On the control of viscoelastic damped swelling porous elastic soils with internal delay feedbacks

Abstract We consider a swelling porous-elastic system with viscoelastic damping and delay feedbacks acting on the fluid equation. Using the multiplier method and under the well-known assumption on the weight of delay term, we unexpectedly establish a general decay result without imposing the usual condition of equal wave speeds of the system, unlike the case of Timoshenko and porous systems where damping on only one of the equations requires equal wave speeds propagation. Our coupling and the result give new contributions to the theory associated with asymptotic behaviors of swelling porous elastic soils.

[1]  Yamina Laskri,et al.  A stability result of a Timoshenko system with a delay term in the internal feedback , 2010, Appl. Math. Comput..

[2]  T. Karalis.  On the elastic deformation of non-saturated swelling soils , 1990 .

[3]  S. Boulite,et al.  Feedback stabilization of a class of evolution equations with delay , 2009 .

[4]  T. Apalara.  Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks , 2016 .

[5]  Kazi Moinul Islam,et al.  Forensic Investigations into Recurrent Pavement Heave from Underlying Expansive Soil Deposits , 2018 .

[6]  R. Quintanilla,et al.  Anti-plane shear deformations of swelling porous elastic soils , 2003 .

[7]  A. Cemal Eringen,et al.  A continuum theory of swelling porous elastic soils , 1994 .

[8]  D. S. Drumheller,et al.  Theories of immiscible and structured mixtures , 1983 .

[9]  Bao-Zhu Guo,et al.  On the stability of swelling porous elastic soils with fluid saturation by one internal damping , 2006 .

[10]  Z. Bien,et al.  Use of time-delay actions in the controller design , 1980 .

[11]  R. Quintanilla.  Existence and exponential decay in the linear theory of viscoelastic mixtures , 2005 .

[12]  D. Ieşan.  ON THE THEORY OF MIXTURES OF THERMOELASTIC SOLIDS , 1991 .

[13]  R. Quintanilla,et al.  Letter to the Editor: exponential stability for one-dimensional problem of swelling porous elastic soils with fluid saturation , 2002 .

[14]  John H. Cushman,et al.  Thermomechanical theories for swelling porous media with microstructure , 2000 .

[15]  Tijani A. Apalara,et al.  General stability result of swelling porous elastic soils with a viscoelastic damping , 2020, Zeitschrift für angewandte Mathematik und Physik.

[16]  Jean-Pierre Richard,et al.  Time-delay systems: an overview of some recent advances and open problems , 2003, Autom..

[17]  A. Ramos,et al.  Stability results for elastic porous media swelling with nonlinear damping , 2020 .

[18]  R. Quintanilla.  On the linear problem of swelling porous elastic soils with incompressible fluid , 2002 .

[19]  Tijani A. Apalara,et al.  On the stabilization of laminated beams with delay , 2021, Applications of Mathematics.

[20]  Serge Nicaise,et al.  Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks , 2006, SIAM J. Control. Optim..

[21]  Michael P. Polis,et al.  An example on the effect of time delays in boundary feedback stabilization of wave equations , 1986 .

[22]  R. Quintanilla.  Exponential Stability of Solutions of Swelling Porous Elastic Soils , 2004 .