[Paper Review] Exponential stability to the Bresse system with boundary dissipation conditions

Margareth S. Alves, V. Octavio Vera+2 more|arXiv (Cornell University)|Jun 4, 2015

Stability and Controllability of Differential Equations12 references4 citations

TL;DR

This paper establishes exponential stability for the Bresse system with three boundary dissipation conditions using semigroup theory and a frequency domain approach. It proves that the energy decays exponentially over time when three dissipative effects—on longitudinal, shear angle, and vertical displacements—are applied at the boundary, resolving a key challenge in boundary stabilization of this structural model.

ABSTRACT

We consider the Bresse model with three control boundary conditions. We prove the exponential stability of the system using the semigroup theory of linear operators and a result obtained by Prüss.

Motivation & Objective

• To investigate the long-term stability of the Bresse system under three boundary dissipation conditions.
• To determine whether boundary damping alone can yield exponential energy decay, despite the system's inherent complexity.
• To extend existing results on Timoshenko-type systems to the more intricate Bresse model with boundary controls.
• To overcome mathematical challenges in proving exponential stability when only boundary dissipation is present.
• To establish a foundation for future work on complete boundary stabilization of the Bresse system.

Proposed method

• Formulates the Bresse system as a first-order evolution equation in a Hilbert space using semigroup theory.
• Imposes three boundary dissipation conditions: one on longitudinal displacement, one on shear angle, and one on vertical displacement.
• Applies the frequency domain method to analyze the resolvent operator and verify the spectrum determined growth assumption.
• Uses a multiplier technique with a specific test function $ q(x) = x - \ell $ to derive energy estimates.
• Applies the Lumer-Philips theorem and a result by Prüss to conclude exponential stability from the resolvent estimate.
• Establishes a key inequality involving the $ L^2 $-norms of the state variables and their derivatives to bound the energy decay.

Experimental results

Research questions

• RQ1 Can exponential stability be achieved in the Bresse system when only three boundary dissipation effects are applied?
• RQ2 What conditions on the boundary damping parameters $ \gamma_1, \gamma_2, \gamma_3 $ ensure exponential decay of the energy?
• RQ3 How does the presence of three boundary controls compare to internal or distributed damping in terms of stability properties?
• RQ4 Is the Bresse system exponentially stable under boundary dissipation when wave speeds differ?
• RQ5 What mathematical techniques are necessary to overcome the challenges in proving exponential stability for boundary-controlled Bresse systems?

Key findings

• The semigroup generated by the system is exponentially stable, meaning the energy decays as $ \|\mathcal{S}_{\mathcal{A}}(t)\|_{\mathcal{L}(\mathcal{H})} \leq M e^{-\mu t} $ for some $ M, \mu > 0 $.
• Exponential stability is achieved despite the system's inherent non-uniform wave speeds, which typically prevent such decay.
• The proof relies on a frequency domain approach and a resolvent estimate that holds uniformly for all real frequencies $ \lambda $.
• The energy estimate is derived using a carefully chosen multiplier function $ q(x) = x - \ell $, which allows control over boundary and internal terms.
• The method establishes a bound on the $ \mathcal{H} $-norm of the resolvent operator, confirming the spectrum determined growth property.
• The result holds under the assumption that the boundary damping coefficients $ \gamma_1, \gamma_2, \gamma_3 $ are all positive, ensuring effective energy dissipation at the boundary.