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Inverse problems for evolution equations with time dependent operator-coefficients
Observability of $N$-dimensional integro-differential systems
1. | Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sezione di Matematica, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma |
2. | Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via Antonio Scarpa 16 I-00161 Roma, Italy |
References:
[1] |
G. Gripenberg, S. O. Londen and O. J. Staffans, Volterra Integral and Functional Equations, Encyclopedia Math. Appl. 34, Cambridge Univ. Press, Cambridge, 1990.
doi: 10.1017/CBO9780511662805. |
[2] |
A. Hanyga, Dispersion and attenuation for an acoustic wave equation consistent with viscoelasticity, J. Comput. Acoust., 22 (2014), 1450006, 22 pp.
doi: 10.1142/S0218396X14500064. |
[3] |
A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379.
doi: 10.1007/BF01180426. |
[4] |
V. Komornik and P. Loreti, Ingham type theorems for vector-valued functions and observability of coupled linear system, SIAM J. Control Optim., 37 (1999), 461-485.
doi: 10.1137/S0363012997317505. |
[5] |
V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monogr. Math., Springer-Verlag, New York, 2005.
doi: 10.1007/b139040. |
[6] |
J. E. Lagnese and J.-L. Lions, Modelling Analysis and Control of Thin Plates, Rech. Math. Appl., Masson, Paris, 1988. |
[7] |
I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Appl. Math. Optim., 19 (1989), 243-290.
doi: 10.1007/BF01448201. |
[8] |
J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, with appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] 8, Masson, Paris, 1988. |
[9] |
J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2. Perturbations, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] 9, Masson, Paris, 1988. |
[10] |
P. Loreti and D. Sforza, Reachability problems for a class of integro-differential equations, J. Differential Equations, 248 (2010), 1711-1755.
doi: 10.1016/j.jde.2009.09.016. |
[11] |
P. Loreti and D. Sforza, Multidimensional controllability problems with memory, in Modern Aspects of the Theory of Partial Differential Equations (eds. M. Ruzhansky and J. Wirth), Operator Theory: Advances and Applications 216, Birkhäuser/Springer, Basel, (2011), 261-274.
doi: 10.1007/978-3-0348-0069-3_15. |
[12] |
P. Loreti and D. Sforza, Control problems for weakly coupled systems with memory, J. Differential Equations, 257 (2014), 1879-1938.
doi: 10.1016/j.jde.2014.05.016. |
[13] |
J. E. McDonald, Maxwellian Intepretation of the Laplacian, Am. J. Phys., 33 (1965), 706-711. |
[14] |
J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[15] |
M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Pitman Monogr. Pure Appl. Math., 35, Longman Sci. Tech., Harlow, Essex, 1987. |
[16] |
D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., 20 (1978), 639-739.
doi: 10.1137/1020095. |
[17] |
R. Triggiani, Exact boundary controllability on $L_2(\Omega)\times H^{-1}(\Omega)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary $\partial\Omega$, and related problems, Appl. Math. Optim., 18 (1988), 241-277.
doi: 10.1007/BF01443625. |
show all references
References:
[1] |
G. Gripenberg, S. O. Londen and O. J. Staffans, Volterra Integral and Functional Equations, Encyclopedia Math. Appl. 34, Cambridge Univ. Press, Cambridge, 1990.
doi: 10.1017/CBO9780511662805. |
[2] |
A. Hanyga, Dispersion and attenuation for an acoustic wave equation consistent with viscoelasticity, J. Comput. Acoust., 22 (2014), 1450006, 22 pp.
doi: 10.1142/S0218396X14500064. |
[3] |
A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379.
doi: 10.1007/BF01180426. |
[4] |
V. Komornik and P. Loreti, Ingham type theorems for vector-valued functions and observability of coupled linear system, SIAM J. Control Optim., 37 (1999), 461-485.
doi: 10.1137/S0363012997317505. |
[5] |
V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monogr. Math., Springer-Verlag, New York, 2005.
doi: 10.1007/b139040. |
[6] |
J. E. Lagnese and J.-L. Lions, Modelling Analysis and Control of Thin Plates, Rech. Math. Appl., Masson, Paris, 1988. |
[7] |
I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Appl. Math. Optim., 19 (1989), 243-290.
doi: 10.1007/BF01448201. |
[8] |
J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, with appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] 8, Masson, Paris, 1988. |
[9] |
J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2. Perturbations, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] 9, Masson, Paris, 1988. |
[10] |
P. Loreti and D. Sforza, Reachability problems for a class of integro-differential equations, J. Differential Equations, 248 (2010), 1711-1755.
doi: 10.1016/j.jde.2009.09.016. |
[11] |
P. Loreti and D. Sforza, Multidimensional controllability problems with memory, in Modern Aspects of the Theory of Partial Differential Equations (eds. M. Ruzhansky and J. Wirth), Operator Theory: Advances and Applications 216, Birkhäuser/Springer, Basel, (2011), 261-274.
doi: 10.1007/978-3-0348-0069-3_15. |
[12] |
P. Loreti and D. Sforza, Control problems for weakly coupled systems with memory, J. Differential Equations, 257 (2014), 1879-1938.
doi: 10.1016/j.jde.2014.05.016. |
[13] |
J. E. McDonald, Maxwellian Intepretation of the Laplacian, Am. J. Phys., 33 (1965), 706-711. |
[14] |
J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[15] |
M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Pitman Monogr. Pure Appl. Math., 35, Longman Sci. Tech., Harlow, Essex, 1987. |
[16] |
D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., 20 (1978), 639-739.
doi: 10.1137/1020095. |
[17] |
R. Triggiani, Exact boundary controllability on $L_2(\Omega)\times H^{-1}(\Omega)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary $\partial\Omega$, and related problems, Appl. Math. Optim., 18 (1988), 241-277.
doi: 10.1007/BF01443625. |
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