Polynomial decay rate estimate for bilinear parabolic systems under weak observability condition

Tsouli, A.; Boutoulout, Ali

doi:10.1007/s12215-015-0204-z

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…Furthermore, we set

φ false(x, t false) = normalcos false(t false), \forall false(x, t false) \in false(0, 1 false) \times false[- \frac{π}{2}, 0 false],

which satisfies the hypothesis

false(scriptH false) .

The operator A is an infinitesimal generator of a semigroup of contractions in L 2 (0,1). The spectrum of A is given by the simple eigenvalues λ j =− π 2 ( j −1) 2 , j ∈ IN * and associated to the eigenfunctions defined by ϕ 1 ( x )=1 and

ϕ_{j} false(x false) = \sqrt{2} normalcos false(false(j - 1 false) π x false), \forall j \geq 2,

(see, e.g., [13]). Taking the compact operator of the form

B z = true \sum_{j = 1}^{+ \infty} \frac{1}{j^{2}} ⟨ z, ϕ_{j} ⟩ ϕ_{j} .

The family ( ϕ j ) j ≥ 1 is an orthonormal basis of L 2 (0,1) and the solution of the system () can be written as:

z false(x, t false) = true \sum_{j = 1}^{+ \infty} {⟨ z false(t false), ϕ_{j} ⟩}_{L^{2} false(0, 1 false)} ϕ_{j} false(x false), \forall false(x, t false) \in false(0, 1 false) \times false[0, + \infty false) .

The operator A generates the semigroup of contractions S ( t ) defined by

S false(t false) z = true \sum_{j = 1}^{+ \infty} e^{λ_{j} t} ⟨ z, ϕ_{j} ⟩ ϕ_{j},

and we have

⟨ B S (t

…”

Section: Numerical Examplesmentioning

confidence: 99%

Exponential and weak stabilization for distributed bilinear systems with time delay via bounded feedback control

Tsouli

Houch

2019

Asian Journal of Control

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In this paper, we deal with the problem of exponential and weak stabilization for a class of distributed bilinear systems with time delay in a Hilbert space by using bounded feedback control. In the case of exponential stabilization, an explicit decay rate estimate of the stabilized state is given provided that a non‐standard observability inequality condition is satisfied. The compactness hypothesis of the control operator prevents the validity of the non‐standard observability inequality, thus it can be relaxed to a weaker sufficient condition to show the weak stabilization result by employing the same feedback control. Finally, numerical examples and simulations to hyperbolic and parabolic partial functional differential equations are considered to illustrate the effectiveness of the obtained theoretical results.

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“…Furthermore, we set

φ false(x, t false) = normalcos false(t false), \forall false(x, t false) \in false(0, 1 false) \times false[- \frac{π}{2}, 0 false],

which satisfies the hypothesis

false(scriptH false) .

ϕ_{j} false(x false) = \sqrt{2} normalcos false(false(j - 1 false) π x false), \forall j \geq 2,

(see, e.g., [13]). Taking the compact operator of the form

B z = true \sum_{j = 1}^{+ \infty} \frac{1}{j^{2}} ⟨ z, ϕ_{j} ⟩ ϕ_{j} .

The family ( ϕ j ) j ≥ 1 is an orthonormal basis of L 2 (0,1) and the solution of the system () can be written as:

z false(x, t false) = true \sum_{j = 1}^{+ \infty} {⟨ z false(t false), ϕ_{j} ⟩}_{L^{2} false(0, 1 false)} ϕ_{j} false(x false), \forall false(x, t false) \in false(0, 1 false) \times false[0, + \infty false) .

The operator A generates the semigroup of contractions S ( t ) defined by

S false(t false) z = true \sum_{j = 1}^{+ \infty} e^{λ_{j} t} ⟨ z, ϕ_{j} ⟩ ϕ_{j},

and we have

⟨ B S (t

…”

Section: Numerical Examplesmentioning

confidence: 99%

Exponential and weak stabilization for distributed bilinear systems with time delay via bounded feedback control

Tsouli

Houch

2019

Asian Journal of Control

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show abstract

“…with 𝐵 𝑢 = 𝑃 𝑢 𝐵𝑃 𝑢 and 𝐵 𝑠 = 𝑃 𝑠 𝐵𝑃 𝑠 (see [20]). It has been shown in [6], under the spectrum growth assumption:…”

Section: Introductionmentioning

confidence: 99%

Archives of Control Sciences

Tsouli,

Ouarit

2022

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In this paper we deal with the problem of uniform exponential stabilization for a class of distributed bilinear parabolic systems with time delay in a Hilbert space by means of a bounded feedback control. The uniform exponential stabilization problem of such a system reduces to stabilizing only its projection on a suitable finite dimensional subspace. Furthermore, the stabilizing feedback control depends only on the state projection on the finite dimensional subspace. An explicit decay rate estimate of the stabilized state is given provided that a nonstandard weaker observability condition is satisfied. Illustrative examples for partial functional differential equations are displayed.

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Stabilization for distributed semilinear systems governed by optimal feedback control

Tsouli

Benslimane

2018

Int. J. Dynam. Control

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Polynomial decay rate estimate for bilinear parabolic systems under weak observability condition

Cited by 3 publications

References 17 publications

Exponential and weak stabilization for distributed bilinear systems with time delay via bounded feedback control

Exponential and weak stabilization for distributed bilinear systems with time delay via bounded feedback control

Archives of Control Sciences

Stabilization for distributed semilinear systems governed by optimal feedback control

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