American Institute of Mathematical Sciences

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2015, 2015(special): 588-595. doi: 10.3934/proc.2015.0588

Noncontrollability for the Colemann-Gurtin model in several dimensions

Andrei Halanay 1, and  Luciano Pandolfi 2,

1. 

Department of Mathematics and Informatics, University Politehnica of Bucharest, 313 Splaiul Independentei, 060042 Bucharest
2. 

Dipartimento di Scienze Matematiche "Giuseppe Luigi Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received  September 2014 Revised  August 2015 Published  November 2015

It is proved that, for every smooth kernel $M(t)$, the corresponding Colemann-Gurtin model in several spatial dimensions cannot be controlled to zero.
Keywords: moment methods, exponential families., lack of controllability, Heat equation with memory, controllability to zero.
Mathematics Subject Classification: Primary: 35Q93, 45K05, 93B0.
Citation: Andrei Halanay, Luciano Pandolfi. Noncontrollability for the Colemann-Gurtin model in several dimensions. Conference Publications, 2015, 2015 (special) : 588-595. doi: 10.3934/proc.2015.0588
References:
[1]

S.A. Avdonin, S.A. Ivanov, Families of exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, New-York, 1995. Google Scholar

[2]

S. Avdonin and L. Pandolfi, Simultaneous temperature and flux controllability for heat equation with memory,, Quarterly Appl. Math., (): 0033.   Google Scholar

[3]

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

[4]

A. Bensoussan, G. Da Prato, M. C. Delfour, and S.K. Mitter, Representation and control of infinite dimensional systems, Birkhäuser Boston, MA, 2007. Google Scholar

[5]

B. D. Colemann, M. E. Gurtin, Equipresence and constitutive equations for heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208. Google Scholar

[6]

H. O. Fattorini, D. L. Rusell, Exact Controllability Theorems for Linear Parabolic Equations in One Space Dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292. Google Scholar

[7]

G. Gripenberg, S. O. Londen, O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and Its Applications, 34, Cambridge University Press, Cambridge, 1990. Google Scholar

[8]

S. Guerrero, O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 288-300. Google Scholar

[9]

A. Halanay, L. Pandolfi, Lack of controllability of the heat equation with memory, Systems & Control Letters, 61 (2012), 999-1002. Google Scholar

[10]

A. Halanay, L. Pandolfi, Lack of controllability of thermal systems with memory, Evol. Eq. Control Theory, 3(3) (2014), 485-497. Google Scholar

[11]

A. Halanay, L. Pandolfi, Approximate controllability and lack of controllability to zero of the heat equation with memory, J. Math. Anal. Appl., 425 (2015), 194-211. Google Scholar

[12]

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

[13]

I. Lasiecka, R. Triggiani, Control theory for partial differential equations: continuous and approximation theories. II. Abstract hyperbolic-like systems over a finite time horizon, Encyclopedia of Mathematics and Its Applications, 75. Cambridge University Press, Cambridge 2000. Google Scholar

[14]

V. Lakshmikantham, R. M. Rao, Theory of integro-differential equations, Gordon & Breach, Lausanne 1995. Google Scholar

[15]

J. L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Dunod, Paris 1968. Google Scholar

[16]

V. P. Mikhailov, Partial Differential Equations, Mir, Moscou 1978. Google Scholar

[17]

L. Pandolfi, Riesz systems and controllability of heat equations with memory, Int. Eq. Operator Theory, 64 (2009), 429-453. Google Scholar

[18]

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

[19]

L. Pandolfi, Sharp control time in viscoelasticity,, in preparation., ().   Google Scholar

[20]

D. D. Joseph, L. Preziosi, Heat waves, Rev. Modern Phys., 61 (1989), 41-73; Addendum to the paper: "Heat waves", Rev. Modern Phys., 62 (1990), 375-391. Google Scholar

[21]

L. Schwartz, Etude des sommes d'exponentielles. Hermann, Paris 1959. Google Scholar

[22]

M. Tucsnak, G. Weiss:, Observation and control for operator semigroups, \/ Birkhäuser, Base, 2009. Google Scholar

show all references

References:
[1]

S.A. Avdonin, S.A. Ivanov, Families of exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, New-York, 1995. Google Scholar

[2]

S. Avdonin and L. Pandolfi, Simultaneous temperature and flux controllability for heat equation with memory,, Quarterly Appl. Math., (): 0033.   Google Scholar

[3]

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

[4]

A. Bensoussan, G. Da Prato, M. C. Delfour, and S.K. Mitter, Representation and control of infinite dimensional systems, Birkhäuser Boston, MA, 2007. Google Scholar

[5]

B. D. Colemann, M. E. Gurtin, Equipresence and constitutive equations for heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208. Google Scholar

[6]

H. O. Fattorini, D. L. Rusell, Exact Controllability Theorems for Linear Parabolic Equations in One Space Dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292. Google Scholar

[7]

G. Gripenberg, S. O. Londen, O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and Its Applications, 34, Cambridge University Press, Cambridge, 1990. Google Scholar

[8]

S. Guerrero, O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 288-300. Google Scholar

[9]

A. Halanay, L. Pandolfi, Lack of controllability of the heat equation with memory, Systems & Control Letters, 61 (2012), 999-1002. Google Scholar

[10]

A. Halanay, L. Pandolfi, Lack of controllability of thermal systems with memory, Evol. Eq. Control Theory, 3(3) (2014), 485-497. Google Scholar

[11]

A. Halanay, L. Pandolfi, Approximate controllability and lack of controllability to zero of the heat equation with memory, J. Math. Anal. Appl., 425 (2015), 194-211. Google Scholar

[12]

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

[13]

I. Lasiecka, R. Triggiani, Control theory for partial differential equations: continuous and approximation theories. II. Abstract hyperbolic-like systems over a finite time horizon, Encyclopedia of Mathematics and Its Applications, 75. Cambridge University Press, Cambridge 2000. Google Scholar

[14]

V. Lakshmikantham, R. M. Rao, Theory of integro-differential equations, Gordon & Breach, Lausanne 1995. Google Scholar

[15]

J. L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Dunod, Paris 1968. Google Scholar

[16]

V. P. Mikhailov, Partial Differential Equations, Mir, Moscou 1978. Google Scholar

[17]

L. Pandolfi, Riesz systems and controllability of heat equations with memory, Int. Eq. Operator Theory, 64 (2009), 429-453. Google Scholar

[18]

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

[19]

L. Pandolfi, Sharp control time in viscoelasticity,, in preparation., ().   Google Scholar

[20]

D. D. Joseph, L. Preziosi, Heat waves, Rev. Modern Phys., 61 (1989), 41-73; Addendum to the paper: "Heat waves", Rev. Modern Phys., 62 (1990), 375-391. Google Scholar

[21]

L. Schwartz, Etude des sommes d'exponentielles. Hermann, Paris 1959. Google Scholar

[22]

M. Tucsnak, G. Weiss:, Observation and control for operator semigroups, \/ Birkhäuser, Base, 2009. Google Scholar

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