[Paper Review] Stability results of a distributed problem involving Bresse system with history and/or Cattaneo law under fully Dirichlet or mixed boundary conditions

Farah Abdallah, Mouhammad Ghader+1 more|arXiv (Cornell University)|Jan 2, 2016

Stability and Controllability of Differential Equations23 references21 citations

TL;DR

This paper investigates the stability of a one-dimensional Bresse system with infinite memory and/or Cattaneo's law for heat conduction under fully Dirichlet or mixed boundary conditions. Using spectral analysis and the frequency domain method, it establishes exponential stability when wave speeds are equal, and polynomial decay of order $ t^{-1/2} $ when speeds differ, extending prior results to more general boundary conditions with improved kernel conditions.

ABSTRACT

In this paper, we study the stability of a one-dimensional Bresse system with infinite memory type control and/or with heat conduction given by Cattaneo's law acting in the shear angle displacement. When the thermal effect vanishes, the system becomes elastic with memory term acting on one equation. Unlike [6], [10], and [22], we consider the interesting case of fully Dirichlet boundary conditions. Indeed, under equal speed of propagation condition, we establish the exponential stability of the system. However, in the natural physical case when the speeds of propagation are different, using a spectrum method, we show that the Bresse system is not uniformly stable. In this case, we establish a polynomial energy decay rate. Our study is valid for all other mixed boundary conditions and generalizes that of [6], [10], and [22].

Motivation & Objective

• To analyze the long-term behavior of a Bresse system with infinite memory and Cattaneo-type heat conduction under general boundary conditions.
• To extend existing stability results—previously limited to specific boundary conditions—by considering fully Dirichlet and mixed boundary conditions.
• To improve the kernel condition for the memory term by replacing a standard decay assumption with a stronger exponential decay hypothesis.
• To determine whether Cattaneo's law enhances energy decay compared to Fourier law or memory-only damping.
• To establish sharp decay rates under non-uniform wave speed conditions.

Proposed method

• Formulates the Bresse system with history term and Cattaneo's law for heat flux, incorporating physical parameters such as density, elasticity, and relaxation times.
• Applies the frequency domain method to analyze spectral properties and derive energy decay estimates.
• Uses a spectral analysis approach to study the resolvent operator and prove non-uniform stability when wave speeds differ.
• Implements a cut-off function technique to localize analysis and control the growth of solutions in the frequency domain.
• Derives energy estimates by taking inner products in the Hilbert space and leveraging the improved kernel condition $ g' \leq -c g $.
• Establishes decay rates by analyzing the asymptotic behavior of eigenvalues and resolvent norms as the frequency tends to infinity.

Experimental results

Research questions

• RQ1 Does the Bresse system with both infinite memory and Cattaneo's law exhibit exponential stability under fully Dirichlet boundary conditions when wave speeds are equal?
• RQ2 What is the energy decay rate when wave speeds are unequal, and does the system remain uniformly stable?
• RQ3 How does the improved kernel condition $ g' \leq -c g $ affect the stability analysis compared to previous works?
• RQ4 Can the stability results be generalized to mixed boundary conditions such as Dirichlet-Neumann or Neumann-Dirichlet?
• RQ5 Does the use of Cattaneo's law significantly improve the decay rate compared to Fourier's law or memory-only damping?

Key findings

• Under equal wave speed conditions, the system exhibits exponential stability, confirming that the energy decays at an exponential rate.
• When wave speeds differ, the system is not uniformly stable, and the energy decays polynomially at rate $ t^{-1/2} $.
• The improved kernel condition $ g' \leq -c g $ allows for stronger decay estimates and generalizes previous results that required weaker assumptions.
• The spectral method successfully identifies the optimal decay rate and proves that Cattaneo's law does not accelerate decay beyond the $ t^{-1/2} $ rate.
• The results hold for all mixed boundary conditions, including Dirichlet-Neumann and Neumann-Dirichlet, extending the scope of earlier studies.
• The paper leaves open the question of whether the polynomial decay rate is optimal and whether Cattaneo's law can be made to enhance stability beyond current bounds.