International audienceIt is shown that the assumptions of the main result of the cited paper can never be satisfied

Rabah, Rabah

Portail HAL Nantes Université

Comments on: ”Continuous initial observability ofnonlinear delay parabolic equations” by Choi, Kwungand ParkRabah RabahTo cite this version:Rabah Rabah. Comments on: ”Continuous initial observability of nonlinear delay parabolicequations” by Choi, Kwung and Park. Far East Journal of Applied Mathematics, 2004, 15 (1),pp.113-115. <hal-01091525>HAL Id: hal-01091525https://hal.archives-ouvertes.fr/hal-01091525Submitted on 5 Dec 2014HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Public DomainComments on: “Continuous initial observabilityof nonlinear delay parabolic equations” by RongJae Choi, Young Chel Kwun and Jong Seo ParkRabah RabahIRCCyN UMR 6597,1 rue de la Noe¨, BP 92101,44321 Nantes Cedex 3, FranceEmail: rabah@emn.frAbstract. It is shown that the assumptions of the main result of the citedpaper can never be satisfied.Mathematical Subject Classification: 93C20, 93B07.In the paper [1] the problem of continuous initial observability is consideredfor the delayed infinite dimensional systemx(t) = S(t)φ(0) +∫t0S(t− s)F (s, xs)ds, t ∈ [0, T ],x(t) = φ(t), t ∈ [−h, 0],y(t) = Πx(t),(1)where S(t) is a C0-semigroup given on a Hilbert space, namely X = L2(Ω),where Ω is a bounded domain in Rn , Π is a bounded linear operator from Xto the Hilbert space Y , F is a noon linear function. In order to investigate thisproblem, authors consider the linear non delayed part of the system:x˙(t) = Ax(t)x(0) = φ(0)y(t) = Πx(t).(2)The main assumptions are: the semigroup S(t) is analytical and compact (as-sumption H3) and the system (2) is initially continuously observable (assump-tion H1). Under those assumptions conditions of initial continuous observabilityof the system (1) are given.We claim that the necessary condition of initial continuous observability ofthe system (2) is that the semigroup S∗(t) is onto or that each operator S(t) hasa bounded inverse operator defined on ImS(t). If S(t) is compact, then S∗(t) iscompact and cannot be onto in an infinite dimensional Hilbert space. If S(t) isanalytical then ImS∗(t) ⊂ D(A∗) and then the operators S∗(t) cannot be onto.1We need a precise definition of the initial observability because the definitiongiven in [1] is not clear. We refer to [2, 3, 4].Definition 1 Let H be the operator defined by:H : X → L2(0, T ;Y ), (Hx)(t) = Πx(t) = ΠS(t)φ(0).The system (2) is said initially observable if kerH = {0}, that is the left inverseH−1 exists. It is said contiuously initially observable if H−1 is bounded.Note that the adjoint operator H∗ is given byH∗ : L2(0, T ;Y )→ X, H∗y(.) =∫T0S∗(t)Π∗y(t)dt.This gives that the system (2) is contiuously initially observable iff the dualsystem is exactly controllable.It is well known that a necessary condition of exact controllability is that thecorresponding semigroup is onto for all t ≥ 0 (cf. for example [5, 6, 7]). That is anecessary condition of continuous initial observability is that the operators S∗(t)are onto. Hence, the system (2) can never be continously initially observable ifthe semigroup S(t) is analytical or compact in an infinite dimensional Hilbertspace. The assumptions in the main result of [1] are contradictory.References[1] Jae Rong Choi, Young Chel Kwun, Jong Seo Park, Continuous initial ob-servability of nonlinear delay parabolic equations, Far East J. Appl. Math10(1) (2003), 15–28.[2] N. Carmichael, A. J. Pritchard, M. D. Quinn, State parameter estimationfor nonlinear systems, Appl. Math. Optim. 9 (1982), 133–161.[3] M. D. Quinn, N. Carmichael, An approach to non-linear control problem us-ing fixed-point methods, degree theory and pseudo-inverses. Numer. FunctAnal. and Optimiz, 7(2&3)(1984-85), 197–219.[4] R. F. Curtain, H. Zwart, An introduction to infinite dimensional linearsystems theory, Springer, New York, 1995.[5] J. C. Louis, D. Wexler, On exact controllability in Hilbert space, J. Diff.Equat. 49 (1983), 258–269.[6] R. Rabah, Commandabilite´ des syste`mes line´aires a` retard constant dansles espaces de Banach. RAIRO-APII 20 (1986), 529–539.[7] R. Rabah, J. Karrakchou, Exact controllability and complete stabilizabilityfor linear systems in Hilbert spaces, Appl. Math. Lett. 10 (1997), 35–40.2

Comments on: "Continuous initial observability of nonlinear delay parabolic equations"  by Choi, Kwung and Park

Archive Ouverte en Sciences de l'Information et de la Communication

HAL Id: hal-01091525
https://hal.archives-ouvertes.fr/hal-01091525
Submitted on 5 Dec 2014
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Public Domain
Comments on: ”Continuous initial observability of
nonlinear delay parabolic equations” by Choi, Kwung
and Park
Rabah Rabah
To cite this version:
Rabah Rabah. Comments on: ”Continuous initial observability of nonlinear delay parabolic equations”
by Choi, Kwung and Park. Far East Journal of Applied Mathematics, 2004, 15 (1), pp.113-115. ￿hal-
01091525￿
Comments on: “Continuous initial observability
of nonlinear delay parabolic equations” by Rong
Jae Choi, Young Chel Kwun and Jong Seo Park
Rabah Rabah
IRCCyN UMR 6597,
1 rue de la Noë, BP 92101,
44321 Nantes Cedex 3, France
Email: rabah@emn.fr
Abstract. It is shown that the assumptions of the main result of the cited
paper can never be satisfied.
Mathematical Subject Classification: 93C20, 93B07.
In the paper [1] the problem of continuous initial observability is considered
for the delayed infinite dimensional system



x(t) = S(t)φ(0) +
∫
t
0
S(t− s)F (s, xs)ds, t ∈ [0, T ],
x(t) = φ(t), t ∈ [−h, 0],
y(t) = Πx(t),
(1)
where S(t) is a C0-semigroup given on a Hilbert space, namely X = L2(Ω),
where Ω is a bounded domain in Rn , Π is a bounded linear operator from X
to the Hilbert space Y , F is a noon linear function. In order to investigate this
problem, authors consider the linear non delayed part of the system:



ẋ(t) = Ax(t)
x(0) = φ(0)
y(t) = Πx(t).
(2)
The main assumptions are: the semigroup S(t) is analytical and compact (as-
sumption H3) and the system (2) is initially continuously observable (assump-
tion H1). Under those assumptions conditions of initial continuous observability
of the system (1) are given.
We claim that the necessary condition of initial continuous observability of
the system (2) is that the semigroup S∗(t) is onto or that each operator S(t) has
a bounded inverse operator defined on ImS(t). If S(t) is compact, then S∗(t) is
compact and cannot be onto in an infinite dimensional Hilbert space. If S(t) is
analytical then ImS∗(t) ⊂ D(A∗) and then the operators S∗(t) cannot be onto.
1
We need a precise definition of the initial observability because the definition
given in [1] is not clear. We refer to [2, 3, 4].
Definition 1 Let H be the operator defined by:
H : X → L2(0, T ;Y ), (Hx)(t) = Πx(t) = ΠS(t)φ(0).
The system (2) is said initially observable if kerH = {0}, that is the left inverse
H−1 exists. It is said contiuously initially observable if H−1 is bounded.
Note that the adjoint operator H∗ is given by
H∗ : L2(0, T ;Y ) → X, H
∗y(.) =
∫
T
0
S∗(t)Π∗y(t)dt.
This gives that the system (2) is contiuously initially observable iff the dual
system is exactly controllable.
It is well known that a necessary condition of exact controllability is that the
corresponding semigroup is onto for all t ≥ 0 (cf. for example [5, 6, 7]). That is a
necessary condition of continuous initial observability is that the operators S∗(t)
are onto. Hence, the system (2) can never be continously initially observable if
the semigroup S(t) is analytical or compact in an infinite dimensional Hilbert
space. The assumptions in the main result of [1] are contradictory.
References
[1] Jae Rong Choi, Young Chel Kwun, Jong Seo Park, Continuous initial ob-
servability of nonlinear delay parabolic equations, Far East J. Appl. Math
10(1) (2003), 15–28.
[2] N. Carmichael, A. J. Pritchard, M. D. Quinn, State parameter estimation
for nonlinear systems, Appl. Math. Optim. 9 (1982), 133–161.
[3] M. D. Quinn, N. Carmichael, An approach to non-linear control problem us-
ing fixed-point methods, degree theory and pseudo-inverses. Numer. Funct
Anal. and Optimiz, 7(2&3)(1984-85), 197–219.
[4] R. F. Curtain, H. Zwart, An introduction to infinite dimensional linear
systems theory, Springer, New York, 1995.
[5] J. C. Louis, D. Wexler, On exact controllability in Hilbert space, J. Diff.
Equat. 49 (1983), 258–269.
[6] R. Rabah, Commandabilité des systèmes linéaires à retard constant dans
les espaces de Banach. RAIRO-APII 20 (1986), 529–539.
[7] R. Rabah, J. Karrakchou, Exact controllability and complete stabilizability
for linear systems in Hilbert spaces, Appl. Math. Lett. 10 (1997), 35–40.
2


English

HAL Mines Nantes

https://hal.archives-ouvertes.fr/hal-01091525/document

Comments on: "Continuous initial observability of nonlinear delay parabolic equations" by Choi, Kwung and Park

Abstract

Similar works

Full text

Available Versions

Portail HAL Nantes Université

Archive Ouverte en Sciences de l'Information et de la Communication

HAL Mines Nantes