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Published in: Meccanica 6/2023

16-07-2022 | General

Boundary controllability for the 1D Moore–Gibson–Thompson equation

Authors: Carlos Lizama, Sebastián Zamorano

Published in: Meccanica | Issue 6/2023

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Abstract

This article addresses the boundary controllability problem for a class of third order in time PDE, known as Moore–Gibson–Thompson equation, with a control supported on the boundary. It is shown that it is not spectrally controllable, which means that nontrivial finite linear combination of eigenvectors can be driven to zero in finite time. This implies that the Moore–Gibson–Thompson equation is not exact and null controllable. However, the approximate controllability will be proved.

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Literature
1.
go back to reference Kaltenbacher B, Lasiecka I, Marchand R (2011) Wellposedness and exponential decay rates for the Moore–Gibson–Thompson equation arising in high intensity ultrasound. Control Cybern 40(4):971–988MathSciNetMATH Kaltenbacher B, Lasiecka I, Marchand R (2011) Wellposedness and exponential decay rates for the Moore–Gibson–Thompson equation arising in high intensity ultrasound. Control Cybern 40(4):971–988MathSciNetMATH
2.
go back to reference Marchand R, McDevitt T, Triggiani R (2012) An abstract semigroup approach to the third-order Moore–Gibson–Thompson partial differential equation arising in high-intensity ultrasound: structural decomposition, spectral analysis, exponential stability. Math Methods Appl Sci 35(15):1896–1929MathSciNetCrossRefMATH Marchand R, McDevitt T, Triggiani R (2012) An abstract semigroup approach to the third-order Moore–Gibson–Thompson partial differential equation arising in high-intensity ultrasound: structural decomposition, spectral analysis, exponential stability. Math Methods Appl Sci 35(15):1896–1929MathSciNetCrossRefMATH
3.
go back to reference Fernández C, Lizama C, Poblete V (2011) Regularity of solutions for a third order differential equation in Hilbert spaces. Appl Math Comput 217(21):8522–8533MathSciNetMATH Fernández C, Lizama C, Poblete V (2011) Regularity of solutions for a third order differential equation in Hilbert spaces. Appl Math Comput 217(21):8522–8533MathSciNetMATH
4.
go back to reference Dell’Oro F, Pata V (2017) On the Moore–Gibson–Thompson equation and its relation to linear viscoelasticity. Appl Math Optim 76(3):641–655MathSciNetCrossRefMATH Dell’Oro F, Pata V (2017) On the Moore–Gibson–Thompson equation and its relation to linear viscoelasticity. Appl Math Optim 76(3):641–655MathSciNetCrossRefMATH
5.
go back to reference Prüss J (1993) Evolutionary integral equations and applications. Monographs in mathematics, vol 87. Birkhäuser, Basel, p 366CrossRefMATH Prüss J (1993) Evolutionary integral equations and applications. Monographs in mathematics, vol 87. Birkhäuser, Basel, p 366CrossRefMATH
6.
go back to reference Caixeta AH, Lasiecka I, Cavalcanti VND (2016) Global attractors for a third order in time nonlinear dynamics. J Differ Equ 261(1):113–147MathSciNetCrossRefMATH Caixeta AH, Lasiecka I, Cavalcanti VND (2016) Global attractors for a third order in time nonlinear dynamics. J Differ Equ 261(1):113–147MathSciNetCrossRefMATH
7.
go back to reference de Andrade B, Lizama C (2011) Existence of asymptotically almost periodic solutions for damped wave equations. J Math Anal Appl 382(2):761–771MathSciNetCrossRefMATH de Andrade B, Lizama C (2011) Existence of asymptotically almost periodic solutions for damped wave equations. J Math Anal Appl 382(2):761–771MathSciNetCrossRefMATH
8.
go back to reference Araya D, Lizama C (2012) Existence of asymptotically almost automorphic solutions for a third order differential equation. Electron J Qual Theory Differ Equ 53:1–20MathSciNetCrossRefMATH Araya D, Lizama C (2012) Existence of asymptotically almost automorphic solutions for a third order differential equation. Electron J Qual Theory Differ Equ 53:1–20MathSciNetCrossRefMATH
9.
go back to reference Conejero JA, Lizama C, Rodenas F (2015) Chaotic behaviour of the solutions of the Moore–Gibson–Thompson equation. Appl Math Inf Sci 9(5):2233–2238MathSciNet Conejero JA, Lizama C, Rodenas F (2015) Chaotic behaviour of the solutions of the Moore–Gibson–Thompson equation. Appl Math Inf Sci 9(5):2233–2238MathSciNet
10.
go back to reference Cai G, Bu S (2016) Periodic solutions of third-order degenerate differential equations in vector-valued functional spaces. Isr J Math 212(1):163–188MathSciNetCrossRefMATH Cai G, Bu S (2016) Periodic solutions of third-order degenerate differential equations in vector-valued functional spaces. Isr J Math 212(1):163–188MathSciNetCrossRefMATH
11.
go back to reference Kalantarov VK, Yilmaz Y (2013) Decay and growth estimates for solutions of second-order and third-order differential-operator equations. Nonlinear Anal TMA 88:1–7MathSciNetCrossRefMATH Kalantarov VK, Yilmaz Y (2013) Decay and growth estimates for solutions of second-order and third-order differential-operator equations. Nonlinear Anal TMA 88:1–7MathSciNetCrossRefMATH
12.
go back to reference Liu S, Triggiani R (2013) An inverse problem for a third order PDE arising in high-intensity ultrasound: global uniqueness and stability by one boundary measurement. J Inverse Ill-Posed Probl 21(6):825–869MathSciNetCrossRefMATH Liu S, Triggiani R (2013) An inverse problem for a third order PDE arising in high-intensity ultrasound: global uniqueness and stability by one boundary measurement. J Inverse Ill-Posed Probl 21(6):825–869MathSciNetCrossRefMATH
13.
go back to reference Liu S, Triggiani R (2014) Inverse problem for a linearized Jordan–Moore–Gibson–Thompson equation. Springer INdAM Ser 10:305–351MathSciNetCrossRefMATH Liu S, Triggiani R (2014) Inverse problem for a linearized Jordan–Moore–Gibson–Thompson equation. Springer INdAM Ser 10:305–351MathSciNetCrossRefMATH
14.
go back to reference Lasiecka I, Wang X (2015) Moore–Gibson–Thompson equation with memory, part II: general decay of energy. J Differ Equ 259(12):7610–7635MathSciNetCrossRefMATH Lasiecka I, Wang X (2015) Moore–Gibson–Thompson equation with memory, part II: general decay of energy. J Differ Equ 259(12):7610–7635MathSciNetCrossRefMATH
15.
go back to reference Dell’Oro F, Lasiecka I, Pata V (2016) The Moore–Gibson–Thompson equation with memory in the critical case. J Differ Equ 261(7):4188–4222MathSciNetCrossRefMATH Dell’Oro F, Lasiecka I, Pata V (2016) The Moore–Gibson–Thompson equation with memory in the critical case. J Differ Equ 261(7):4188–4222MathSciNetCrossRefMATH
16.
go back to reference Lasiecka I (2017) Global solvability of Moore–Gibson–Thompson equation with memory arising in nonlinear acoustics. J Evol Equ 17(1):411–441MathSciNetCrossRefMATH Lasiecka I (2017) Global solvability of Moore–Gibson–Thompson equation with memory arising in nonlinear acoustics. J Evol Equ 17(1):411–441MathSciNetCrossRefMATH
18.
go back to reference Nikolić V, Said-Houari B (2021) Asymptotic behavior of nonlinear sound waves in inviscid media with thermal and molecular relaxation. Nonlinear Anal Real World Appl 62:103384–38MathSciNetCrossRefMATH Nikolić V, Said-Houari B (2021) Asymptotic behavior of nonlinear sound waves in inviscid media with thermal and molecular relaxation. Nonlinear Anal Real World Appl 62:103384–38MathSciNetCrossRefMATH
19.
go back to reference Bongarti M, Charoenphon S, Lasiecka I (2021) Vanishing relaxation time dynamics of the Jordan Moore–Gibson–Thompson equation arising in nonlinear acoustics. J Evol Equ 21(3):3553–3584MathSciNetCrossRefMATH Bongarti M, Charoenphon S, Lasiecka I (2021) Vanishing relaxation time dynamics of the Jordan Moore–Gibson–Thompson equation arising in nonlinear acoustics. J Evol Equ 21(3):3553–3584MathSciNetCrossRefMATH
20.
go back to reference Nikolić V, Said-Houari B (2021) On the Jordan–Moore–Gibson–Thompson wave equation in hereditary fluids with quadratic gradient nonlinearity. J Math Fluid Mech 23(1):3–24MathSciNetCrossRefMATH Nikolić V, Said-Houari B (2021) On the Jordan–Moore–Gibson–Thompson wave equation in hereditary fluids with quadratic gradient nonlinearity. J Math Fluid Mech 23(1):3–24MathSciNetCrossRefMATH
21.
go back to reference Bounadja H, Said Houari B (2021) Decay rates for the Moore–Gibson–Thompson equation with memory. Evol Equ Control Theory 10(3):431–460MathSciNetCrossRefMATH Bounadja H, Said Houari B (2021) Decay rates for the Moore–Gibson–Thompson equation with memory. Evol Equ Control Theory 10(3):431–460MathSciNetCrossRefMATH
22.
go back to reference Kaltenbacher B, Nikolić V (2021) The inviscid limit of third-order linear and nonlinear acoustic equations. SIAM J Appl Math 81(4):1461–1482MathSciNetCrossRefMATH Kaltenbacher B, Nikolić V (2021) The inviscid limit of third-order linear and nonlinear acoustic equations. SIAM J Appl Math 81(4):1461–1482MathSciNetCrossRefMATH
26.
go back to reference Lizama C, Zamorano S (2019) Controllability results for the Moore–Gibson–Thompson equation arising in nonlinear acoustics. J Differ Equ 266(12):7813–7843MathSciNetCrossRefMATH Lizama C, Zamorano S (2019) Controllability results for the Moore–Gibson–Thompson equation arising in nonlinear acoustics. J Differ Equ 266(12):7813–7843MathSciNetCrossRefMATH
27.
go back to reference Bucci F, Lasiecka I (2019) Feedback control of the acoustic pressure in ultrasonic wave propagation. Optimization 68(10):1811–1854MathSciNetCrossRefMATH Bucci F, Lasiecka I (2019) Feedback control of the acoustic pressure in ultrasonic wave propagation. Optimization 68(10):1811–1854MathSciNetCrossRefMATH
28.
go back to reference Russell D (1986) Mathematical models for the elastic beam and their control-theoretic implications. Semigroups Theory Appl 2(152):177–216MathSciNet Russell D (1986) Mathematical models for the elastic beam and their control-theoretic implications. Semigroups Theory Appl 2(152):177–216MathSciNet
29.
go back to reference Leugering G, Schmidt E, Meister E (1989) Boundary control of a vibrating plate with internal damping. Math Methods Appl Sci 11(5):573–586MathSciNetCrossRefMATH Leugering G, Schmidt E, Meister E (1989) Boundary control of a vibrating plate with internal damping. Math Methods Appl Sci 11(5):573–586MathSciNetCrossRefMATH
30.
31.
go back to reference Martin P, Rosier L, Rouchon P (2013) Null controllability of the structurally damped wave equation with moving control. SIAM J Control Optim 51(1):660–684MathSciNetCrossRefMATH Martin P, Rosier L, Rouchon P (2013) Null controllability of the structurally damped wave equation with moving control. SIAM J Control Optim 51(1):660–684MathSciNetCrossRefMATH
32.
go back to reference Rosier L, Rouchon P (2007) On the controllability of a wave equation with structural damping. Int J Tomogr Stat 5(W07):79–84MathSciNet Rosier L, Rouchon P (2007) On the controllability of a wave equation with structural damping. Int J Tomogr Stat 5(W07):79–84MathSciNet
33.
go back to reference Sheu TWH, Solovchuck MA, Chen AWJ, Thiriet M (2011) On an acoustic-thermal-fluid coupling model for the prediction of temperature elevation of liver tumor. Int J Heat Mass Transf 54:4117–4126CrossRefMATH Sheu TWH, Solovchuck MA, Chen AWJ, Thiriet M (2011) On an acoustic-thermal-fluid coupling model for the prediction of temperature elevation of liver tumor. Int J Heat Mass Transf 54:4117–4126CrossRefMATH
34.
go back to reference Pellicer M, Solà-Morales J (2019) Optimal scalar products in the Moore–Gibson–Thompson equation. Evol Equ Control Theory 8(1):203–220MathSciNetCrossRefMATH Pellicer M, Solà-Morales J (2019) Optimal scalar products in the Moore–Gibson–Thompson equation. Evol Equ Control Theory 8(1):203–220MathSciNetCrossRefMATH
35.
go back to reference Bucci F, Pandolfi L (2020) On the regularity of solutions to the Moore–Gibson–Thompson equation: a perspective via wave equations with memory. J Evol Equ 20(3):837–867MathSciNetCrossRefMATH Bucci F, Pandolfi L (2020) On the regularity of solutions to the Moore–Gibson–Thompson equation: a perspective via wave equations with memory. J Evol Equ 20(3):837–867MathSciNetCrossRefMATH
36.
go back to reference Lions, J.L.: Contrôlabilité Exacte Perturbations et Stabilisation de Systèmes Distribués. Tome 1, Contrôlabilité exacte, vol 8. Recherches en mathematiques appliquées, Masson (1988) Lions, J.L.: Contrôlabilité Exacte Perturbations et Stabilisation de Systèmes Distribués. Tome 1, Contrôlabilité exacte, vol 8. Recherches en mathematiques appliquées, Masson (1988)
38.
go back to reference Rudin W (1991) Functional analysis, 2nd edn. International series in pure and applied mathematics. McGraw-Hill, Inc., New York, p 424 Rudin W (1991) Functional analysis, 2nd edn. International series in pure and applied mathematics. McGraw-Hill, Inc., New York, p 424
39.
go back to reference Brezis H (2011) Functional analysis, Sobolev spaces and partial differential equations. Springer, New YorkCrossRefMATH Brezis H (2011) Functional analysis, Sobolev spaces and partial differential equations. Springer, New YorkCrossRefMATH
40.
go back to reference Chaves-Silva FW, Rosier L, Zuazua E (2014) Null controllability of a system of viscoelasticity with a moving control. Journal de Mathématiques Pures et Appliquées 101(2):198–222MathSciNetCrossRefMATH Chaves-Silva FW, Rosier L, Zuazua E (2014) Null controllability of a system of viscoelasticity with a moving control. Journal de Mathématiques Pures et Appliquées 101(2):198–222MathSciNetCrossRefMATH
41.
go back to reference Rosier L, Zhang B-Y (2013) Unique continuation property and control for the Benjamin–Bona–Mahony equation on a periodic domain. J Differ Equ 254(1):141–178MathSciNetCrossRefMATH Rosier L, Zhang B-Y (2013) Unique continuation property and control for the Benjamin–Bona–Mahony equation on a periodic domain. J Differ Equ 254(1):141–178MathSciNetCrossRefMATH
42.
Metadata
Title
Boundary controllability for the 1D Moore–Gibson–Thompson equation
Authors
Carlos Lizama
Sebastián Zamorano
Publication date
16-07-2022
Publisher
Springer Netherlands
Published in
Meccanica / Issue 6/2023
Print ISSN: 0025-6455
Electronic ISSN: 1572-9648
DOI
https://doi.org/10.1007/s11012-022-01551-3

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