On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

Valein, Julie

doi:10.3934/mcrf.2021039

Cited by 11 publications

(16 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This result improves that of Vasconcellos and Silva, 18 where no delay occurs in the equation and also that of Valein, 25 where the equation is of third order. More importantly, unlike Valein, 25 we manage to obtain our stability results (see Theorem 2 and Theorem 3) without any smallness condition on the length

\ell

.…”

Section: Introductionsupporting

confidence: 85%

“…Next, we choose

{\lambda}_1

and

{\lambda}_2

small satisfying () to ensure that

{B}_1

and

{B}_2

are negative. Finally, we choose

\gamma

as in () so that

{B}_3&#x0002B;{B}_4

and

{B}_7

are negative. It is worth mentioning that, contrary to Valein 25 for the KdV equation, the stability result stated in Theorem 2 does not require any smallness condition on the length

\ell

. This is expectable since the equation has a damping term. The condition A2 plays an important role in the previous study.…”

Section: The Problem (11) With ω⊂Trueω^$$ \Omega \Subset \Hat{\omega...mentioning

confidence: 92%

“…This implies that the dissipation property () of the system's energy is lost. To overcome this difficulty, we shall adopt, as in Valein, 25 a perturbation argument developed in Nicaise and Pignotti 29 by considering the following auxiliary problem

({, \begin{matrix} left left \\ array & array \partial_{t} u (x, t) + \partial_{x}^{3} u (x, t) - η \partial_{x}^{5} u (x, t) + u (x, t) \partial_{x} u (x, t) + α \partial_{x} u (x, t) \\ array + (a (x) + ξ b (x)) u (x, t) + b (x) u (x, t - τ) = 0, & array (x, t) \in Ω \times (0, \infty), \\ array u (0, t) = u (ℓ, t) = \partial_{x} u (0, t) = \partial_{x} u (ℓ, t) = \partial_{x}^{2} u (ℓ, t) = 0, & array t > 0, \\ array u (x, 0) = u_{0} (x), & array x \in Ω, \\ array u (x, t) = z_{0} (x, t), & array x \in Ω, t \in T, \end{matrix})

…”

Section: The Problem (11) With ω⊈Trueω^$$ \Omega \Nsubseteq \Hat{\om...mentioning

confidence: 99%

“…and then choose 25 for the KdV equation, the stability result stated in Theorem 2 does not require any smallness condition on the length 𝓁. This is expectable since the equation has a damping term.…”

Section: The Trivial Solution Of the Problem (11) Is Uniformly Expone...mentioning

confidence: 99%

“…To be more precise, the damping control proposed in Vasconcellos and Silva 18 is supposed to have a delay term. This idea is used for the Korteweg–de Vries equation 25 . Of course, the presence of a time‐delay in the equation is motivated by the fact that in control systems, the sensors and actuators act under a delay 26–29 .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Well‐posedness and exponential stability of the Kawahara equation with a time‐delayed localized damping

Chentouf

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

The aim of this article is to investigate the well‐posedness and stability problems of the so‐called Kawahara equation under the presence of an interior localized delayed damping. The system is shown to be well‐posed. Furthermore, we prove that the trivial solution is exponentially stable in spite of the delay effect. Specifically, local and semi‐global stability results are established according to the properties of the spatial distribution of the delay term.

show abstract

\ell

.…”

Section: Introductionsupporting

confidence: 85%

“…Next, we choose

{\lambda}_1

and

{\lambda}_2

small satisfying () to ensure that

{B}_1

and

{B}_2

are negative. Finally, we choose

\gamma

as in () so that

{B}_3&#x0002B;{B}_4

and

{B}_7

are negative. It is worth mentioning that, contrary to Valein 25 for the KdV equation, the stability result stated in Theorem 2 does not require any smallness condition on the length

\ell

. This is expectable since the equation has a damping term. The condition A2 plays an important role in the previous study.…”

Section: The Problem (11) With ω⊂Trueω^$$ \Omega \Subset \Hat{\omega...mentioning

confidence: 92%

({, \begin{matrix} left left \\ array & array \partial_{t} u (x, t) + \partial_{x}^{3} u (x, t) - η \partial_{x}^{5} u (x, t) + u (x, t) \partial_{x} u (x, t) + α \partial_{x} u (x, t) \\ array + (a (x) + ξ b (x)) u (x, t) + b (x) u (x, t - τ) = 0, & array (x, t) \in Ω \times (0, \infty), \\ array u (0, t) = u (ℓ, t) = \partial_{x} u (0, t) = \partial_{x} u (ℓ, t) = \partial_{x}^{2} u (ℓ, t) = 0, & array t > 0, \\ array u (x, 0) = u_{0} (x), & array x \in Ω, \\ array u (x, t) = z_{0} (x, t), & array x \in Ω, t \in T, \end{matrix})

…”

Section: The Problem (11) With ω⊈Trueω^$$ \Omega \Nsubseteq \Hat{\om...mentioning

confidence: 99%

Section: The Trivial Solution Of the Problem (11) Is Uniformly Expone...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Well‐posedness and exponential stability of the Kawahara equation with a time‐delayed localized damping

Chentouf

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

Asymptotic behavior of the linearized compressible barotropic Navier‐Stokes system with a time varying delay term in the boundary or internal feedback

Majumdar

2023

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we consider the linearized compressible barotropic Navier‐Stokes system in a bounded interval with a time‐varying delay term acting in the Dirichlet boundary or internal feedback of the hyperbolic component. Assuming some suitable conditions on the time‐dependent delay term and the coefficients of feedback (delayed or not), we study the exponential stability of the concerned hyperbolic‐parabolic system. Due to the presence of the time‐varying delay term, the corresponding spatial operator is also time dependent. Using classical semigroup theory with Kato's variable norm approach, we first show the existence and uniqueness of the Navier‐Stokes system with time delay, acting in the boundary or interior. Next, we prove the two stabilization results by means of interior delay feedback and boundary delay. In both cases, we establish the exponential stability results by introducing some suitable functional energy and using the Lyapunov function approach.

show abstract

Delayed stabilization of the Korteweg–de Vries equation on a star-shaped network

Parada

Crépeau

Prieur

2022

Math. Control Signals Syst.

View full text Add to dashboard Cite

In this work we deal with the exponential stability of the nonlinear Kortewegde Vries (KdV) equation on a finite star-shaped network in the presence of delayed internal feedback. We start by proving the well-posedness of the system and some regularity results. Then we state an exponential stabilization result using a Lyapunov function by imposing small initial data and a restriction over the lengths. In this part also, we are able to obtain an explicit expression for the rate of decay. Then we prove the exponential stability of the solutions without restriction on the lengths and for small initial data, this result is based on an observability inequality. After that, we obtain a semi-global stabilization result working directly with the nonlinear system. Next we study the case where it may happen that a control domain with delay is outside of the control domain without delay. In that case, we obtain also a local exponential stabilization result. Finally, we present some numerical simulations in order to illustrate the stabilization.

show abstract

On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

Cited by 11 publications

References 33 publications

Well‐posedness and exponential stability of the Kawahara equation with a time‐delayed localized damping

Well‐posedness and exponential stability of the Kawahara equation with a time‐delayed localized damping

Asymptotic behavior of the linearized compressible barotropic Navier‐Stokes system with a time varying delay term in the boundary or internal feedback

Delayed stabilization of the Korteweg–de Vries equation on a star-shaped network

Contact Info

Product

Resources

About