allows the determination of the regional boundary initial condition y
01on Γ , based on the internal regional observability.
The method is an extension of those given in (El Jai and Pritchard, 1988; El Jai et al., 1993; Kobayashi 1980).
Let us consider the system (3) with the output (4) on the same assumptions as in Section 2, and let ω be an open subset of Ω, regular and of positive Lebesgue mea- sure such that Γ ⊂ ∂Ω ∩ ∂ω. If the system (3) together
Table 2. Zonal sensor.
Sensor location Non -strategic cases
?f uniformly distributed on D µ
1/a ∈ Q or ∃ k ∈ N
∗| 2kη
1/a is odd.
?f symmetric with respect to (η
1, 0) or (η
1, d)
D =]α
1, α
2[×{0} or η
1/a ∈ Q
=]α
1, α
2[×{d} ?f symmetric with respect to the axis x = η
1∃ k ∈ N
?| 2kη
1/a is odd
?f uniformly distributed on D D = {0}×]β
1, β
2[ or µ
2/d ∈ Q or ∃ l ∈ N
∗| 2lη
2/d is odd.
= {a}×]β
1, β
2[ ?f symmetric with respect to (0, η
2) or (a, η
2),
η
2/d ∈ Q
?f symmetric with respect to the axis y = η
2∃ l ∈ N
?| 2lη
2/d is odd
?f uniformly distributed on D µ
1/a ∈ Q or µ
2/d ∈ Q or
∃ k ∈ N
?| 2kη
1/a is odd or D =]α
1, α
2[×]β
1, β
2[ ∃ l ∈ N
?| 2lη
2/d is odd
?f symmetric with respect to (η
1, η
2) η
1/a ∈ Q or η
2/d ∈ Q
?f symmetric with respect to the axis x = η
1(resp. y = η
2)
∃ k ∈ N
?| 2kη
1/a is odd (resp. ∃ l ∈ N
?| 2lη
2/d is odd).
with (4) is approximately observable on ω, then it is ap- proximately B-observable on Γ (Zerrik et al., 1999).
This result links the internal regional observability on ω developed by El Jai et al. in (1993) to the regional B-observability on Γ (which is part of ∂ω).
Let the initial state be decomposed in the following form:
y
0=
y
02on ω, y
03on Ω \ ω.
The problem consists in reconstructing the initial state y
02on ω and determining its trace y
10on Γ. Let us go further
in the state reconstruction by considering various types of
sensors.
4.1. Pointwise Measurements
In this case the output function is given by
z(t) = y(b, t), (7)
where b ∈ Ω denotes the given location of the sensor.
The problem consists in constructing the component y
02of the initial state on ω with the knowledge of (3)–(7).
For that purpose, we consider the set
G = g ∈ H
1(Ω) such that g = 0 on Ω\ω . (8) For a given ϕ
0∈ G , the system
∂ϕ
∂t (x, t) = Aϕ(x, t) in Q,
∂ϕ
∂ν
A(ξ, t) = 0 on Σ, ϕ(x, 0) = ϕ
0(x) in Ω
(9)
has a unique solution ϕ ∈ L
2(0, T ; H
1(Ω)) ∩ C
0(0, T ; L
2(Ω)) and the mapping
ϕ
0∈ G → kϕ
0k
2G= Z
T0
ϕ
2(b, t) dt (10) defines a semi-norm on G.
If the system (9) is approximately observable on ω, the mapping (10) defines a norm on G (Zerrik et al., 1999). We also denote by G the completion of G.
For ϕ
0∈ G, (9) gives ϕ, which allows us to consider the system
∂Ψ
1∂t (x, t) = −A
∗Ψ
1(x, t)
−ϕ(b, t)δ(x − b) in Q,
∂Ψ
1∂ν
A∗(ξ, t) = 0 on Σ,
Ψ
1(x, T ) = 0 in Ω.
(11)
Let Ψ
1be the solution of ( 11) and consider the operator G −→ G
∗,
∧ : ϕ
0−→ P (Ψ
1(0)), (12)
where P denotes the projection on G
∗. Now consider the system
∂Ψ
2∂t (x, t) = −A
∗Ψ
2(x, t)
−y(b, t)δ(x − b) in Q,
∂Ψ
2∂ν
A∗(ξ, t) = 0 on Σ,
Ψ
2(x, T ) = 0 in Ω.
(13)
If ϕ
0is such that ϕ leads to Ψ
1(0) = Ψ
2(0) on ω, then the system (13) looks like the adjoint of the system to be observed (3)–(7) and, consequently, the observation problem on ω is equivalent to solving the equation
∧ ϕ
0= P Ψ
2(0). (14) If we assume that the operator A has a complete set of eigenfunctions (ϕ
i) in H
1(Ω), then we have the fol- lowing result:
Proposition 2. If the system (3) together with the output function (7) is approximately observable on ω, then (14) has a unique solution ϕ
0∈ G and the regional boundary initial state to be observed on Γ is given by y
01= χγ
0ϕ
0. Sketch of the proof. The proof proceeds in the following two steps:
Step 1. We show that the map ϕ
0−→ kϕ
0k
2Gdefines a norm on G using the fact that the sensor (b, δ
b) is ω-strategic (Amouroux et al., 1994).
Step 2. We prove that the operator ∧ is an isomor- phism. It is sufficient to multiply the result by ϕ and integrate it on Q. Using the Green formula, we ob- tain hϕ
0, ∧ϕ
0i = kϕ
0k
2G, which proves that ∧ is an isomorphism and then (14) has a unique solution.
For more details, see (Zerrik et al., 1999).
4.2. Case of Zone Measurements
Let us come back to the system (3) and suppose that the measurements are given by an internal zone sensor defined by (D, f ) with D ⊂ Ω and f ∈ L
2(D). The system is augmented with the output function
z(t) = Z
D
y(x, t)f (x) dx. (15) In this case we consider the system (9), G being given by (8), and the mapping
ϕ
0∈ G → kϕ
0k
2G= Z
T0
hf, ϕ(t)i
2L2(D)dt, (16)
which is a semi-norm on G. Thus with the system
∂Ψ
1∂t (x, t) = −A
∗Ψ
1(x, t) − hf, ϕ(x, t)i
2L2(D)×f (x)χ
Din Q,
∂Ψ
1∂ν
A∗(ξ, t) = 0 on Σ, Ψ
1(x, T ) = 0 in Ω
(17)
Regional boundary observability: A numerical approach 149
we introduce the operator ∧: ϕ
0∈ G −→ P (Ψ
1(0)) and consider the system
∂Ψ
2∂t (x, t) = −A
∗Ψ
2(x, t)
−z(t)f (x)χ
Din Q,
∂Ψ
2∂ν
A∗(ξ, t) = 0 on Σ,
Ψ
2(x, T ) = 0 in Ω.
(18)
The observation problem on ω reduces to solving the equation
∧ ϕ
0= P (Ψ
2(0)). (19) Proposition 3. If the system (3) together with the output function (15) is approximately observable on ω, then (19) has a unique solution ϕ
0∈ G and the regional boundary initial state to be observed on Γ is given by y
01= χγ
0ϕ
0. The proof is (with some minor technical modifica- tion) similar to the pointwise case. For more details, we refer the reader to (Zerrik et al., 1999).
5. Simulations
We have seen that the regional boundary observability is equivalent, in all cases, to solving the equation
∧ ϕ
0= P Ψ
2(0). (20) The numerical approximation of (20) is realized easily when one can have a basis ( ˜ ϕ
i) of H
1(Ω) and the idea is to calculate the components ∧
ijof the operator ∧. Then we approximate the solution of (20) by the linear system
N
X
j=0
∧
ijϕ
0,j= Ψ
2,
ifor i = 0, . . . , N, (21)
where N is the order of approximation and Ψ
2,
iare the components of P (Ψ
2(0)) in the basis ( ˜ ϕ
i) considered.
Assume that ( ˜ ϕ
i) is the set of eigenfunctions of the oper- ator A
∗associated with the eigenvalues λ
iof multiplic- ity one.
In the case of pointwise measurements we have
h∧ϕ
0, ϕ
0i=
∞
X
i,j=0
h ˜ ϕ
i, ϕ
0i
H1(ω)h ˜ ϕ
j, ϕ
0i
H1(ω)× −1 + e
(λj+λi)Tλ
j+ λ
iϕ ˜
i(b) ˜ ϕ
j(b).
Then the components of ∧ are given by
∧
ij= −1 + e
(λj+λi)Tλ
j+ λ
i˜
ϕ
i(b) ˜ ϕ
j(b). (22)
In the zonal case the same developments lead to the com- ponents of ∧:
∧
ij= e
(λi+λj)T− 1 λ
j+ λ
ihf, ˜ ϕ
ii
L2(D)hf, ˜ ϕ
ji
L2(D). (23) Summing up, in the pointwise case, the regional recon- struction is obtained via the following simplified algo- rithm:
Step 1. Choose a sensor location b, an error test ε, an initial state y
0.
Step 2. Choose an approximation order N . Step 3. Solve (13) to obtain Ψ
2(x, 0).
Step 4. Solve (21) to obtain ϕ
0where ∧
ij’s are given by (22).
Step 5. If ky
0− ϕ
0k
2L2(ω)> ε, go to Step 2, oth- erwise ϕ
0corresponds to the initial state to be ob- served on ω.
This algorithm converges since the developments are based on a Dirichlet series. To avoid instabilities in nu- merical calculations, which are suspected to arise, we must take some care with the numerical method for solv- ing the linear system (21) and also with the choice of the sensor location.
5.1. Example
For an illustrative application, consider the parabolic sys- tem on Ω =]0, 1[×]0, 1[ given by
∂y
∂t (x
1, x
2, t) = 0.01 ∂
2y
∂x
21(x
1, x
2, t) + ∂
2y
∂x
22(x
1, x
2, t)
in Q,
∂y
∂ν (ξ, η, t) = 0 on Σ,
y(x
1, x
2, 0) = y
0(x
1, x
2) in Ω.
(24)
The measurements are given by a pointwise sensor z(t) = y(b, t), with b = (0.30, 0.65) and T = 2. Here the boundedness of multiplicities r
mdoes not hold, but it holds for Ω
?=]0, 1 + ε[×]0, 1[ (ε 6∈ Q small enough), which constitutes a good approximation of Ω (Michelitti, 1976)), and for which the outlined results are applicable.
We note that numerically an irrational number does not exist but it can be considered as irrational if the truncation number exceeds the desired precision.
Let Γ = {0} × [0, 1] and
y
10(η) = 2 η
33 − η
22 + 0.1
E. Zerrik et al.
150
Fig. 4. True initial state y
20on ω.
Fig. 5. Estimated state y
20,eon ω.
be the initial state to be observed on Γ. Let ω = ]0, 0.24[×]0, 1[ and
y
02(x
1, x
2) = 2 x
313 − x
212 + 0.1 x
323 − x
222 + 0.1
be an extension of y
01to ω. Applying the delineated ap- proach, we obtain the results given in Figs. 4 and 5.
Figure 6 shows that the estimated boundary state is very close to the true initial boundary state on Γ. The initial state y
10is estimated with the reconstruction error ky
01− y
0,e1k
2L2(Γ)= 4.951 × 10
−7.
5.2. Subregion-Pointwise Actuator
The following simulation results show the evolution of the estimated state error with respect to the sensor location b = (b
1, b
2) when b
1is fixed at 0.81 and b
2∈]0, 1[.
Figure 7 reveals the following facts:
• For a given subregion Γ, there is an optimal sensor location (optimal in the sense that it leads to a solu- tion which is very close to the initial boundary state).
• When a sensor is located sufficiently far from the subregion Γ, the estimated state error is constant for any locations.
0 ,0 0 ,2 0 ,4 0 ,6 0 ,8 1 ,0
-0 ,0 2 -0 ,0 1 0 ,0 0 0 ,0 1 0 ,0 2
Fig. 6. True (y
10) (dashed line) and estimated y
0,e1(continuous line) initial state on Γ.
0 ,0 0 ,1 0 ,2 0 ,3 0 ,4 0 ,5 0 ,6 0 ,7 0 ,8 0 ,9 1 ,0
0 ,0 0 ,5 1 ,0 1 ,5
