American Institute of Mathematical Sciences

Advanced
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, (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.  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, (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.  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.   Google Scholar

[6]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems,, Birkhäuser Boston, (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.   Google Scholar

[8]

B. D. Colemann and M. E. Gurtin, Equipresence and constitutive equations for heat conductors,, Z. Angew. Math. Phys., 18 (1967), 199.  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.  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.  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, (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, 19 (2013), 288.  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.  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.  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.  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.  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.  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, (2000).   Google Scholar

[19]

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

[20]

J. L. Lions, Contrôlabilitè Exacte Perturbations et Stabilization de Systémes Distribuès,, (French) [Exact Controllability, (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.  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.  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.  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.  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.  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, 14 (2010), 1487.  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.   Google Scholar

[30]

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

[6]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems,, Birkhäuser Boston, (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.   Google Scholar

[8]

B. D. Colemann and M. E. Gurtin, Equipresence and constitutive equations for heat conductors,, Z. Angew. Math. Phys., 18 (1967), 199.  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.  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.  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, (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, 19 (2013), 288.  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.  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.  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.  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.  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.  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, (2000).   Google Scholar

[19]

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

[20]

J. L. Lions, Contrôlabilitè Exacte Perturbations et Stabilization de Systémes Distribuès,, (French) [Exact Controllability, (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.  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.  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.  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.  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.  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, 14 (2010), 1487.  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.   Google Scholar

[30]

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

[1]

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
[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]

Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020011
[4]

Víctor Hernández-Santamaría, Liliana Peralta. Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 161-190. doi: 10.3934/dcdsb.2019177
[5]

Enrique Fernández-Cara, Arnaud Münch. Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods. Mathematical Control & Related Fields, 2012, 2 (3) : 217-246. doi: 10.3934/mcrf.2012.2.217
[6]

Ping Lin. Feedback controllability for blowup points of semilinear heat equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1425-1434. doi: 10.3934/dcdsb.2017068
[7]

Víctor Hernández-Santamaría, Luz de Teresa. Robust Stackelberg controllability for linear and semilinear heat equations. Evolution Equations & Control Theory, 2018, 7 (2) : 247-273. doi: 10.3934/eect.2018012
[8]

Dario Pighin, Enrique Zuazua. Controllability under positivity constraints of semilinear heat equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 935-964. doi: 10.3934/mcrf.2018041
[9]

Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control & Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011
[10]

Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002
[11]

Sergei A. Avdonin, Boris P. Belinskiy. On controllability of a linear elastic beam with memory under longitudinal load. Evolution Equations & Control Theory, 2014, 3 (2) : 231-245. doi: 10.3934/eect.2014.3.231
[12]

Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183
[13]

Jonathan Touboul. Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control & Related Fields, 2012, 2 (4) : 429-455. doi: 10.3934/mcrf.2012.2.429
[14]

Jonathan Touboul. Erratum on: Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control & Related Fields, 2019, 9 (1) : 221-222. doi: 10.3934/mcrf.2019006
[15]

Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305
[16]

Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations & Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020
[17]

Xiangqing Zhao, Bing-Yu Zhang. Global controllability and stabilizability of Kawahara equation on a periodic domain. Mathematical Control & Related Fields, 2015, 5 (2) : 335-358. doi: 10.3934/mcrf.2015.5.335
[18]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299
[19]

Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations & Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039
[20]

José R. Quintero, Alex M. Montes. On the exact controllability and the stabilization for the Benney-Luke equation. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019039

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences