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September  2014, 3(3): 485-497. doi: 10.3934/eect.2014.3.485

Lack of controllability of thermal systems with memory

Andrei Halanay 1, and  Luciano Pandolfi 2,

1. 

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

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

Received  May 2013 Revised  January 2014 Published  August 2014

Heat equations with memory of Gurtin-Pipkin type (i.e. Eq. (1) with $ \alpha=0 $) have controllability properties which strongly resemble those of the wave equation. Instead, recent counterexamples show that when $ \alpha>0 $ the control properties do not parallel those of the (memoryless) heat equation, in the sense that there are square integrable initial conditions which cannot be controlled to zero. The proof of this fact, in previous papers, consists in the construction of two quite special examples of systems with memory which cannot be controlled to zero. Here we prove that lack of controllability holds in general, for every smooth memory kernel $ M(t) $.
Keywords: controllability to zero, exponential families., lack of controllability, Heat equation with memory, moment methods.
Mathematics Subject Classification: Primary: 35Q93, 45K05; Secondary: 93B0.
Citation: Andrei Halanay, Luciano Pandolfi. Lack of controllability of thermal systems with memory. Evolution Equations & Control Theory, 2014, 3 (3) : 485-497. doi: 10.3934/eect.2014.3.485
References:
[1]

G. Amendola, M. Fabrizio and G. M. Golden, Thermodynamics of Materials with Memory. Theory and Applications, Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0.  Google Scholar

[2]

S. Avdonin and B. P. Belinskiy, On controllability of an homogeneous string with memory, J. Mathematical Analysis Appl., 398 (2013), 254-269. doi: 10.1016/j.jmaa.2012.08.037.  Google Scholar

[3]

S. A. Avdonin and 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

[4]

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

[5]

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

[6]

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

[7]

F. W. Chavez-Silva, L. Rosier and E. Zuazua, Null controllability of a system of viscoelasticity with a moving control, J. Mathematiques Pure Appl.,101 (2014), 198-222.  Google Scholar

[8]

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

[9]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292. doi: 10.1007/BF00250466.  Google Scholar

[10]

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

[11]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.  Google Scholar

[12]

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

[13]

M. E. Gurtin and A. G. 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

[14]

A. Halanay and L. Pandolfi, Lack of controllability of the heat equation with memory, Systems & Control Letters, 61 (2012), 999-1002. doi: 10.1016/j.sysconle.2012.07.002.  Google Scholar

[15]

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

[16]

D. D. Joseph and L. Preziosi, Heat waves, Rev. Modern Phys., 61 (1989), 41-73; Addendum to the paper: "Heat Waves'', Rev. Modern Phys., 62 (1990), 375-391. doi: 10.1103/RevModPhys.62.375.  Google Scholar

[17]

J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim., 31 (1993), 101-110. doi: 10.1137/0331008.  Google Scholar

[18]

I. Lasiecka and R. Triggiani, Control Theory for Parital Differential Equations: Continuous and Approxiamtion Theory. I, Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications 74, Cambridge University Press, Cambridge, 2000.  Google Scholar

[19]

V. Lakshmikantham and M. R. Rama, Theory of Integro-Differential Equations, Gordon & Breach, Lausanne, 1995.  Google Scholar

[20]

J. L. Lions, Contrôlabilitè Exacte Perturbations et Stabilization de Systémes Distribuès, (French) [Exact Controllability, Perturbation and Stabilization of Distributed Systems], Masson, Paris, 1988.  Google Scholar

[21]

P. Loreti, L. 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

[22]

P. Martin, L. Rosier and P. Rouchon, Null controllability of the structurally damped wave equation with moving control, SIAM. J. Control Optim., 51 (2013), 660-684. doi: 10.1137/110856150.  Google Scholar

[23]

S. Micu and I. Roventa, Uniform controllability of the linear one dimensional Schrodinger equation with vanishing viscosity, ESAIM Control Optim. Calc. Var., 18 (2012), 277-293. doi: 10.1051/cocv/2010055.  Google Scholar

[24]

L. Pandolfi, The controllability of the Gurtin-Pipkin equation: A cosine operator approach, Applied Mathematics and Optimization, 52 (2005), 143-165; Erratum to: The controllability of the Gurtin-Pipkin equation: A Cosine Operator Approach, Appl. Math. Optim., 64 (2011), 467-468. doi: 10.1007/s00245-005-0819-0.  Google Scholar

[25]

L. Pandolfi, Riesz systems and the controllability of heat equations with memory, Int. Eq. Operator Theory, 64 (2009), 429-453. doi: 10.1007/s00020-009-1682-1.  Google Scholar

[26]

L. Pandolfi, Riesz systems and moment method in the study of heat equations with memory in one space dimension, Discr. Cont. Dynamical Systems, Ser. B., 14 (2010), 1487-1510. doi: 10.3934/dcdsb.2010.14.1487.  Google Scholar

[27]

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

[28]

L. Pandolfi, Distributed Systems with Persistent Memory: Control and Moment Problems,, Springer, ().   Google Scholar

[29]

L. Rosier and P. Rouchon, On the controllability of a wave equation with structural damping, Internat. J. Tomogr. Statist, 5 (2007), 79-84.  Google Scholar

[30]

L. Schwartz, Etude des Sommes d'Exponentielles, Hermann, Paris, 1959.  Google Scholar

show all references

References:
[1]

G. Amendola, M. Fabrizio and G. M. Golden, Thermodynamics of Materials with Memory. Theory and Applications, Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0.  Google Scholar

[2]

S. Avdonin and B. P. Belinskiy, On controllability of an homogeneous string with memory, J. Mathematical Analysis Appl., 398 (2013), 254-269. doi: 10.1016/j.jmaa.2012.08.037.  Google Scholar

[3]

S. A. Avdonin and 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

[4]

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

[5]

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

[6]

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

[7]

F. W. Chavez-Silva, L. Rosier and E. Zuazua, Null controllability of a system of viscoelasticity with a moving control, J. Mathematiques Pure Appl.,101 (2014), 198-222.  Google Scholar

[8]

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

[9]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292. doi: 10.1007/BF00250466.  Google Scholar

[10]

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

[11]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.  Google Scholar

[12]

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

[13]

M. E. Gurtin and A. G. 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

[14]

A. Halanay and L. Pandolfi, Lack of controllability of the heat equation with memory, Systems & Control Letters, 61 (2012), 999-1002. doi: 10.1016/j.sysconle.2012.07.002.  Google Scholar

[15]

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

[16]

D. D. Joseph and L. Preziosi, Heat waves, Rev. Modern Phys., 61 (1989), 41-73; Addendum to the paper: "Heat Waves'', Rev. Modern Phys., 62 (1990), 375-391. doi: 10.1103/RevModPhys.62.375.  Google Scholar

[17]

J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim., 31 (1993), 101-110. doi: 10.1137/0331008.  Google Scholar

[18]

I. Lasiecka and R. Triggiani, Control Theory for Parital Differential Equations: Continuous and Approxiamtion Theory. I, Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications 74, Cambridge University Press, Cambridge, 2000.  Google Scholar

[19]

V. Lakshmikantham and M. R. Rama, Theory of Integro-Differential Equations, Gordon & Breach, Lausanne, 1995.  Google Scholar

[20]

J. L. Lions, Contrôlabilitè Exacte Perturbations et Stabilization de Systémes Distribuès, (French) [Exact Controllability, Perturbation and Stabilization of Distributed Systems], Masson, Paris, 1988.  Google Scholar

[21]

P. Loreti, L. 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

[22]

P. Martin, L. Rosier and P. Rouchon, Null controllability of the structurally damped wave equation with moving control, SIAM. J. Control Optim., 51 (2013), 660-684. doi: 10.1137/110856150.  Google Scholar

[23]

S. Micu and I. Roventa, Uniform controllability of the linear one dimensional Schrodinger equation with vanishing viscosity, ESAIM Control Optim. Calc. Var., 18 (2012), 277-293. doi: 10.1051/cocv/2010055.  Google Scholar

[24]

L. Pandolfi, The controllability of the Gurtin-Pipkin equation: A cosine operator approach, Applied Mathematics and Optimization, 52 (2005), 143-165; Erratum to: The controllability of the Gurtin-Pipkin equation: A Cosine Operator Approach, Appl. Math. Optim., 64 (2011), 467-468. doi: 10.1007/s00245-005-0819-0.  Google Scholar

[25]

L. Pandolfi, Riesz systems and the controllability of heat equations with memory, Int. Eq. Operator Theory, 64 (2009), 429-453. doi: 10.1007/s00020-009-1682-1.  Google Scholar

[26]

L. Pandolfi, Riesz systems and moment method in the study of heat equations with memory in one space dimension, Discr. Cont. Dynamical Systems, Ser. B., 14 (2010), 1487-1510. doi: 10.3934/dcdsb.2010.14.1487.  Google Scholar

[27]

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

[28]

L. Pandolfi, Distributed Systems with Persistent Memory: Control and Moment Problems,, Springer, ().   Google Scholar

[29]

L. Rosier and P. Rouchon, On the controllability of a wave equation with structural damping, Internat. J. Tomogr. Statist, 5 (2007), 79-84.  Google Scholar

[30]

L. Schwartz, Etude des Sommes d'Exponentielles, Hermann, Paris, 1959.  Google Scholar

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