American Institute of Mathematical Sciences

Advanced
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.

[2]

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

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[19]

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

[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.

[21]

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

[22]

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

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.

[2]

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

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[19]

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

[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.

[21]

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

[22]

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

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