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Etude qualitative de quelques EDPs en temps avec amortissement

Etude qualitative de quelques EDPs en temps avec amortissement

LAKEHAL, IBRAHIM
URI: https://dspace.univ-bba.dz:443/xmlui/handle/123456789/5056
Date: 2024

Abstract:

 
This thesis is devoted to the study of two problems related to the theory of control of PDE. In a first and second time, we study two nonlinear Euler-Bernoulli beams with a neutral type delay and viscoelastic, using controls acting on the free boundaries. By using the method of Faedo-Galerkin, we prove the existence and uniqueness of the solution for each problem. After that using the energy method and constructing an appropriate Lyapunov function, under certain conditions on the neutral delay term kernel and the viscoelastic term, we show that although, the destructive nature of delay in general, which is a very general degrading energy problem.
 
This thesis is devoted to the study of two problems related to the theory of control of PDE. In a first and second time, we study two nonlinear Euler-Bernoulli beams with a neutral type delay and viscoelastic, using controls acting on the free boundaries. By using the method of Faedo-Galerkin, we prove the existence and uniqueness of the solution for each problem. After that using the energy method and constructing an appropriate Lyapunov function, under certain conditions on the neutral delay term kernel and the viscoelastic term, we show that although, the destructive nature of delay in general, which is a very general degrading energy problem.
 

Description:

In conclusion, the qualitative exploration of selected Partial Differential Equations (PDEs) concerning temporal dynamics with damping, particularly focusing on the examination of two non-linear Euler-Bernoulli beams featuring neutral-type delays and viscoelasticity, has yielded significant insights into their behaviors. The investigation has uncovered intricate dynamics within the Euler-Bernoulli beams, shedding light on the intricate interplay between nonlinearity, damping effects, and temporal delays. Incorporating viscoelastic properties has introduced additional layers of complexity, influencing the overall system responses in nuanced ways. Stability analysis has played a pivotal role in comprehending the long-term behaviors of the systems under scrutiny. By scrutinizing spectral properties and employing advanced analytical techniques such as Lyapunov functionals, criteria for stability have been delineated, offering valuable discernment into the conditions dictating system stability or instability. The implications of this study extend beyond theoretical realms, offering practical insights applicable to various engineering domains, including vibration control, structural health monitoring, and the design of damping systems. Understanding the complexities inherent in such systems is paramount for ensuring the reliability and performance of engineered structures. While significant strides have been made, numerous avenues for future exploration remain open. These include delving into more intricate beam configurations, exploring diverse damping mechanisms, accounting for uncertainties and parameter variations, and broadening the analysis to encompass other classes of PDEs sharing similar charac- 68 teristics. We believe that it would be interesting to study in future the following Timoshenko beam with thermodiffusion effects:  ρh3 12 ϕtt −ϕxx + k(ϕ +ψx) − δ1ϱx − δ2Px = 0, ρh ψtt − k(ϕ +ψx)x − [ψx ηx + 1 2ψ2 x ]x = 0, ρhηtt − ηx + 1 2ψ2 x x = 0, cϱt + dPt − ∞R 0 ω1(s)ϱxx(t − s)ds − δ1ϕtx = 0, dϱt + rPt − ∞R 0 ω2(s)Pxx(t − s)ds − δ2ϕtx = 0.
 
In conclusion, the qualitative exploration of selected Partial Differential Equations (PDEs) concerning temporal dynamics with damping, particularly focusing on the examination of two non-linear Euler-Bernoulli beams featuring neutral-type delays and viscoelasticity, has yielded significant insights into their behaviors. The investigation has uncovered intricate dynamics within the Euler-Bernoulli beams, shedding light on the intricate interplay between nonlinearity, damping effects, and temporal delays. Incorporating viscoelastic properties has introduced additional layers of complexity, influencing the overall system responses in nuanced ways. Stability analysis has played a pivotal role in comprehending the long-term behaviors of the systems under scrutiny. By scrutinizing spectral properties and employing advanced analytical techniques such as Lyapunov functionals, criteria for stability have been delineated, offering valuable discernment into the conditions dictating system stability or instability. The implications of this study extend beyond theoretical realms, offering practical insights applicable to various engineering domains, including vibration control, structural health monitoring, and the design of damping systems. Understanding the complexities inherent in such systems is paramount for ensuring the reliability and performance of engineered structures. While significant strides have been made, numerous avenues for future exploration remain open. These include delving into more intricate beam configurations, exploring diverse damping mechanisms, accounting for uncertainties and parameter variations, and broadening the analysis to encompass other classes of PDEs sharing similar charac- 68 teristics. We believe that it would be interesting to study in future the following Timoshenko beam with thermodiffusion effects:  ρh3 12 ϕtt −ϕxx + k(ϕ +ψx) − δ1ϱx − δ2Px = 0, ρh ψtt − k(ϕ +ψx)x − [ψx ηx + 1 2ψ2 x ]x = 0, ρhηtt − ηx + 1 2ψ2 x x = 0, cϱt + dPt − ∞R 0 ω1(s)ϱxx(t − s)ds − δ1ϕtx = 0, dϱt + rPt − ∞R 0 ω2(s)Pxx(t − s)ds − δ2ϕtx = 0.
 

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