# American Institute of Mathematical Sciences

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

## Lack of controllability of thermal systems with memory

 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 $M(t)$.
Citation: Andrei Halanay, Luciano Pandolfi. Lack of controllability of thermal systems with memory. Evolution Equations and 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. [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. [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. [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. [5] V. Barbu and M. Iannelli, Controllability of the heat equation with memory, Diff. Integral Eq., 13 (2000), 1393-1412. [6] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser Boston, MA, 2007. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim., 31 (1993), 101-110. doi: 10.1137/0331008. [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. [19] V. Lakshmikantham and M. R. Rama, Theory of Integro-Differential Equations, Gordon & Breach, Lausanne, 1995. [20] J. L. Lions, Contrôlabilitè Exacte Perturbations et Stabilization de Systémes Distribuès, (French) [Exact Controllability, Perturbation and Stabilization of Distributed Systems], Masson, Paris, 1988. [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. [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. [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. [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. [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. [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. [27] L. Pandolfi, Sharp control time in viscoelasticity,, submitted., (). [28] L. Pandolfi, Distributed Systems with Persistent Memory: Control and Moment Problems,, Springer, (). [29] L. Rosier and P. Rouchon, On the controllability of a wave equation with structural damping, Internat. J. Tomogr. Statist, 5 (2007), 79-84. [30] L. Schwartz, Etude des Sommes d'Exponentielles, Hermann, Paris, 1959.

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. [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. [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. [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. [5] V. Barbu and M. Iannelli, Controllability of the heat equation with memory, Diff. Integral Eq., 13 (2000), 1393-1412. [6] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser Boston, MA, 2007. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim., 31 (1993), 101-110. doi: 10.1137/0331008. [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. [19] V. Lakshmikantham and M. R. Rama, Theory of Integro-Differential Equations, Gordon & Breach, Lausanne, 1995. [20] J. L. Lions, Contrôlabilitè Exacte Perturbations et Stabilization de Systémes Distribuès, (French) [Exact Controllability, Perturbation and Stabilization of Distributed Systems], Masson, Paris, 1988. [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. [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. [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. [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. [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. [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. [27] L. Pandolfi, Sharp control time in viscoelasticity,, submitted., (). [28] L. Pandolfi, Distributed Systems with Persistent Memory: Control and Moment Problems,, Springer, (). [29] L. Rosier and P. Rouchon, On the controllability of a wave equation with structural damping, Internat. J. Tomogr. Statist, 5 (2007), 79-84. [30] L. Schwartz, Etude des Sommes d'Exponentielles, Hermann, Paris, 1959.
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