Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications

Abdallah, Farah; Nicaise, Serge; Valein, Julie; Wehbe, Ali

doi:10.1051/cocv/2012036

Cited by 13 publications

(24 citation statements)

References 33 publications

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“…The results have been generalized in [34]. In [1], one can find similar results under appropriate spectral gap conditions (in this reference, we can also find results on polynomial decay, see further). In [36,37], viscosity operators are used for the plate equation, and for general classes of second-order evolution equations under appropriate spectral gap assumptions, combined with the uniform Huang-Prüss conditions derived in [32].…”

Section: Theorem 3 ([21]supporting

confidence: 59%

“…It easily follows that the function V defined on IR n by V (y) = y ⊤ P y is a Lyapunov function for the closed-loop systemẏ(t) = (A + BK)y(t). Now, for the semilinear system (1) in closed-loop with u(t) = Ky(t), we have d dt V (y(t)) = − y(t) 2 + y(t) ⊤ P F (y(t)) −C 1 y(t) 2 −C 2 V (y(t)),…”

Section: In Finite Dimensionmentioning

confidence: 99%

“…which is continuously embedded in L ∞ ((0, T ) × (0, L)). 1 First of all, in order to end up with a Dirichlet problem, we set…”

Section: Parabolic Pde'smentioning

confidence: 99%

“…In such cases, it is interesting to investigate the question of proving a uniform polynomial decay rate for space and/or time semi-discrete approximations. In [1], such results are stated for second-order linear equations, with appropriate viscosity terms, and under adequate spectral gap conditions. The extension to a more general framework (weaker assumptions, full discretizations), and to semilinear equations, seems to be open.…”

Section: Uniform Polynomial Energy Decay Without Observabilitymentioning

confidence: 99%

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Stabilization of Semilinear PDEs, and Uniform Decay under Discretization

Trélat

Evolution Equations

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These notes are issued from a short course given by the author in a summer school in Chambéry in June 2015.We consider general semilinear PDE's and we address the following two questions: 1) How to design an efficient feedback control locally stabilizing the equation asymptotically to 0? 2) How to construct such a stabilizing feedback from approximation schemes?To address these issues, we distinguish between parabolic and hyperbolic semilinear PDE's. By parabolic, we mean that the linear operator underlying the system generates an analytic semi-group. By hyperbolic, we mean that this operator is skew-adjoint.We first recall general results allowing one to consider the nonlinear term as a perturbation that can be absorbed when one is able to construct a Lyapunov function for the linear part. We recall in particular some known results borrowed from the Riccati theory.However, since the numerical implementation of Riccati operators is computationally demanding, we focus then on the question of being able to design "simple" feedbacks. For parabolic equations, we describe a method consisting of designing a stabilizing feedback, based on a small finite-dimensional (spectral) approximation of the whole system. For hyperbolic equations, we focus on simple linear or nonlinear feedbacks and we investigate the question of obtaining sharp decay results.When considering discretization schemes, the decay obtained in the continuous model cannot in general be preserved for the discrete model, and we address the question of adding appropriate viscosity terms in the numerical scheme, in order to recover a uniform decay. We consider space, time and then full discretizations and we report in particular on the most recent results obtained in the literature.Finally, we describe several open problems and issues.

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Section: Theorem 3 ([21]supporting

confidence: 59%

Section: In Finite Dimensionmentioning

confidence: 99%

“…which is continuously embedded in L ∞ ((0, T ) × (0, L)). 1 First of all, in order to end up with a Dirichlet problem, we set…”

Section: Parabolic Pde'smentioning

confidence: 99%

Section: Uniform Polynomial Energy Decay Without Observabilitymentioning

confidence: 99%

See 2 more Smart Citations

Stabilization of Semilinear PDEs, and Uniform Decay under Discretization

Trélat

Evolution Equations

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“…In that case, it would be of interest to establish a uniform polynomial decay rate for space and/or time semi-discrete and full discrete approximations of (1). In [1], such results are stated for second-order linear equations (certain examples being taken from [3,4]) , with appropriate viscosity terms, and under adequate spectral gap conditions.…”

Section: Do Microlocalization and Discretization Commute?mentioning

confidence: 99%

Nonlinear damped partial differential equations and their uniform discretizations

Alabau‐Boussouira

Privat

Trélat

2017

Journal of Functional Analysis

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We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time discretization parameters, by adding appropriate numerical viscosity terms. Our main arguments use the optimal-weight convexity method and uniform observability inequalities with respect to the discretization parameters. We establish our results, first in the continuous setting, then for space semi-discrete models, and then for time semi-discrete models. The full discretization is inferred from the previous results.Our results cover, for instance, the Schrödinger equation with nonlinear damping, the nonlinear wave equation, the nonlinear plate equation, the nonlinear transport equation, as well as certain classes of equations with nonlocal terms.

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Uniform exponential stability approximations of semi‐discretization schemes for two hybrid systems

Zhang,

Zheng,

Wang

et al. 2024

Math Methods in App Sciences

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The uniform exponential stabilities (UESs) of two hybrid control systems comprised of a wave equation and a second‐order ordinary differential equation are investigated in this study. Linear feedback law and local viscosity are considered, as are nonlinear feedback law and internal anti‐damping. The hybrid system is first reduced to a first‐order port‐Hamiltonian system with dynamical boundary conditions, and the resulting system is discretized using the average central‐difference scheme. Second, the UES of the discrete system is obtained without prior knowledge of the exponential stability of the continuous system. The frequency domain characterization of UES for a family of contractive semigroups and the discrete multiplier approach are used to validate the main conclusions. Finally, the Trotter–Kato theorem is used to perform a convergence study on the numerical approximation approach. Most notably, the exponential stability of the continuous system is derived by the convergence of energy and UES, which is a novel approach to studying the exponential stability of some complex systems. Numerical simulation is used to validate the effectiveness of the numerical approximating strategy.

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Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications

Cited by 13 publications

References 33 publications

Stabilization of Semilinear PDEs, and Uniform Decay under Discretization

Stabilization of Semilinear PDEs, and Uniform Decay under Discretization

Nonlinear damped partial differential equations and their uniform discretizations

Uniform exponential stability approximations of semi‐discretization schemes for two hybrid systems

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