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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 |
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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