doi: 10.3934/mcrf.2021051
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Controllability to rest of the Gurtin-Pipkin model

School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, 524048, China

* Corresponding author: Xiuxiang Zhou

Received  April 2021 Revised  August 2021 Early access October 2021

Fund Project: The first author is supported by the National Natural Science Foundation of China under grants 11926331, 11926337 and 11601213, and by the Foundation for Talents of Lingnan Normal University under grant ZL1612. The second author is supported by the Natural Science Foundation of Guangdong Province under grant 2018A0303070012

This paper is devoted to analyzing the controllability to rest of the Gurtin-Pipkin model, which is a class of differential equations with memory terms. The goal is not only to derive the state to vanish at some time but also to require the memory term to vanish at the same time, ensuring that the controlled system is controllable to rest. In order to get rid of the influence of memory, the controllability result is obtained by means of the Fourier type approach and the moment theory.

Citation: Xiuxiang Zhou, Shu Luan. Controllability to rest of the Gurtin-Pipkin model. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021051
References:
[1] S. A. Avdonin and S. A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, Cambridge, 1995.   Google Scholar
[2]

S. Avdonin and L. Pandolfi, Simultaneous temperature and flux controllability for heat equations with memory, Quart. Appl. Math., 71 (2013), 339-368.  doi: 10.1090/S0033-569X-2012-01287-7.  Google Scholar

[3]

V. Barbu and M. Iannelli, Controllability of the heat equation with memory, Differential Integral Equations, 13 (2000), 1393-1412.   Google Scholar

[4]

A. Belleni-Morante, An integro-differential equation arising from the theory of heat conduction in rigid materials with memory, Boll. Un. Mat. Ital. B, 15 (1978), 470-482.   Google Scholar

[5]

F. W. Chaves-SilvaX. Zhang and E. Zuazua, Controllability of evolution equations with memory, SIAM J. Control Optim., 55 (2017), 2437-2459.  doi: 10.1137/151004239.  Google Scholar

[6]

B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208.  doi: 10.1007/BF01596912.  Google Scholar

[7]

X. FuJ. Yong and X. Zhang, Controllability and observability of a heat equation with hyperbolic memory kernel, J. Differential Equations, 247 (2009), 2395-2439.  doi: 10.1016/j.jde.2009.07.026.  Google Scholar

[8]

S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM Control Optim. Calc. Var., 19 (2013), 288-300.  doi: 10.1051/cocv/2012013.  Google Scholar

[9]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rat. Mech. Anal., 31 (1968), 113-126.  doi: 10.1007/BF00281373.  Google Scholar

[10]

S. Ivanov and L. Pandolfi, Heat equation with memory: Lack of controllability to rest, J. Math. Anal. Appl., 355 (2009), 1-11.  doi: 10.1016/j.jmaa.2009.01.008.  Google Scholar

[11]

G. Knowles, Some problems in the control of distributed systems, and their numerical solution, SIAM J. Control Optim., 17 (1979), 5-22.  doi: 10.1137/0317002.  Google Scholar

[12]

P. Loreti and V. Komornik, Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005.  Google Scholar

[13]

P. LoretiL. Pandolfi and D. Sforza, Boundary controllability and observability of a viscoelastic string, SIAM J. Control Optim., 50 (2012), 820-844.  doi: 10.1137/110827740.  Google Scholar

[14]

Q. LüX. Zhang and E. Zuazua, Null controllability for wave equations with memory, J. Math. Pures Appl., 108 (2017), 500-531.  doi: 10.1016/j.matpur.2017.05.001.  Google Scholar

[15]

S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation, SIAM J. Control. Optim., 44 (2006), 1950-1972.  doi: 10.1137/S036301290444263X.  Google Scholar

[16]

R. K. Miller, Asymptotic stability properties of linear Volterra integrodifferential equations, J. Differential Equations, 10 (1971), 485-506.  doi: 10.1016/0022-0396(71)90008-8.  Google Scholar

[17]

L. Pandolfi, The controllability of the Gurtin-Pipkin equation: A cosine operator approach, Appl. Math. Optim., 52 (2005), 143-165.  doi: 10.1007/s00245-005-0819-0.  Google Scholar

[18]

L. Pandolfi, Riesz systems and moment method in the study of heat equations with memory in one space dimension, Discrete Continuous Dynam. Systems - B, 14 (2010), 1487-1510.   Google Scholar

[19]

L. Pandolfi, Distributed Systems with Persistent Memory: Control and Moment Problems, SpringerBriefs in Electrical and Computer Engineering. SpringerBriefs in Control, Automation and Robotics. Springer, Cham, 2014. doi: 10.1007/978-3-319-12247-2.  Google Scholar

[20]

J. Prüss, Evolutionary Integral Equations and Applications, vol. 87 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[21]

I. Romanov and A. Shamaev, Noncontrollability to rest of the two-dimensional distributed system governed by the integrodifferential equation, J. Optim. Theory Appl., 170 (2016), 772-782.  doi: 10.1007/s10957-016-0945-7.  Google Scholar

[22]

Q. TaoH. GaoB. Zhang and Z. Yao, Approximate controllability of a parabolic integro-differential equation, Math. Meth. Appl. Sci., 37 (2014), 2236-2244.  doi: 10.1002/mma.2970.  Google Scholar

[23] R. M. Young, An Introduction to Nonharmonic Fourier Series, Applied Mathematics, 93. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.   Google Scholar
[24]

X. Zhou, Integral-type approximate controllability of linear parabolic integro-differential equations, Systems Control Lett., 105 (2017), 44-47.  doi: 10.1016/j.sysconle.2017.04.007.  Google Scholar

show all references

References:
[1] S. A. Avdonin and S. A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, Cambridge, 1995.   Google Scholar
[2]

S. Avdonin and L. Pandolfi, Simultaneous temperature and flux controllability for heat equations with memory, Quart. Appl. Math., 71 (2013), 339-368.  doi: 10.1090/S0033-569X-2012-01287-7.  Google Scholar

[3]

V. Barbu and M. Iannelli, Controllability of the heat equation with memory, Differential Integral Equations, 13 (2000), 1393-1412.   Google Scholar

[4]

A. Belleni-Morante, An integro-differential equation arising from the theory of heat conduction in rigid materials with memory, Boll. Un. Mat. Ital. B, 15 (1978), 470-482.   Google Scholar

[5]

F. W. Chaves-SilvaX. Zhang and E. Zuazua, Controllability of evolution equations with memory, SIAM J. Control Optim., 55 (2017), 2437-2459.  doi: 10.1137/151004239.  Google Scholar

[6]

B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208.  doi: 10.1007/BF01596912.  Google Scholar

[7]

X. FuJ. Yong and X. Zhang, Controllability and observability of a heat equation with hyperbolic memory kernel, J. Differential Equations, 247 (2009), 2395-2439.  doi: 10.1016/j.jde.2009.07.026.  Google Scholar

[8]

S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM Control Optim. Calc. Var., 19 (2013), 288-300.  doi: 10.1051/cocv/2012013.  Google Scholar

[9]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rat. Mech. Anal., 31 (1968), 113-126.  doi: 10.1007/BF00281373.  Google Scholar

[10]

S. Ivanov and L. Pandolfi, Heat equation with memory: Lack of controllability to rest, J. Math. Anal. Appl., 355 (2009), 1-11.  doi: 10.1016/j.jmaa.2009.01.008.  Google Scholar

[11]

G. Knowles, Some problems in the control of distributed systems, and their numerical solution, SIAM J. Control Optim., 17 (1979), 5-22.  doi: 10.1137/0317002.  Google Scholar

[12]

P. Loreti and V. Komornik, Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005.  Google Scholar

[13]

P. LoretiL. Pandolfi and D. Sforza, Boundary controllability and observability of a viscoelastic string, SIAM J. Control Optim., 50 (2012), 820-844.  doi: 10.1137/110827740.  Google Scholar

[14]

Q. LüX. Zhang and E. Zuazua, Null controllability for wave equations with memory, J. Math. Pures Appl., 108 (2017), 500-531.  doi: 10.1016/j.matpur.2017.05.001.  Google Scholar

[15]

S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation, SIAM J. Control. Optim., 44 (2006), 1950-1972.  doi: 10.1137/S036301290444263X.  Google Scholar

[16]

R. K. Miller, Asymptotic stability properties of linear Volterra integrodifferential equations, J. Differential Equations, 10 (1971), 485-506.  doi: 10.1016/0022-0396(71)90008-8.  Google Scholar

[17]

L. Pandolfi, The controllability of the Gurtin-Pipkin equation: A cosine operator approach, Appl. Math. Optim., 52 (2005), 143-165.  doi: 10.1007/s00245-005-0819-0.  Google Scholar

[18]

L. Pandolfi, Riesz systems and moment method in the study of heat equations with memory in one space dimension, Discrete Continuous Dynam. Systems - B, 14 (2010), 1487-1510.   Google Scholar

[19]

L. Pandolfi, Distributed Systems with Persistent Memory: Control and Moment Problems, SpringerBriefs in Electrical and Computer Engineering. SpringerBriefs in Control, Automation and Robotics. Springer, Cham, 2014. doi: 10.1007/978-3-319-12247-2.  Google Scholar

[20]

J. Prüss, Evolutionary Integral Equations and Applications, vol. 87 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[21]

I. Romanov and A. Shamaev, Noncontrollability to rest of the two-dimensional distributed system governed by the integrodifferential equation, J. Optim. Theory Appl., 170 (2016), 772-782.  doi: 10.1007/s10957-016-0945-7.  Google Scholar

[22]

Q. TaoH. GaoB. Zhang and Z. Yao, Approximate controllability of a parabolic integro-differential equation, Math. Meth. Appl. Sci., 37 (2014), 2236-2244.  doi: 10.1002/mma.2970.  Google Scholar

[23] R. M. Young, An Introduction to Nonharmonic Fourier Series, Applied Mathematics, 93. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.   Google Scholar
[24]

X. Zhou, Integral-type approximate controllability of linear parabolic integro-differential equations, Systems Control Lett., 105 (2017), 44-47.  doi: 10.1016/j.sysconle.2017.04.007.  Google Scholar

[1]

Luciano Pandolfi. Joint identification via deconvolution of the flux and energy relaxation kernels of the Gurtin-Pipkin model in thermodynamics with memory. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1589-1599. doi: 10.3934/dcdss.2020090

[2]

Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143

[3]

Luciano Pandolfi. Riesz systems, spectral controllability and a source identification problem for heat equations with memory. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 745-759. doi: 10.3934/dcdss.2011.4.745

[4]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953

[5]

Abdelaziz Khoutaibi, Lahcen Maniar. Null controllability for a heat equation with dynamic boundary conditions and drift terms. Evolution Equations & Control Theory, 2020, 9 (2) : 535-559. doi: 10.3934/eect.2020023

[6]

Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations & Control Theory, 2020, 9 (2) : 399-430. doi: 10.3934/eect.2020011

[7]

Eduardo Cerpa. Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 91-102. doi: 10.3934/cpaa.2010.9.91

[8]

André da Rocha Lopes, Juan Límaco. Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021024

[9]

Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1

[10]

Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks & Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695

[11]

El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations & Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441

[12]

Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020

[13]

Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control & Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031

[14]

Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001

[15]

Debayan Maity. On the null controllability of the Lotka-Mckendrick system. Mathematical Control & Related Fields, 2019, 9 (4) : 719-728. doi: 10.3934/mcrf.2019048

[16]

Peng Gao. Null controllability with constraints on the state for the 1-D Kuramoto-Sivashinsky equation. Evolution Equations & Control Theory, 2015, 4 (3) : 281-296. doi: 10.3934/eect.2015.4.281

[17]

Idriss Boutaayamou, Lahcen Maniar, Omar Oukdach. Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021044

[18]

Sergei Avdonin, Julian Edward. Controllability for a string with attached masses and Riesz bases for asymmetric spaces. Mathematical Control & Related Fields, 2019, 9 (3) : 453-494. doi: 10.3934/mcrf.2019021

[19]

Orazio Arena. A problem of boundary controllability for a plate. Evolution Equations & Control Theory, 2013, 2 (4) : 557-562. doi: 10.3934/eect.2013.2.557

[20]

Rajesh Dhayal, Muslim Malik, Syed Abbas, Anil Kumar, Rathinasamy Sakthivel. Approximation theorems for controllability problem governed by fractional differential equation. Evolution Equations & Control Theory, 2021, 10 (2) : 411-429. doi: 10.3934/eect.2020073

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (35)
  • HTML views (33)
  • Cited by (0)

Other articles
by authors

[Back to Top]