- Email: [email protected]
Comments on “Controllability of the wave equation with bilinear controls”
European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Contents lists available at ScienceDirect
European Journal of Control journal homepage: www.elsevier.com/locate/ejcon
Discussion
Comments on “Controllability of the wave equation with bilinear controls” Mohamed Ouzahra Department of Mathematics & Informatics, ENS, University of Sidi Mohamed Ben Abdellah, P.O. Box 5206, Fès, Morocco
art ic l e i nf o
a b s t r a c t
Keywords: Bilinear control Wave equation Approximate controllability
During the proof of Theorem 1 in the paper “controllability of the wave equation with bilinear controls”, we have worked with a constant λ that satisfies the inequality just before (14) (see Ouzahra, 2014 [3, p. 59]). However, the right term of this inequality depends implicitly on λ, so the existence of λis not obvious. The aim of this technical note is twofold: firstly, we will give necessary details that clarify the solvability of the above-mentioned inequality. Secondly, we will show that the step of building such a λ is unnecessary, and thus can be removed from the proof. That way, we will optimize the initial proof of Theorem 1 of Ouzahra (2014) [3]. & 2014 European Control Association. Published by Elsevier Ltd. All rights reserved.
1. Introduction and the statement of Theorem 1 of [3] In [3], approximate and exact controllability was investigated for the following n-dimensional wave equation: 8 > < wtt ¼ Δw þvðx; tÞw þ γðtÞgðxÞwt w¼0 > : wðx; 0Þ ¼ w ; w ðx; 0Þ ¼ w 0
t
1
ðK1 Þ : g A L1 ðΩÞ; g Z 0; a:e: Ω and satisfies the following assumption: Z T ‖gφt ‖2H1 dt Z δ‖ðφ0 ; φ1 Þ‖2H1 ; 8 ðφ0 ; φ1 Þ A H1 ; ð2Þ 0
where δ; T 4 0 and φ is a solution of in Ω; t 4 0 on ∂Ω
φtt ¼ Δφ þ aφ;
ð1Þ
φð0Þ ¼ φ0 A V 1 ;
φt ð0Þ ¼ φ1 A H 1 :
ð3Þ
in Ω
where Ω is a bounded open set of Rn ; n Z 1 with smooth boundary ∂Ω, and g Z 0, a.e. on Ω with support ω. The real valued coefficients vðx; tÞ and γðtÞ are the multiplicative controls. We will keep the notations of [3]. Let us consider the space H 1 ¼ L2 ðΩÞ, endowed with its natural inner product 〈; 〉H1 , and define the unbounded operator A ¼ Δ with domain DðAÞ ¼ H 10 ðΩÞ \ H 2 ðΩÞ. Let V 1 ¼ H 10 ðΩÞ and let us consider the Hilbert space H1 ¼ V 1 H 1 with the inner product:
The Theorem 1, as stated in [3], is as follows: Theorem 1. Assume that assumptions ðK0 Þ and ðK1 Þ hold. Then for all initial states ðw0 ; w1 Þ A H1 and for every ϵ 4 0, there are a time T 1 40 and controls vðx; tÞ and γðtÞ such that the solution w of (1) satisfies ‖wðT 1 Þ θ‖V 1 þ ‖wt ðT 1 Þ‖H1 o ϵ
ð4Þ
〈ðu1 ; u2 Þ; ðv1 ; v2 Þ〉H1 ¼ 〈A1=2 u1 ; A1=2 v1 〉H1 þ〈u2 ; v2 〉H1
2. A new version of the proof of Theorem 1
and corresponding norm ‖ ‖H1 . Let θ A H 10 ðΩÞ \ H 2 ðΩÞ and let a≔ ðΔθ=θÞ1Λ ; where Λ ¼ fx A Ω=θðxÞ a0g and 1Λ is the characteristic function of Λ; and let us consider the following assumptions:
First, let us remark that the choice of the control vðx; tÞ ¼ aðxÞ λ in the initial proof of Theorem 1 of [3] was made in order to get a coercive operator A aI λI, so that one can apply the stabilization result of [10,22] in [3]. Here, we will see that the coercivity is not needed and so one can consider the control vðx; tÞ ¼ aðxÞ rather than vðx; tÞ ¼ aðxÞ λ, which amounts to λ ¼0 in the initial proof of [3]. This will be clarified in the sequel.
ðK0 Þ : a A L1 ðΩÞ and A aI is a positive operator,
E-mail address: [email protected]
Proof of Theorem 1. Let y ¼ w θ. Then, as explained in [3], it follows from the definition of a that system (1) can be reduced to
http://dx.doi.org/10.1016/j.ejcon.2014.09.002 0947-3580/& 2014 European Control Association. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: L. Jetto, et al., Accurate output tracking for nonminimum phase nonhyperbolic and near nonhyperbolic systems, European Journal of Control (2014), http://dx.doi.org/10.1016/j.ejcon.2014.09.001i
M. Ouzahra / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
the following one: ytt ¼ Δy þ aðxÞy ρgyt ;
x A Ω; t 4 0; yð0Þ ¼ y0 ; yt ð0Þ ¼ y1 :
Eq. (3) can be written in the form ! ! ! φ 0 I ¼ : φt φt A þ aI 0 φ
ð5Þ
y ð6Þ
t
If Sa(t) denotes the contraction semigroup associated to Eq. (6), then we have Sa ðtÞh ¼ ðφðtÞ; φt ðtÞÞ, for all h ¼ ðφ0 ; φ1 Þ. Thus, from (2) there exists δ~ 4 0 such that Z T 2 ~ j〈BSa ðtÞh; Sa ðtÞh〉H1 j dt Z δ‖h‖ 8 h ¼ ðφ0 ; φ1 Þ A H1 ; ð7Þ H1 ; 0
where B ¼
0 0 0 gI
. Then, using a classical stability result (see for
instance [1,2]), we deduce that for ρ 4 0 small enough, there exist M 0 ; σ 0 4 0 such that the semigroup T(t) generated on H1 by 0 I A þ aI ρgI satisfies the following estimate: ‖TðtÞz0 ‖H1 r M 0 e σ0 t ‖z0 ‖H1 ;
8 t Z 0; 8 z0 ¼ ðy0 ; y1 Þ A H1 :
ð8Þ
Thus (4) holds for any time T 1 4 0 such that M 0 e σ0 T 1 ‖z0 ‖H1 o ϵ: This concludes the proof.
where the constants σ and M depend on λ, so the existence of λ in (9) is not guaranteed, thus the change will concern the part between (13) and (14) in [3]. Let us consider the following equation:
□
yt
! ¼ t
0 A þ ðaðxÞ λÞI
I ρgI
!
y yt
! :
ð10Þ
To justify the solvability of (9), we will provide explicit values for the constants M and σ. ≔SðtÞz0 of Eq. (10) can be written in the The mild solution yyðtÞ t ðtÞ form Z SðtÞz0 ¼ TðtÞz0 λ
t
Tðt sÞESðsÞz0 ds;
0
8 z0 ¼ ðy0 ; y1 Þ A H1 ;
ð11Þ
0 0 where E¼ I 0 and T(t) is the semigroup generated on H1 by 0 I A þ aI ρgI . Using the estimate (8), it follows from (11) and Gronwall inequality that J SðtÞz0 J r M 0 e ðσ 0 λM0 Þt J z0 J ;
8 t Z 0; z0 A H:
Hence, we can take M ¼ M 0 and σ ¼ σ 0 λM 0 in (9) for 0 o λ o σ 0 =M 0 . With these explicit values of M and σ, it is easy to see that (9) holds for λ small enough.
Remark 2. Of course, one may replace λ by 0 in (40) and (42) of [3]. 3. Correction of the proof of Theorem 1 of [3] The subject of this section is to bridge the gap in the proof of Theorem 1 of [3]. The gap was in the following inequality (see [3] p. 59): δ σϵ 0 o λ o inf 1; ; ; ð9Þ Tα 2M‖θ‖V 1
References [1] R.F. Curtain, G. Weiss, Exponential stabilization of well-posed systems by colocated feedback, SIAM J. Control Optim. 45 (1) (2006) 273–297. [2] I. Lasiecka, R. Triggiani, L2 ðΣÞ-regularity of the boundary to boundary operator Bn L for hyperbolic and Petrowski PDEs. Abstr. Appl. Anal. (19) (2003) 1061–1139. [3] M. Ouzahra, Controllability of the wave equation with bilinear controls, Eur. J. Control 20 (2) (2014) 57–63.
Please cite this article as: L. Jetto, et al., Accurate output tracking for nonminimum phase nonhyperbolic and near nonhyperbolic systems, European Journal of Control (2014), http://dx.doi.org/10.1016/j.ejcon.2014.09.001i
Contents lists available at ScienceDirect
European Journal of Control journal homepage: www.elsevier.com/locate/ejcon
Discussion
Comments on “Controllability of the wave equation with bilinear controls” Mohamed Ouzahra Department of Mathematics & Informatics, ENS, University of Sidi Mohamed Ben Abdellah, P.O. Box 5206, Fès, Morocco
art ic l e i nf o
a b s t r a c t
Keywords: Bilinear control Wave equation Approximate controllability
During the proof of Theorem 1 in the paper “controllability of the wave equation with bilinear controls”, we have worked with a constant λ that satisfies the inequality just before (14) (see Ouzahra, 2014 [3, p. 59]). However, the right term of this inequality depends implicitly on λ, so the existence of λis not obvious. The aim of this technical note is twofold: firstly, we will give necessary details that clarify the solvability of the above-mentioned inequality. Secondly, we will show that the step of building such a λ is unnecessary, and thus can be removed from the proof. That way, we will optimize the initial proof of Theorem 1 of Ouzahra (2014) [3]. & 2014 European Control Association. Published by Elsevier Ltd. All rights reserved.
1. Introduction and the statement of Theorem 1 of [3] In [3], approximate and exact controllability was investigated for the following n-dimensional wave equation: 8 > < wtt ¼ Δw þvðx; tÞw þ γðtÞgðxÞwt w¼0 > : wðx; 0Þ ¼ w ; w ðx; 0Þ ¼ w 0
t
1
ðK1 Þ : g A L1 ðΩÞ; g Z 0; a:e: Ω and satisfies the following assumption: Z T ‖gφt ‖2H1 dt Z δ‖ðφ0 ; φ1 Þ‖2H1 ; 8 ðφ0 ; φ1 Þ A H1 ; ð2Þ 0
where δ; T 4 0 and φ is a solution of in Ω; t 4 0 on ∂Ω
φtt ¼ Δφ þ aφ;
ð1Þ
φð0Þ ¼ φ0 A V 1 ;
φt ð0Þ ¼ φ1 A H 1 :
ð3Þ
in Ω
where Ω is a bounded open set of Rn ; n Z 1 with smooth boundary ∂Ω, and g Z 0, a.e. on Ω with support ω. The real valued coefficients vðx; tÞ and γðtÞ are the multiplicative controls. We will keep the notations of [3]. Let us consider the space H 1 ¼ L2 ðΩÞ, endowed with its natural inner product 〈; 〉H1 , and define the unbounded operator A ¼ Δ with domain DðAÞ ¼ H 10 ðΩÞ \ H 2 ðΩÞ. Let V 1 ¼ H 10 ðΩÞ and let us consider the Hilbert space H1 ¼ V 1 H 1 with the inner product:
The Theorem 1, as stated in [3], is as follows: Theorem 1. Assume that assumptions ðK0 Þ and ðK1 Þ hold. Then for all initial states ðw0 ; w1 Þ A H1 and for every ϵ 4 0, there are a time T 1 40 and controls vðx; tÞ and γðtÞ such that the solution w of (1) satisfies ‖wðT 1 Þ θ‖V 1 þ ‖wt ðT 1 Þ‖H1 o ϵ
ð4Þ
〈ðu1 ; u2 Þ; ðv1 ; v2 Þ〉H1 ¼ 〈A1=2 u1 ; A1=2 v1 〉H1 þ〈u2 ; v2 〉H1
2. A new version of the proof of Theorem 1
and corresponding norm ‖ ‖H1 . Let θ A H 10 ðΩÞ \ H 2 ðΩÞ and let a≔ ðΔθ=θÞ1Λ ; where Λ ¼ fx A Ω=θðxÞ a0g and 1Λ is the characteristic function of Λ; and let us consider the following assumptions:
First, let us remark that the choice of the control vðx; tÞ ¼ aðxÞ λ in the initial proof of Theorem 1 of [3] was made in order to get a coercive operator A aI λI, so that one can apply the stabilization result of [10,22] in [3]. Here, we will see that the coercivity is not needed and so one can consider the control vðx; tÞ ¼ aðxÞ rather than vðx; tÞ ¼ aðxÞ λ, which amounts to λ ¼0 in the initial proof of [3]. This will be clarified in the sequel.
ðK0 Þ : a A L1 ðΩÞ and A aI is a positive operator,
E-mail address: [email protected]
Proof of Theorem 1. Let y ¼ w θ. Then, as explained in [3], it follows from the definition of a that system (1) can be reduced to
http://dx.doi.org/10.1016/j.ejcon.2014.09.002 0947-3580/& 2014 European Control Association. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: L. Jetto, et al., Accurate output tracking for nonminimum phase nonhyperbolic and near nonhyperbolic systems, European Journal of Control (2014), http://dx.doi.org/10.1016/j.ejcon.2014.09.001i
M. Ouzahra / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
the following one: ytt ¼ Δy þ aðxÞy ρgyt ;
x A Ω; t 4 0; yð0Þ ¼ y0 ; yt ð0Þ ¼ y1 :
Eq. (3) can be written in the form ! ! ! φ 0 I ¼ : φt φt A þ aI 0 φ
ð5Þ
y ð6Þ
t
If Sa(t) denotes the contraction semigroup associated to Eq. (6), then we have Sa ðtÞh ¼ ðφðtÞ; φt ðtÞÞ, for all h ¼ ðφ0 ; φ1 Þ. Thus, from (2) there exists δ~ 4 0 such that Z T 2 ~ j〈BSa ðtÞh; Sa ðtÞh〉H1 j dt Z δ‖h‖ 8 h ¼ ðφ0 ; φ1 Þ A H1 ; ð7Þ H1 ; 0
where B ¼
0 0 0 gI
. Then, using a classical stability result (see for
instance [1,2]), we deduce that for ρ 4 0 small enough, there exist M 0 ; σ 0 4 0 such that the semigroup T(t) generated on H1 by 0 I A þ aI ρgI satisfies the following estimate: ‖TðtÞz0 ‖H1 r M 0 e σ0 t ‖z0 ‖H1 ;
8 t Z 0; 8 z0 ¼ ðy0 ; y1 Þ A H1 :
ð8Þ
Thus (4) holds for any time T 1 4 0 such that M 0 e σ0 T 1 ‖z0 ‖H1 o ϵ: This concludes the proof.
where the constants σ and M depend on λ, so the existence of λ in (9) is not guaranteed, thus the change will concern the part between (13) and (14) in [3]. Let us consider the following equation:
□
yt
! ¼ t
0 A þ ðaðxÞ λÞI
I ρgI
!
y yt
! :
ð10Þ
To justify the solvability of (9), we will provide explicit values for the constants M and σ. ≔SðtÞz0 of Eq. (10) can be written in the The mild solution yyðtÞ t ðtÞ form Z SðtÞz0 ¼ TðtÞz0 λ
t
Tðt sÞESðsÞz0 ds;
0
8 z0 ¼ ðy0 ; y1 Þ A H1 ;
ð11Þ
0 0 where E¼ I 0 and T(t) is the semigroup generated on H1 by 0 I A þ aI ρgI . Using the estimate (8), it follows from (11) and Gronwall inequality that J SðtÞz0 J r M 0 e ðσ 0 λM0 Þt J z0 J ;
8 t Z 0; z0 A H:
Hence, we can take M ¼ M 0 and σ ¼ σ 0 λM 0 in (9) for 0 o λ o σ 0 =M 0 . With these explicit values of M and σ, it is easy to see that (9) holds for λ small enough.
Remark 2. Of course, one may replace λ by 0 in (40) and (42) of [3]. 3. Correction of the proof of Theorem 1 of [3] The subject of this section is to bridge the gap in the proof of Theorem 1 of [3]. The gap was in the following inequality (see [3] p. 59): δ σϵ 0 o λ o inf 1; ; ; ð9Þ Tα 2M‖θ‖V 1
References [1] R.F. Curtain, G. Weiss, Exponential stabilization of well-posed systems by colocated feedback, SIAM J. Control Optim. 45 (1) (2006) 273–297. [2] I. Lasiecka, R. Triggiani, L2 ðΣÞ-regularity of the boundary to boundary operator Bn L for hyperbolic and Petrowski PDEs. Abstr. Appl. Anal. (19) (2003) 1061–1139. [3] M. Ouzahra, Controllability of the wave equation with bilinear controls, Eur. J. Control 20 (2) (2014) 57–63.
Please cite this article as: L. Jetto, et al., Accurate output tracking for nonminimum phase nonhyperbolic and near nonhyperbolic systems, European Journal of Control (2014), http://dx.doi.org/10.1016/j.ejcon.2014.09.001i