Sentinels for the Identification of Pollution in Domains with Missing Data

REZZOUG IMAD, ACHAB FATMA

Department of Mathematics and Computer Science,

University of Oum El Bouaghi,

Laboratory of Dynamics systems and control,

ALGERIA

Abstract: Sentinel method introduced in the study of problems with incomplete data, particularly in the context

of distributed systems where pollution terms may arise at the boundary. The idea of sentinel likely involves

constructing a surrogate or placeholder value that helps account for missing data or uncertainties in the system.

Weakly sentinel appears to be a modification or extension of the concept of a sentinel specifically tailored for

estimating pollution terms in distributed systems with missing data. The term weakly might suggest that this

sentinel is not as robust or precise as the ideal sentinel, but it serves a similar purpose in providing estimates or

approximations in situations where complete data is not available.

Key-Words: Distributed system ; Controllability ; Optimal control ; Pollution term ; Missing term ; Sentinel

method.

Received: April 19, 2023. Revised: December 21, 2023. Accepted: December 28, 2023. Published: December 31, 2023.

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Several domains are modeled by dynamic systems. A

dynamic system refers to a system that changes over

time, where the behavior of the system is determined

by its current state as well as its history. These sys-

tems are pervasive in various fields including physics,

engineering, biology, economics, and social sciences.

Characteristics of dynamic systems include :

Change over Time, State Variables, Feedback Loops,

Nonlinearity, Complex Behavior [1], [2], [3], [4], [5].

Examples of dynamic systems include : Mechan-

ical Systems, Electrical Circuits, Biological Systems,

Economic Systems, Social Systems [6], [7], [8], [9],

[10].

Analyzing dynamic systems often involves mathe-

matical modeling, simulation, and numerical methods

to understand their behavior and predict future states.

Control theory is a branch of engineering and mathe-

matics that deals specifically with the control and reg-

ulation of dynamic systems [11], [12], [13], [14].

Bellow we Present the organization of our work.

In the first section, we present the notion of sen-

tinel and optimal control theory :

The Sentinel Method introduced by Lions pro-

vides a powerful framework for solving complex

boundary value problems, especially when traditional

methods based on fixed boundary conditions may not

be applicable or effective. It offers flexibility in han-

dling uncertain or evolving boundary conditions and

has found applications in various fields, including

fluid dynamics, solid mechanics, heat transfer, and

electromagnetic [15], [16], [17], [18], [19], [20], [21],

[22].

It’s important to note that the Sentinel Method

is a sophisticated mathematical technique and may

require advanced knowledge of partial differential

equations, numerical analysis, and functional analysis

for its implementation and understanding [23], [24],

[25], [26], [27], [28], [29], [30].

Optimal control theory finds applications in a

wide range of fields, including aerospace engineer-

ing, robotics, economics, finance, manufacturing, and

process control. It provides a powerful framework

for designing control strategies that optimize perfor-

mance, efficiency, and resource utilization in com-

plex dynamical systems [31], [32], [33], [34], [35],

[36], [37], [38], [39].

While these two concepts might seem unrelated

at first glance, there could be scenarios where opti-

mal control theory could be applied to design con-

trol strategies for processes or systems that are being

monitored using the sentinel method. For example, in

a manufacturing setting, optimal control techniques

could be used to adjust process parameters in real-

time based on feedback from sentinel units to opti-

mize some performance criterion, such as minimizing

defects or maximizing throughput .

In the second section, we introduced the approxi-

mate controllability :

Approximate controllability provides a more re-

laxed notion of controllability that is often more feasi-

ble to achieve in practice, especially for systems with

inherent uncertainties or limitations. It allows for

practical control strategies that can effectively steer

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a system towards desired states while accounting for

real-world constraints and imperfections. [40], [41],

[42], [43].

Let be T > 0,and Ωan open subset of Rnof

smooth boundary ∂Ω=Γand D0⊂Γ, Let O ⊂ Ω,

considered as an observatory. We define Ωαopen

“neighbor” of Ωof boundary ∂Ωα= (Γ − D0)∪ Dα.

Where Dαis defined starting from D0like the lo-

cus of the points Dα={x+αβ (x)ν(x), x ∈ D0}.

We denote by νthe outer normal on Γ,αsmall real

parameter, and βis a C1function on D0with |β(x)| ≤

1, β = 0 on ∂D0.

We consider the parabolic evolution equation :











y′+Ay +h(y)|Qα=Ωα×]0,T [= 0

y|Σ0=Γ0×]0,T [=f+λ

y|Σ\Σ0= 0

y(0) |Ωα=y0+τ

(1)

where y=y(x, t;λ, τ),and where Γ0∩Dτ=.We

assume here that h:R−→ Ris of class C1,the func-

tions y0and fare known with y0∈L2(Ωα).But, the

terms : τ

y0(so-called missing term) and λ

f(so-called

pollution term) are unknown, 

y0and 

fare renormal-

ized and represent the size of missing and pollution

∥

y0∥L2(Ωα)≤1, τ ∈R“small”.

The observation is yon O,for the time T. we de-

note by yobs this observation

yobs =m0∈L2(O × ]0, T [) .(2)

We suppose that (1) has a unique solution denoted

by y(λ, τ) := y(x, t;λ, τ)in some relevant space.

The question is

(q) : how to calculate the pollution term, indepen-

dently from the variation missing term ?.

Least squares. Least squares is a powerful and

versatile technique that is widely used in data analy-

sis and regression modeling due to its simplicity, ro-

bustness, and efficiency. It provides a systematic way

to estimate parameters of mathematical models from

noisy or imperfect data, making it a fundamental tool

in statistical analysis and scientific research.

Sentinels. The Sentinel Method introduced by

Jacques-Louis Lions, a renowned French mathemati-

cian, is a mathematical approach used primarily in the

field of partial differential equations (PDEs) for solv-

ing boundary value problems (BVPs). Jacques-Louis

Lions made significant contributions to various areas

of mathematics, including functional analysis, control

theory, and numerical analysis, and his work laid the

foundation for modern PDE theory and its applica-

tions.

In the context of partial differential equations, the

Sentinel Method introduced by Lions involves a tech-

nique for solving problems where the boundary con-

ditions are unknown or partially known. The method

is particularly useful when the boundary conditions

depend on the solution itself or when the boundary

conditions are uncertain.

The sentinel concept relies on the following three

objects: some state equation (1), some observation

function (2), and some control function uto be de-

termined.

J.L.Lions calls a “sentinel”, a functional S(.)

which is the scalar product of the measure yobs and

a function u. It is built to get some information on the

pollution term.

2Presentation of the Method

Proposition 1 (definition, existence and uniqueness

of the sentinel)

We now consider the sentinel method of Lions

which is an other attempt and brings better answer

to question (q), as we will explain now :

Let h0be some function in L2(O × ]0, T [). Let on

the other hand ωbe some open and non empty subset

of Ω.

For a control function uϵ∈L2(ω×]0; T[), we

define the functional

S(λ, τ) = Q

(h0χO+uϵχω)y(λ, τ)dxdt (3)

where y(λ, τ) = y(x, t;λ, τ)is the solution of

(3), and the function uϵare to be found in such a way

that

for all ϵ > 0there exists uϵ∈L2(ω×]0; T[) such

∥uϵ∥L2(ω×]0;T[) =min ∥v∥;v∈ U,(4)

where U=v∈L2(ω×]0; T[) ; ∂

∂τ S(0,0) = 0.



∂

∂τ S(0,0)≤ϵ, ∀

y0,05 (5)

∂

∂τ S(0,0) = 0,∀

y0; (ϵ−→ 0) .(6)

Then S(λ, τ)defined by (3) (4) (6) exists and is

unique (that means the existence and uniqueness of

the function uϵ).

It will take two steps :

1/ The conditions (4) (6) will be rewritten into a

control problem,

2/ An weakly controllability result will be proved,

First step :

We consider the functions y0which solve problem

(1) for λ= 0 and τ= 0 :











∂

∂t y0+Ay0+h(y0)|Qα=Ωα×]0,T [= 0,

y0|Σ0=Γ0×]0,T [=f,

y0|Σ\Σ0= 0,

y0(0) |Ωα=y0,

(7)

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because of (3) we can write

S(0,0) = Q

(h0χO+uϵχω)y0(x, t)dxdt,

is known. One carries out a development of Taylor

of Sin the vicinity of (0,0)

S(λ, τ)≃ S(0,0) + λ∂S

∂λ (0,0) + τ∂S

∂τ (0,0) ,

for τsmall.

And

∂S

∂λ (0,0) = Q

(h0χO+uϵχω)yλ(x, t)dxdt,

∂S

∂τ (0,0) = Q

(h0χO+uϵχω)yτ(x, t)dxdt,

and yλ(x, t)is the solution of











∂

∂t yλ+Ayλ+h′(y0)yλ= 0,

yλ|Σ0=

yλ|Σ\Σ0= 0,

yλ(0) = 0,

(8)

and yτ(x, t)is the solution of











∂

∂t yτ+Ayτ+h′(y0)yτ|Ω×]0,T [= 0,

yτ|D0×]0,T [=−β∂y0

∂ν ,

yτ|Σ\D0×]0,T [= 0,

yτ(0) = 0.

(9)

To build the sentinel, one must determine uϵwhich

ensures the condition (4), (6) for a given positive ϵ.

Adjoint state :

Assume that ∂y

∂τ can be defined for λ=τ= 0.

Then, the yτsolves the problem (9).

If yτand y0solve respectively (9) and (7), then the

insensibility condition (6) is equivalent to

∂S

∂τ (0,0) = Q

(h0χO+uϵχω)yτ(x, t)dxdt,

(10)

∀

y0,∥

y0∥L2(Ωα)≤1.

Let q=q(x, t)be the solution of the following

adjoint problem :







−q′+A∗q+h′(y0)q=h0χO+uϵχω,

q|Σ= 0,

q(T) = 0.

(11)

As for the problem (9), the problem (11) has a

unique solution q. The function qdepends on the con-

trol uϵthat we shall determine :

Indeed, if we multiply the first equation in (11) by

yτ,and we integrate by parts, lead to

It is seen that the conditions (10), and for ϵ−→ 0

one gets :

∂S

∂τ (0,0) = −Σ

∂q

∂ν∗

yτdΣ(12)

=D0×]0,T [

β∂y0

∂ν

∂q

∂ν∗

dΣ

= 0.

This equality must take place for any regular func-

tion α, with |α(x)| ≤ 1, α = 0 on ∂D0.That is

equivalent to

T

∂y0

∂ν

∂q

∂ν∗

dt = 0; ∀x∈ D0.(13)

The problem thus now to find uϵin U=

L2(ω×]0; T[) .

Such that one has (12), et (6).

This is a controllability problem.

Equivalent controllability problem :

For that one breaks up the system (11) into two

systems:







−q′

0+A∗q0+h′(y0)q0=h0χO,

q0(T) = 0,

q0|Σ= 0,

(14)

and 





−z′+A∗z+h′(y0)z=uϵχω,

z(T) = 0,

z|Σ= 0.

(15)

Thus q=q0+zsuch as q0is thus given. Then

one seeks uϵso that z=z(uϵ)who checks

T

∂y0

∂ν

∂q

∂ν∗

dt =−T

∂y0

∂ν

∂q0

∂ν∗

dt, (16)

on D0.

If it is considered here that

uϵ=function of control.

z=state of one (new) system.

That is to say q0(0) the desired state given by the

resolution of the system (14), the problem of regional

controllability consists in finding, for all ϵ > 0a con-

trol uϵof the space of control U=L2(O × (0, T ))

allowing to approach with ϵmeadows, in a time fin-

ished, the state z(t)of the system (15) of an initial

state z(T) = 0, in a desired final state q0(0) on Ω

(see: [44]).

Second step :

Penalization and system of optimality

For ϑ > 0, consider the function Jϑdefined by

Jϑ(uϵ, z) = 1

2T

0ω

ϵdxdt +(17)

2ϑ∥Ξ∥2

L2(Ω×]0,T [) ,

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such that Ξ = −z′+A∗z+h′(y0)z−uϵχω.

Where one posed z′=∂z/∂t

In (17), one considers all zsuch that











−z′+A∗z+h′(y0)z∈L2(Ω ×(0, T )) ,

z(T) = 0; z|Σ= 0,

T

∂y0

∂ν

∂q

∂ν∗dt |D0=−T

∂y0

∂ν

∂q0

∂ν∗dt.

(18)

Let uϑ

ϵ, zϑthe solution of

inf Jϑ(uϵ, z).

One poses moreover

ρϑ=1

ϑ−∂zϑ/∂t +A∗zϑ+h′(y0)zϑ−χωuϑ

ϵ.

The couple uϑ

ϵ, zϑis characterized by :

 ω×(0,T )

uϑ

ϵ.

uϵdxdt +(19)

 Ω×(0,T )

ρϑ(Ξ) dxdt

= 0,

such that Ξ = −∂

z/∂t +A∗

z+h′(y0)

z−χω

uϵ.

∀

uϵand ∀

zsuch that

T

∂y0

∂ν

∂

∂ν∗

dt |D0= 0.(20)

One thus has



−∂ρϑ/∂t +Aρϑ+h′(y0)ρϑ= 0,

ρϑ(0) = 0,

ρϑ|Σ\D0×]0,T [= 0.

(21)

And

∂ρϑ

∂ν =σϑ∂y0

∂ν (x.t)|D0×]0,T [.(22)

For any σϑ.So that (19) becomes

uϑ

ϵ=χωρϑ.(23)

System of optimality : (ϑ→0) :

For ρ0∈L2(Ω) and for σregular function, one

defines ρsolution of



ρ′+Aρ +h′(y0)ρ|Ω×(0,T )= 0,

ρ(0) = 0,

ρ|D0×]0,T [=σ(x)∂y0

∂ν ,

ρ|Σ\D0×]0,T [= 0,

(24)

One defines then zby



−z′+A∗z+h′(y0)z=χωρ,

z(T) = 0,

z|Σ= 0.

(25)

One seeks σso that

T

∂y0

∂ν

∂z

∂ν∗

dt |D0=−T

∂y0

∂ν

∂q0

∂ν∗

dt. (26)

We now define a linear operator Λby

Λσ|D0=T

∂y0

∂ν

∂z

∂ν∗

dt. (27)

And

Mh0=T

∂y0

∂ν

∂q0

∂ν∗

dt.

It remains to solve (26).

Multiplying (25) by ρ, we obtain after integrating

by part

⟨Λσ, σ⟩= ω×(0,T )

ρ2dxdt.

What results in introducing

∥σ∥F= ω×(0,T )

ρ2dxdt1

.(28)

One indicates by Fthe space of Hilbert separate

and supplemented regular functions σfor the norm

(28).

Λ∈ L (F, F ′)is an isomorphism of Fon F′, and

Λ∗= Λ ;F′being the dual space of F.

The equation (26) is written

Λσ=−Mh0,

from where

σ=−Λ−1Mh0,(29)

subject checking that

Mh0∈F′.(30)

But if one multiplies (14) by ρ, one sees that

⟨Mh0, σ⟩= (h0, χωρ)L2(Q).(31)

from where (30) with

∥Mh0∥F′≤ ∥h0∥L2(Q).

therefore, the sought sentinel is given by

S(λ, τ) = Q

(h0χO+uϵχω)y(λ, τ)dxdt

=Q

Ξy(λ, τ)dxdt,

such that Ξ = h0χO− M∗Λ−1Mh0χω.

In what follows we apply the preceding result to

estimate the term of pollution of the system (1).

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3A use of the concept of sentinel:

The identification of the unknown

pollution term

Remark 2 If the semigroup S∗(t)generated by the

operator A∗is compact in L2(Ω) ,the system (15) is

not exactly controllable [44].

Remark 3 There are systems which are weakly con-

trollable but they are not exactly controllable.

Example 4 Ωan open subset of Rnof smooth bound-

ary ∂Ω, we consider here the state equation:







y′−∆y|Q=v,

y(x, 0) |Ω= 0,

y(x, t)|Σ= 0.

The system above is a particular case of system

(15); indeed, it is enough to take A∗= ∆ when

y∈D(Ω) = H2(Ω) ∩H1

0(Ω) ,O= Ω, v =uϵ∈

L2(U). This system cannot be exactly controllable

in L2(Ω) because the semigroup S∗(t)generated by

A∗= ∆ is compact, but it is exactly controllable in

0(Ω) [44].

These two remarks led us to introduce the notion

of the sentinel to estimate the term of pollution inde-

pendently of the missing term. It is supposed that the

system (15) is not exactly controllable thus the fol-

lowing theorem shows the interest of weakly control-

lability in the construction industry of the sentinels.

Theorem 5 If the system (15) is weakly controllable

then for all ϵpositive it exists a function uϵ∈

L2(ω×(0, T )) who checks the conditions (4), (6) of

the proposition (1).

its shows already.

Theorem 6 Since the system (15) is weakly control-

lable on Ωthen one has

Σ0

∂

∂ν∗q(h0)λ

fdΣ≤

Q(h0χO+uϵχω)|m0−y0|dxdt +τϵ,

where y0(x, t)is the solution of (7) and m0is the

state observed on Oduring the interval of time (0, T ).

that is to say S(λ, τ)the sentinel defined by h0

thus

λ∂S

∂λ (0,0)

=λQ(h0χO+uϵχω)yλ(x, t)dxdt

=S(λ, τ)− S (0,0) −τ∂S

∂τ (0,0) .

And on the observatory Oone poses y=m0then

λ∂S

∂λ (0,0)

=Q(h0χO+uϵχω) (m0−y0)dxdt −

τ∂S

∂τ (0,0) ,

where yλ(x, t)is the solution of (8).

Now, we designate as q(h0)the unique solution of

(11) depending on h0.

Multiplying (11) by yλ,we obtain after integrating

by part

Σ0

∂

∂ν∗q(h0)λ

fdΣ

=λQ(h0χO+uϵχω)yλ(x, t)dxdt,

and in addition one has

∂S

∂τ (0,0)

=Q(h0χO+uϵχω)yτ(x, t)dxdt

= 0,for ϵ−→ 0.

It results that the unknown pollution term λ

fcan

be defined as follows

Σ0

∂

∂ν∗q(h0)λ

fdΣ

=S(λ, τ)− S (0,0) −τ∂S

∂τ (0,0)

≤Q(h0χO+uϵχω)|m0−y0|dxdt +τϵ

≤Qh0χO− M∗Λ−1Mh0χω|m0−y0|dxdt+

τϵ,

thus, the proof of Theorem.

Acknowledgements

The authors are grateful to the reviewers for their

valuable and insightful comments.

Availability of data and materials

This paper does not use data and materials.

Ethics approval and consent to

participate

The authors reveal that there is no ethical problem in

the production of this paper.

Consent for publication

The authors want to publish this paper in this journal.

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