Nonlinear boundary control with gain scheduled control for the porous elastic system

Mauro L. Santos; Mauro L. Santos

doi:10.3934/cam.2026021

Communications in Analysis and Mechanics

2026, Volume 18, Issue 3: 511-534. doi: 10.3934/cam.2026021

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Research article

Nonlinear boundary control with gain scheduled control for the porous elastic system

Mauro L. Santos ^,

Institute of Exact and Natural Sciences, Doctoral Program in Mathematics, Federal University of Pará, Belém-PA, 66075-110, Brazil

Received: 17 September 2025 Revised: 05 February 2026 Accepted: 26 March 2026 Published: 01 July 2026
65N30, 76S05, 70K15

In this paper, nonlinear boundary controls associated with two control gain for a one-dimensional continuous porous elastic system are designed, with the primary goal of uniformly stabilizing the model. The nonlinear boundary controllers were designed based on the spillover effect generated by high-frequency problems that generate so-called spurious oscillations. The existence and uniqueness of the solutions are proven using the Galerkin approximation method with a special basis constructed using the Gram-Schmidt process. Uniform stabilization is achieved using the method introduced by Lasiecka-Tataru.
- Galerkin method,
- porous elastic materials,
- uniform decay rates
Citation: Mauro L. Santos. Nonlinear boundary control with gain scheduled control for the porous elastic system[J]. Communications in Analysis and Mechanics, 2026, 18(3): 511-534. doi: 10.3934/cam.2026021

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Abstract

In this paper, nonlinear boundary controls associated with two control gain for a one-dimensional continuous porous elastic system are designed, with the primary goal of uniformly stabilizing the model. The nonlinear boundary controllers were designed based on the spillover effect generated by high-frequency problems that generate so-called spurious oscillations. The existence and uniqueness of the solutions are proven using the Galerkin approximation method with a special basis constructed using the Gram-Schmidt process. Uniform stabilization is achieved using the method introduced by Lasiecka-Tataru.

References

[1]	S. C. Cowin, J. W. Nunziato, Linear elastic materials with voids, J. Elasticity, 13 (1983), 125–147. https://doi.org/10.1007/BF00041230 doi: 10.1007/BF00041230
[2]	J. W. Nunziato, S. C. Cowin, A nonlinear theory of elastic materials with voids, Arch. Rational Mech. Anal., 72 (1979), 175–201. https://doi.org/10.1007/BF00249363 doi: 10.1007/BF00249363
[3]	S. C. Cowin, The viscoelastic behavior of linear elastic materials with voids, J. Elasticity, 15 (1985), 185–191. https://doi.org/10.1007/BF00041992 doi: 10.1007/BF00041992
[4]	M. Ciarletta, D. Iesan, Non-classical elastic solids, In: Pitman research notes in mathematics series, New York, 1993.
[5]	M. L. P. Nascimento, A. J. A. Ramos, M. M. Freitas, C. A. S. Nonato, Structural and microthermal effects on the exponential stabilization of poroelastic systems, Stud. Appl. Math., 155 (2025), e70089. https://doi.org/10.1111/sapm.70089 doi: 10.1111/sapm.70089
[6]	X. Xue, G. Shen, X. Wu, Y. Xiong, J. Liao, H. Bai, Thermo-mechanical performances of elastic–porous materials with metallic wire mesh structures, Compos Struct, 297 (2022), 115918. https://doi.org/10.1016/j.compstruct.2022.115918 doi: 10.1016/j.compstruct.2022.115918
[7]	A. Magaña, R. Quintanilla, On the time decay of solutions in one-dimensional theories of porous materials, Int. J. Solids Struct., 43 (2006), 3414–3427. https://doi.org/10.1016/j.ijsolstr.2005.06.077 doi: 10.1016/j.ijsolstr.2005.06.077
[8]	P. S. Casas, R. Quintanilla, Exponential decay in one–dimensional porous-thermo-elasticity, Mech. Research Commun., 32 (2005), 652–658. https://doi.org/10.1016/j.mechrescom.2005.02.015 doi: 10.1016/j.mechrescom.2005.02.015
[9]	R. Quintanilla, Slow decay for one-dimensional porous dissipation elasticity, Appl. Math. Lett., 16 (2003), 487–491. https://doi.org/10.1016/S0893-9659(03)00025-9 doi: 10.1016/S0893-9659(03)00025-9
[10]	M. L. Santos, A. D. S. Campelo, D. S. Almeida Júnior, Rates of decay for porous elastic system weakly dissipative, Acta Appl. Math., 151 (2017), 1–26. https://doi.org/10.1007/s10440-017-0100-y doi: 10.1007/s10440-017-0100-y
[11]	P. Xavier Pamplona, J. E. Muñoz Rivera, R. Quintanilla, Analyticity in porous-thermoelasticity with microtemperatures, J. Math. Anal. Appl., 394 (2012), 645–655. https://doi.org/10.1016/j.jmaa.2012.04.024 doi: 10.1016/j.jmaa.2012.04.024
[12]	J. E. Muñoz Rivera, R. Quintanilla, On the time polynomial decay in elastic solids with voids, J. Math. Anal. Appl., 338 (2008), 1296–1309. https://doi.org/10.1016/j.jmaa.2007.06.005 doi: 10.1016/j.jmaa.2007.06.005
[13]	J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847–857. https://doi.org/10.1090/S0002-9947-1984-0743749-9 doi: 10.1090/S0002-9947-1984-0743749-9
[14]	Z. Liu, S. Zheng, Semigroups associated with dissipative systems, Chapman and Hall/CRC, Boca Raton, 1999.
[15]	A. Soufyane, Energy decay for porous-thermo-elasticity systems of memory type, Appl. Anal., 87 (2008), 451–464. https://doi.org/10.1080/00036810802035634 doi: 10.1080/00036810802035634
[16]	A. Soufyane, M. Afilal, M. Chacha, General decay of solutions of a linear one-dimensional porous-thermoelasticity system with a boundary control of memory type, Nonlinear Anal., 72 (2010), 3903–3910. https://doi.org/10.1016/j.na.2010.01.004 doi: 10.1016/j.na.2010.01.004
[17]	A. Soufyane, M. Afilal, M. Chacha, Boundary stabilization of memory type for the porous-thermo-elasticity system, Abstr. Appl. Anal., 2009 (2009), 280790. https://doi.org/10.1155/2009/280790 doi: 10.1155/2009/280790
[18]	P. X. Pamplona, J. E. Muñoz Rivera, R. Quintanilla, Stabilization in elastic solids with voids, J. Math. Anal. Appl., 350 (2011), 37–49. https://doi.org/10.1016/j.jmaa.2008.09.026 doi: 10.1016/j.jmaa.2008.09.026
[19]	M. M. Freitas, M. L. Santos, José A. Langa, Porous elastic system with nonlinear damping and sources terms, J. Differ. Equs., 264 (2018), 2970–3051. https://doi.org/10.1016/j.jde.2017.11.006 doi: 10.1016/j.jde.2017.11.006
[20]	I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differ. Integral Equ., 6 (1993), 507–533. https://doi.org/10.57262/die/1370378427 doi: 10.57262/die/1370378427
[21]	M. L. Santos, D. S. Almeida Júnior, On porous-elastic system with localized damping, Z. Angew. Math. Phys., 67 (2016), 63. https://doi.org/10.1007/s00033-016-0622-6 doi: 10.1007/s00033-016-0622-6
[22]	M. L. Santos, D. S. Almeida Júnior, S. M. S. Cordeiro, Energy decay for a porous-elastic system with nonlinear localized damping, Z. Angew. Math. Phys., 73 (2022), 7. https://doi.org/10.1007/s00033-021-01636-1 doi: 10.1007/s00033-021-01636-1
[23]	M. M. Cavalcanti, V. N. D. Cavalcanti, I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differ. Equ., 236 (2007), 407–459. https://doi.org/10.1016/j.jde.2007.02.004 doi: 10.1016/j.jde.2007.02.004
[24]	M. L. Santos, D. S. Almeida Júnior, S. M. S. Cordeiro, R. F. C. Lobato, Double-wall carbon nanotubes model with nonlinear localized damping: Asymptotic stability, Adv. Differential Equations, 28 (2023), 753–777. https://doi.org/10.57262/ade028-0910-753 doi: 10.57262/ade028-0910-753
[25]	J. L. Lions, Quelques Mèthodes de resolution de problèmes aux limites non lineaires, Dunod Gauthiers Villars, Paris, 1969.
[26]	J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61–90. https://doi.org/10.1016/0022-247X(73)90121-2 doi: 10.1016/0022-247X(73)90121-2
[27]	Luc Jaouen, A. Renault, M. Deverge, Elastic and damping characterizations of acoustical porous materials: Available experimental methods and applications to a melamine foam, Appl. Acoust., 69 (2008), 1129–1140. https://doi.org/10.1016/j.apacoust.2007.11.008 doi: 10.1016/j.apacoust.2007.11.008

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