Comparison between the two HUM and no-regret control methods

B28$), N$',$ M(5$%7, N(65,1( L$0<$ R(==28* I0$'

Department of Mathematics and Computer Science

University of Oum El Bouaghi,

Laboratory of Dynamics systems and control

$/*(5,$

Abstract: We present in our work the steps of the Hilbert Uniqueness Method (HUM) and characterization of the

low regret and no regret control. At the end of this article, we compare these methods.

Key-Words: Distributed system ; Incomplete data ; Low-regret control ; no-regret control ; HUM method ;

Optimality system.

Received: January 2, 2023. Revised: September 4, 2023. Accepted: October 9, 2023. Published: November 8, 2023.

1 Introduction

Several domains are modeled by dynamic or station-

ary systems[1], the sentinel theory2, 3, 4, 5, 6, 7,

8, 9, 10, 11, 12, 13]is an important tool for the iden-

tification of some system data based on control the-

ory[14, 15, 16, 17], control plays an interesting role

in resolving the different systems in the different do-

mains.

Bellow we Present the organization of our mem-

ory.

In the first section, we present a description of the

HUM method for solving the problem of the control-

lability system.

In the second section, we present the standard

optimal control theory and we consecrated to study

the notion of no-regret control and low-regret of dis-

tributed system[18, 19, 20].

Finally, we conclude by comparison between the

HUM method and the low regrets control method.

2 Hilbert Uniqueness Method

The construction of the Hilbertian spaces adapted to

the building of the system according to the criteria of

the specific uniqueness of the homogeneous system

associated with it, and the method adopted for that

is Hibert Uniqueness Method (HUM), the following

algorithm describes the basics of applying the HUM

method to solving the problem of exact system con-

trollability.

The basic idea is the following :

Assuming that the system is exactly controllable,

characterize the control that minimizes the associated

cost function among the set of admissible controls by

an optimality system.

2.1 Exact controllability and penalization

2.1.1 Orientation

Let be Ωa bounded domain of Rn,n≥1, at the

border Γof class C2.

We consider the wave equation

y”−∆y= 0,(1)

in Q= Ω ×[0, T ]with T > 0fixed.

We assume that we can act on the system through

the intermediary of the control v=v(x, t)on the edge

Σ = Γ ×[0, T ], so that

y=v, (2)

on Σ.Let the initial data be

y(x, 0) = y0(x); y′(x, 0) = y1(x),(3)

on Ω.Let x0∈Rn,m(x) = x−x0and

R(x0) = max |m(x)|, x ∈¯

Ω.

Consider the usual partition of the boundary

Γ(x0) = x∈Γ/m(x).v(x)>0,

Γ∗(x0) = Γ\Γ(x0),

and

Σ(x0) = Γ(x0)×[0, T ],

Σ∗(x0) = Σ/Σ(x0).

Let be the exact controllability of the following

equation.

If T > T (x0)=2R(x0)for each pair of initial

data (y0, y1)∈L2(Ω) ×H−1(Ω),there is a control

v∈L2(Σ(x0)),such as the solution y=y(v)in

(1 −3) checked y(T, v) = y′(T, v) = 0.

The fact that the control vis defended Σ(x0)must

be interpreted as meaning y=vin Σ(x0), y = 0 in

Σ∗(x0).

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2022.18.37

Bouafi Nadia, Merabti Nesrine Lamya, Rezzoug Imad

E-ISSN: 2224-2856

354

Volume 18, 2023

For each pair of initial data we have y0, y1∈

L2(Ω) ×H−1(Ω).

The set of admissible controls

Uad =nv∈L2(Σ(x0)); y(T, v) = y′(T, v) = 0 |Ωo,

contains an infinity of elements.

We will now show that the control given by HUM

is that realizes the minimum of the cost function

J(v) = 1

2ZΣ(x0)|v|2dΣ,

on all admissible controls Uad we will next char-

acterize the control vusing the optimality system.

2.2 Characterization of control

Theorem 1 For each pair of initial data y0;y1∈

L2(Ω) ×H−1(Ω),the control v∈L2(Σ(x0)) given

by HUM is the one that minimizes the cost function

J(v)on all admissible controls Uad.

2.2.1 First step

We consider the minimization problem

inf J(v), v ∈Uad (4)

the (4) problem is an optimal control problem with

constraint.

Theorem 2 By a penalization method we define the

function

Jϵ(v, z) = 1

2RΣ(x0)|v|2dΣ + 1

2εRQ(z′′ −

∆z)2dxdt,

with ε > 0,v ∈L2(Σ(x0)) and z=z(x, t)a

function such as

z′′ −∆z∈L2(Q),

z(0) = y0, z′(0) = y1in Ω,

zχΣ(x0)=v, (5)

z= 0 in Σ∗(x0), z(T) = z′(T) = 0 in Ω,

recess for each v∈Uad the function y=y(v)of

(1 −3) verifies these condition.

The term 1

2εRQ(z′′ −∆z)2dxdt is a penalty term.

We consider the optimal control problem

inf Jε(v, z),(6)

for each ε > 0there exist a unique solu-

tion {uε, zε}of this problem, i.e. Jε(uε, zε) =

inf Jε(v, z).

2.2.2 Second step

Note that the sequence (uε)ε⟩0is bounded in

L2(Σ(x0)).

Let v∈Uad and y=y(v)the solution of the

problem (1 −3) associated. The couple {v, y(v)}is

admissible for the minimization problem (6) for every

ε > 0and so

Jε(uε, zε)≤Jε(v, y(v)).

But as y(x)verifies

y′′ − △y= 0 in Q.

We see that

Jε(v, y(v)) = J(v),∀ϵ > 0,

Jε(uε, zε)≤J(v),∀ϵ > 0,

and this for each v∈Uad , so we have

Jε(uε, zε)≤inf J(v),∀ε > 0.

Especially

J(uε)≤inf J(v),∀ϵ > 0.

And, if we put

fϵ=1

√ε(z′′

ε− △zε).

We have (fε)bounded in L2(Q).

2.2.3 Third step

Quite to extract subsequences we will have

uε→0b

vin L2(Σ(x0)) weak.

We moreover



z”

ε− △zε

L2(Q)≤C√ε, ∀ε > 0.

We fined (zε)bounded in

L∞(0, T, L2(Ω)) ∩W1.∞(0, T, H−1(Ω)),

on particular

∥zε∥L2(Q)≤C, ∀ε > 0,(7)

and even it means extracting yet another sub-suite

zε →b

y, ε →0(8)

in L2(Q)weak.

According to (5) and (8) we have

y′′ −∆b

y= 0,

y=b

vin Σ(x0),b

y= 0 in Σ∗(x0),

y(T) = b

y′(T) = 0 in Ω,

y(0) = y0;b

y′(0) = y1in Ω,

so we have

Jϵ(uϵ, zϵ)≥J(uϵ),b

v∈Uad,

and after the week lower semi continuity of Jwe

have

J(b

v)≤lim inf J(uε)≤lim inf Jε(uε, zε),

we conclude

J(b

v) = inf J(v).

We have also proved that

lim J(uε) = J(b

v),(ε−→ 0)

which, combined with (7) gives

uε−→ b

vin L2(Σ(x0)) (strong).

2.2.4 Fourth step

Consider the sequel

Pε=1

ε(z′′

ε−∆zε),∀ε > 0,

obviously

Pε=1

εfε,∀ε > 0,

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2022.18.37

Bouafi Nadia, Merabti Nesrine Lamya, Rezzoug Imad

E-ISSN: 2224-2856

355

Volume 18, 2023

we say that (fε)ε>0is bounded in L2(Q)but we

do not yet have of estimate on (Pε)ε>0.

By writing the equation of Euler associated with

the problem of minimization (6). We have

ZΣ(x0)

uϵvdΣ−ZQ

pϵ(ζ′′ −∆ζ)dxdt = 0,(9)

for all solution of

ζ′′ −∆ζ∈L2(Q),

ζ(0) = ζ′(0) = ζ(T) = ζ′(T) = 0

ζ=vin Σ(x0), ζ = 0 in Σ∗(x0),

with v∈L2(Σx0)).

By means of the Green formula we obtain

p′′

ϵ−∆pϵ= 0 in Q,

pϵ= 0 on Σ,

∂pϵ

∂ν =uϵon Σ(x0),

in effect

RQpε(ζ′′ − △ζ)dxdt =

RQ(p′′

ε−pε)ζdxdt

−RΣpε∂ζ

∂v dΣ + RΣ(x0)

∂pε

∂v vdΣ,

and so, after (9)

RΣ(x0)uεv dΣ =

RQ(p′′

ε−pε)ζdxdt

−RΣpε∂ζ

∂v dΣ + RΣ(x0)

∂pε

∂v vdΣ,

which is equivalent.

2.2.5 Fifth step

According to the inverse inequality we obtain

0.5×(T−2R(x0)) n|∇pε(0)|2+|p′

ε(0)|2o≤

0.5×R(x0)RΣ(x0)∂pε

∂v 2dΣ =

R(x0)

2RΣ(x0)|uε|2dΣ,

the sequence (uε)ε>0being bounded in L2(Σ(x0)),

we see that

|∇pε(0)|+|p′

ε(0)| ≤ 0,∀ε > 0,

and according to the law of conservation of energy

,we have

pε→pin L∞((0, T, H1

0(Ω)) ∩

W1.∞(0, T, L2(Ω)),

pε→pon L∞((0, T, H1

0(Ω)) weak,

p′

ε→p′on L∞((0, T, L2(Ω)) weak,

npϵ(0),p′

ε(0)o→np(0), p′(0)oon H1

0(Ω) ×

L2(Ω) weak.

So the function p=p(x, t)solution of

p′′ − △p= 0 in Q,

p= 0 on Σ,

∂p

∂v =−→

von Σ(x0),

p(0) = p;p′(0) = p1in Ω.

2.2.6 Sixth step

We pose Φ = p;Φ0=p0; Φ1=p1and ψ=b

According to to (9) we have

Φ′′ −∆Φ = 0 in Q,

Φ = 0 on Σ,

Φ(0) = Φ0,Φ′(0) = Φ1in Ω.

ψ′′ −∆ψ= 0 in Q.

ψ=∂ψ

∂v on Σ(x0), ψ = 0 on Σ∗(x0).

On the other hand as

ψ(0) = y0;ψ′(0) = y1,

we have

ΛΦ0,Φ1=y1,−y0.

With Λis the isomorphism between H1

0(Ω) ×

L2(Ω) and H−1(Ω) ×L2(Ω) introduced in the ap-

plication of HUM.

We thus see that the control −→

vwhich by construc-

tion minimizes J′(v)on Uad is the control given by

HUM since b

v=∂p

∂v =∂Φ

∂v .

3 Standard optimal control of

distributed system

In this chapter, we will study the optimal control of

linear PDE’s ,( the dimension of space of solution

is infinite) We start by the presentation of the clas-

sical theory of the optimal control when we prove the

existence, uniqueness and characterization of the op-

timum and we give some examples Then we study

the optimal control for a linear system with incom-

plete data by present the notion of no-regret control

[21], and associated with low-regret control which

converges to the no-regret control, then we charac-

terize them and we give example.

3.1 Position of problem

Let Y,Uand Zbe infinite dimensional Hilbert spaces

of states, controls and observation resp. Uad⊂ U is

a subset of admissible controls supposed non empty,

closed and convex.

fis a source function in y. Consider the (10)

well-posed abstract linear partial differential equation

Ay=f+Bv. (10)

Where A∈L(Y)is a linear partial differential

operator stationary or evolutionary (elliptic, parabolic

and hyperbolic ) makes an isomorphism on Y′identi-

fied toY,B ∈ L(U,Y)is the control operator.

Our optimal control problem consists in looking

for a control function u∈ Uad which minimizes the

following cost function

J(v) = ∥Cy(v)−yd∥2

Z+N∥v∥2

U∀v∈ Uad,

(11)

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2022.18.37

Bouafi Nadia, Merabti Nesrine Lamya, Rezzoug Imad

E-ISSN: 2224-2856

356

Volume 18, 2023

Jis convex function from Uad ⊂ U to R∪{+∞},

C ∈ L(Y,Z) : the observation operator and Nis a

symmetric definite positive operator bounded in U.

ydis the fixed observation in Z.

we search usolution of : find

u∈ Uad (12)

such that J(u) = inf J(v), v ∈ Uad

Theorem 3 ”Existence and uniqueness of optimal

control” Let Uad ⊂ U closed and nonempty, Jis

lower semi continuous, bounded from below and co-

ercive on Uad. Then there exists a minimize for Jon

Uad. Moreover, if Jis strictly convex the minimize is

unique.

3.2 Optimal systems (Optimal control

characterization)

We have by a first order optimality condition

J′(u) (v−u)≥0∀v∈ Uad,

Jis Gateaux-differentiable function

J′(u) (v−u) = lim t−1(J(u+t(v−u)) −J(u))

for every v∈ Uad, t −→ 0.

with a calculatation we fined

J(u+t(v−u)) = J(u) + t2∥Cy(v−u)∥2

+2t(Cy(u)−yd,Cy(v−u))Z

+t2N∥v−u∥2

U+ 2tN(u, v −u)U,

which gives

t−1(J(u+t(v−u)) −J(u))

=t∥Cy(v−u)∥2

+2(Cy(u)−yd,Cy(v−u))z+tN ∥v−u∥2

U+

2N(u, v −u)U,

when t→0we find

J′(u) (v−u) = 2(C∗(Cy(u)−yd), y(v−u))Y

+2N(u, v −u)U≥0,∀v∈ Uad.

Remark 4 A condition of the (12) from

J′(u) (v−u)is called the variationel inequal-

ity.

C∗∈ L(Z,Y)is the adjoint of C,A∗is the adjoint

operator of Aand introduce the adjoint state p=p(u)

given by

A∗p(u) = C∗(Cy(u)−yd),

then

(C∗(Cy(u)−yd), δy(v−u))Y

= (A∗p(u), δy(v−u))Y

= (B∗p(u),(v−u))U.

Hence,

J′(u) (v−u)=(B∗p(u) + Nu, v −u)U≥

0,∀v∈ Uad.

The optimal control problem (10, 11, 12) has a

unique solution ucharacterized by the following op-

timality system







Ay(u) = f+Bu,

A∗p(u) = C∗(Cy(u)−yd),

(B∗p+Nu, v −u)U≥0,∀v∈ Uad.

(13)

The equation 1 and 2 from (13) must be associated

to some appropriate boundary and initial condition.

We called the pair (u, p(u)) by the optimal pair.

Remark 5 We have no constraints on control, by

space structure of U( if Uad =U)we deduce that

we also have J′(u) (v−u)≤0∀v∈ Uad,

and with the previous condition we get

J′(u) (v−u) = 0 ∀v∈ U,

therefore the optimality system become as follow-

ing

Ay(u) = f+Bu,

A∗p(u) = C∗(Cy(u)−yd),

(B∗p(u) + Nu, v −u)U= 0,∀v∈ U.

3.3 No-regret control and low-regret control

to solve distributed system with missing

data

In this section, we make an initiation to the theory of

the optimal control of problems with incomplete data,

where we introduce this leads to define the notion of

no- regret control, low regret control[22]. Moreover,

we give existence, uniqueness , and prove that it con-

verges to the no-regret control, then we characterize

them via optimality systems and we give example.

3.3.1 Position of problem

We keep the same theorical framework as mentioned

in the last paragraphed, the difference here is the pres-

ence of missing data. For this reason, we define a new

operator β∈ L(F, Y)where

Fis a Hilbert space of uncertainties (missing data),

Gis a non-empty closed subspace of F.

For f∈ Y the abstract equation related to the con-

trol v∈ Uad and the uncertainty g∈Gis given by

Ay(v, g) = f+Bv+βg. (14)

The equation (14) is well posed in Yand has a

unique solution y(v, g),which associate to her the

following cost function :

J(v, g) =∥ Cy(v, g)−yd∥2

Z+N∥v∥2

U,∀v∈ Uad,∀g∈G.

(15)

as usual, we are concerned with the optimal con-

trol of (14) and (15) is to search usolution of

inf J(v, g),∀g∈G, v ∈ Uad

when Gis an infinite dimensional space the prob-

lem (14) has no sene, this problem is solved in[23]

they using many notion like no-regret control and

pareto control[24]there equivalents is proved in[25].

We take

inf (g∈GsupJ(v, g)) , v ∈ Uad,

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2022.18.37

Bouafi Nadia, Merabti Nesrine Lamya, Rezzoug Imad

E-ISSN: 2224-2856

357

Volume 18, 2023

but Gis an infinite dimensional space we can get

g∈GsupJ(v, g) = +∞and by the way the problem

has no sense. So, to avoid this difficulty, we introduce

the concept of “No-regret control”.

Remark 6 If G={0}then J(v, g) = J(v, 0).

Therefore, the problem (14) becomes a standard op-

timal control problem :

find u∈ Uad such that, J(u) = inf J(v), v ∈

Uad.

To avoid difficulties arise when we get sup(v, g) =

+∞, g ∈G, we take only controls such that ∀v∈

Uad :

J(v, g)≤J(0, g),∀g∈G

J(v, g)−J(0, g)≤0,∀g∈G.

Thus, we can say that sup (J(v, g)−J(0, g)) , g ∈

Gexists.

3.4 No-regret control

Definition 7 We say that u∈ Uad is a no-regret con-

trol for (14) and (15) if usolves

inf (sup (J(v, g)−J(0, g))) , v ∈ Uad, g ∈G.

(16)

Remark 8 of course , the next problem is defied only

for controls such that

sup (J(v, g)−J(0, g)) <∞, g ∈G.

Lemma 9 For every u∈ Uad and g∈Gwe have :

J(v, g)−J(0, g) = J(v.0)−J(0,0)+2 (S(v), g)G′,G ,

(17)

where S(v) = β∗ξ(v)and ξ(v)defined for v∈

Uad by

A∗ξ(v) = C∗C(y(v, 0) −y(0,0)).

Ais a linear operator in Y,so :

y(v.g) = y(v, 0) + y(0, g)−y(0,0),

y(0, g) = y(0,0) + y(0, g)−y(0,0),

with y(v, 0) and y(0, g)are a solution of (14) when

g= 0 and v= 0 resp.

By the definition of J(v, g)one obtain

J(v, g) = J(v, 0) + ∥C(y(0, g)−y(0,0)∥2

+2(Cy(v, 0) −yd,C(y(0, g)−y(0,0)))Z,

and

J(0, g) = J(0,0) + ∥C(y(0, g)−y(0,0))∥2

+2(C(y(0,0) −yd,C(y(0, g)−y(0,0)))Z,

then

J(v, g)−J(0, g) = J(v, 0) −J(0,0)

+2(C∗C(y(v, 0) −y(0,0)), y(0, g)−y(0,0))Y.

Introduce an adjoint state ξ(v)given by A∗ξ(v) =

C∗C(y(v, 0) −y(0,0)) to write

J(v, g)−J(0, g) = J(v, 0)−J(0,0)+2(S(v), g)G′,G

(18)

where S(v) = β∗ξ(v),the last equation leads to

(18).

Remark 10 1. By (18) you can see that condition

(17) holds iff v∈k, where

K={v∈ Uad,(S(v), g) = 0∀g∈G},

is a closed subspace of U. Then, uis a no-regret

control iff u∈k.

2. The notion of no-regret control could be gener-

alized to no-regret control related to any a fixed con-

trol u0∈Uad,i.e , we want controls vs.t

J(v, g)≤J(u0,g)∀g∈G

Definition 11 we say that u∈Uad is a no-regret con-

trol related to u∈Uad for (14)-(15) if usolve

inf sup(J(v, g)−J(u0,g).

Unfortunately, the main difficulty with no-regret

control arises when we want to characterize the set

k, for this reason we shall approximate the no-regret

control by a sequence of controls called low regret

controls

3.4.1 Characterization of the no-regret control

The optimality system of no-regret control is given by

Ay=f+Bu,

A∗ζ=C∗Cy(u, 0) −yd,

Aρ=βλ, λ ∈G,

A∗p=C∗(Cy(u, 0) −yd) + C∗Cρ,

(B∗p+Nu, v −u)U≥0∀v∈ Uad.

where y(u, 0) = y, ξ(u) = ξ.

3.5 The low-regret control

One through to relax (16) by making some quadratic

perturbation on J(0, g), in other words, we search

controls vsuch that

J(v, g)≤J(0, g) + γ∥g∥2

G, γ > 0, g ∈G.

Definition 12 We say that uγ∈Uad is a low -regret

control for (14)-(15) if usolves

inf sup(J(v, g)−J(0, g)−γ∥g∥2

G), γ > 0, v ∈

Uad, g ∈G.

So we have the equivalence

inf(J(v, 0) −J(0,0)+

sup(2(S(v), g)G−γ∥g∥2

G)),

v∈ Uad, g ∈G.

Legendre transform, for

sup(2(S(v), g)G−γ∥g∥2

G) = 1

γ∥S(v)∥2

G, g ∈G.

then

inf J′(v), v ∈ Uad

where

J′(v) = J(v, 0) −J(0,0) + 1

γ∥S(v)∥2

Now, we can define the low-regret by

Definition 13 We say that uγ∈ Uad is a low-regret

control for (14) and (15) if usolves

inf sup(J(v, g)−J(0, g)−γ∥g∥2

G, γ > 0),

v∈ Uad, g ∈G.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2022.18.37

Bouafi Nadia, Merabti Nesrine Lamya, Rezzoug Imad

E-ISSN: 2224-2856

358

Volume 18, 2023

Theorem 14 ”Low-regret control: existence and

uniqueness” The problem (14) and (17) with (18)

has a unique solution uγ.

Theorem 15 The unique low-regret control uγis con-

verge weakly when γ→0to the unique no-regret con-

trol uin Uad.

Let uγbe a low-regret control in Uad then for all

v∈ Uad

J′(uγ)≤Jγ(v),

J(uγ,0) −J(0,0) + 1

γ∥β∗ζ(uγ)∥2

≤J(v, 0) −J(0,0) + 1

γ∥β∗ζ(v)∥2

G,∀v∈ Uad,

by implies

J(uγ,0) + 1

γ∥β∗ζ(uγ)∥2

≤J(v, 0) + 1

γ∥β∗ζ(v)∥2

G,∀v∈ Uad,

we choose v= 0 to find

J(uγ,0) + 1

γ∥β∗ζ(uγ)∥2

G=constant,

then

∥uγ∥U≤C,

∥Cy(uγ,0)∥Z≤C,

∥β∗ζ(uγ)∥G≤√γC,

where Cis a constant independent of γ.

(uγ)is bounded in Uad then we can extract a sub-

sequence still be denoting (uγ)converges weakly to

u∈ Uad.

It’s clear that for every v∈ Uad

J(v, g)−J(0, g)−γ∥g∥2

G≤

J(v, g)−J(0, g),∀g∈G,

i.e,

J(v, g)−J(0, g)−γ∥g∥2

G≤

sup (J(v, g)−J(0, g)) ,∀g∈G,

from another side we have

J(uγ, g)−J(0, g)−γ∥g∥2

≤J(v, g)−J(0, g)−γ∥g∥2

J(uγ, g)−J(0, g)−γ∥g∥2

≤sup (J(v, g)−J(0, g)) ,∀g∈G,

when γtend to 0we obtain

J(u, g)−J(0, g)≤

sup (J(v, g)−J(0, g)) ,∀g∈G,

which means that

sup (J(u, g)−J(0, g))

=inf {sup (J(v, g)−J(0, g))}.

In conclusion, uis a no-regret control.

3.5.1 Characterization of the low-regret control

By a first order optimality condition we have

J′(uγ)(v−uγ)≥0,∀v∈ Uad,

where

J′(uγ)(v−uγ) =

lim h−1(J(uγ+h(v−uγ)) −J(uγ)),∀v∈

Uad,

we have

h−1(J(uγ+t(v−uγ)) −J(uγ)) =

h∥Cy(v−uγ,0)∥2

Z+hN ∥v−uγ∥2

γ∥S(v−uγ)∥2

G+ 2(Cy(uγ,0) −yd,Cy(v−

uγ,0))Z

+2N(uγ, v −uγ)U+2

γ(S(uγ), S (v−uγ))G,

when h→0we find

J′(uγ)(v−uγ) = 2(Cy(uγ,0) −yd,Cy(v−

uγ,0))Z

+2N(uγ, v −uγ)U+2

γ(S(uγ), S (v−uγ))G.

By linearity of the operator Cin Zwe have

Jγ′(uγ)(v−uγ) =

2(C∗(Cy(uγ,0) −yd), y(v, 0) −y(uγ,0))Y

+2N(uγ, v −uγ)U+2

γ(S(uγ), S (v−uγ))G,

y(v, 0) −y(uγ,0) = y(v−uγ,0) −y(0,0) ,

then

J′(uγ)(v−uγ) = 2(C∗(Cy(uγ,0) −yd), y(v−

uγ,0) −y(0,0))Y

+2N(uγ, v −uγ)U+2

γ(S(uγ), S (v−uγ))G.

The adjoint state

A∗ξ(uγ) = C∗C(y(uγ,0) −y(0,0)),then

(S(uγ), S (v−uγ))G= (ββ∗ξ(uγ), ξ(v−uγ))Y.

Introduce the state ργ=ρ(uγ)by

Aργ=1

γββ∗ξ(uγ),

this leads to the following equality

(Aργ, ξ(v−uγ))Y= (C∗Cργ, y(v−uγ,0) −

y(0,0))Y,

introducing the new adjoint state pγ=p(uγ)by

A∗pγ=C∗(Cyγ−yd) + C∗Cργ,

to find

(A∗pγ, y (v−uγ,0) −y(0,0))Y= (B∗pγ, v −

uγ)U.

Hence, the optimality condition is given by

J′(uγ)(v−uγ)=(B∗pγ+Nuγ, v −uγ)U≥

0,∀v∈ Uad.

Finally, the low-regret control is characterized by

the following optimality system :

Ayγ=f+Buγ,

A∗ξγ=C∗C(yγ−y(0,0)),

Aργ=1

γββ∗ξγ,

A∗pγ=C∗(Cyγ−yd) + C∗Cργ,

(B∗pγ+Nuγ, v −uγ)U≥0,∀v∈ Uad,

where

y(uγ,0) = yγ, ξ(uγ) = ξγ.

4 Comparison between the controls

calculated through HUM and the

low regrets method

After the comprehensive and in-depth study of the

two methods, we can draw the following comparison

between the HUM method and the low-regret method.

The HUM method has advantages represented in If

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2022.18.37

Bouafi Nadia, Merabti Nesrine Lamya, Rezzoug Imad

E-ISSN: 2224-2856

359

Volume 18, 2023

Uad =Hthe control is identifiable with the con-

joint state of systems for systems satisfying the Mi-

zohata hypotheses and the disadvantages represented

in firstly if Uad is empty this method does not work,

secondly if Uad is not empty (Slater) the method gives

a duality between the control and the conjoint state of

the systems. The advantages of the low-regret method

ensure control existence even in the empty Uad case

and it gives characterization equations for singular

systems, and the disadvantages that are not constric-

tive.

5 Conclusion

Generally, we conclude that the HUM method is used

for the regulars systems, and the no-regret method is

used for the singulars systems.

When Uad =Hor interior non-empty (slater), we

can use the HUM method.

Acknowledgements

The authors thank the referees for their careful

reading and their precious comments. Their help is

much appreciated.

References:

[1] E.H. Zerrik, A. Afifi, A. EL Jai. Systèmes

dynamiques, Analyse régionale des systèmes

linéaires distribués. Tome 2, Presses Universi-

taires de Perpignoom, 2008.

[2] F. Achab and I. Rezzoug. Average no

regret sentinel. Innovative Journal of

Mathematics. v. 1 no. 3, 1-15, (2022).

doi:10.55059/ijm.2022.1.3/25

[3] A. Berhail and I. Rezzoug. Identification of the

source term in Navier-Stokes system with in-

complete data. AIMS Mathematics, 4(3): 516–

526. DOI:10.3934/math.2019.3.516. 2019.

[4] T. Laib. Instantaneous sentinel for the identifica-

tion of the pollution terme in Navier Stockes sys-

tems. Proyecciones. V40. N6. PP. 1489-1505.

2021.

[5] S.M. Kallab Debbih. Approximate solution for

the Heat equation with delay time and missing

data. 2021.

[6] N.L. Merabti and I. Rezzoug. Sentinel method

and distributed systems with missing data. Inno-

vative Journal of Mathematics. v. 1 no. 2, 44-62,

(2022). doi:10.55059/ijm.2022.1.2/26

[7] I. Rezzoug and T.E. Oussaeif. Approxi-

mate Controllability. Wseas Transcations

on Systems. Volume 19, (2020). DOI:

10.37394/23202.2020.19.3

[8] I. Rezzoug et al. Pollution detection for the sin-

gular linear parabolic equation. J. Appl. Math.&

Informatics Vol 41(3), 647-656. 2023.

[9] I. Rezzoug et al. Regularization of a class of ill-

posed Cauchy problems associated with genera-

tors of analytie semi-groups. PJM. V11, N1. PP.

485-495. 2022

[10] I. Rezzoug and T.E. Oussaeif. Solvability of a

solution and controllability of partial fractional

differential systems. JIM. V24. N5. PP 1175-

1200. 2021.

[11] I. Rezzoug. Étude théorique et numérique des

problèmes d’identification des systèmes gou-

vernes par des équations aux dérivées partielles.

Thèse de Doctorat. Université oum el Boughi.

2014.

[12] I. Rezzoug. Identification d’une partie de la

frontière inconnue d’une membrane. Thèse de

Magister. Université oum el Boughi. 2009.

[13] M.B. Iqbal et I. Rezzoug. Pollition de-

tection for the singular linear parabolic

equation. J. Appl. Math. & Informat-

ics Vol. 41, No. 3, pp. 647 - 656. 2023.

https://doi.org/10.14317/jami.2023.647

[14] A. Benbrahim et I. Rezzoug. A revision

of M. Asch notion of Discrete control.

Wseas Transcations on Applied and Theo-

rical Mechanics. Volume 17, (2022). DOI:

10.37394/232011.2022.17.29

[15] J. P. Raymond. Optimal control of partial differ-

ential equations. Université Paul Sabatier, 2013.

[16] L. J. Savage. The foundations of statistics.

Courier Corporation. 1972.

[17] E. Trélat. Contrôle optimal : Théorie et applica-

tions. Université Pierre et Marie Curie (paris 6).

2013

[18] A. Ayadi and A. Hafdallah. Averaged op-

timal control for some evolution problems

with missing parameter. Technical Report.

DOI:10.13140/RG.2.1.3002.8563. 2016.

[19] J.L. Lions. Optimal control of systems governed

by partial differential equations problèmes aux

limites. 1971.

[20] J. Lohéac and E. Zuazua. Averaged control-

lability of parameter dependent conservative

semigroups. Journal of Differential Equations,

262(3), 1540-1574. 2017.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2022.18.37

Bouafi Nadia, Merabti Nesrine Lamya, Rezzoug Imad

E-ISSN: 2224-2856

360

Volume 18, 2023

[21] E. Zuazua. Averaged control. Automatica,

50(12), pp 3077-3087. 2014.

[22] J.L. Lions. Contrôle à moindres regrets des

systèmes distribués. Comptes rendus de

l’Académie des sciences. Série 1, Mathéma-

tique, 315(12), 1253-1257. 1992.

[23] J.L. Lions, Controllability, Penalty and stiffness,

Anal, Nor, Serie 4, Tome 25, N3-4. PP. 547-610,

1997.

[24] J.L. Lions. Contrôlabilité exacte, perturbations

et stabilisation de systèmes distribués. Vol. 1 et

2, Masson, RMA, paris, 1988.

[25] O. Nakoulima and A. Omrane and J. Velin. On

the pareto control and no-regret control for dis-

tributed systems with incomplete data. SIAM

journal on control and optimization, 42(4),

1167-1184. 2003.

Contribution of individual authors to

the creation of a scientific article

(ghostwriting policy)

Author Contributions: Please, indicate the role

and the contribution of each author:

Example

John Smith, Donald Smith carried out the simulation

and the optimization.

George Smith has implemented the Algorithm 1.1

and 1.2 in C++.

Maria Ivanova has organized and executed the ex-

periments of Section 4.

George Nikolov was responsible for the Statistics.

Follow: www.wseas.org/multimedia/contributor-

role-instruction.pdf

Sources of funding for research

presented in a scientific article or

scientific article itself

Report potential sources of funding if there is any

Creative Commons Attribution

License 4.0 (Attribution 4.0

International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en_US

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.e

n_US

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2022.18.37

Bouafi Nadia, Merabti Nesrine Lamya, Rezzoug Imad

E-ISSN: 2224-2856

361

Volume 18, 2023