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Linear evolution equations with strongly measurable families and application to the Dirac equation
Riesz systems, spectral controllability and a source identification problem for heat equations with memory
| 1. | Politecnico di Torino, Dipartimento di Matematica, Corso Duca degli Abruzzi 24, 10129 Torino |
References:
| [1] |
A. Belleni-Morante, An integro-differential equation arising from the theory of heat conduction in rigid materials with memory, Boll. Unione Mat. Ital. B (5), 15 (1978), 470-482. |
| [2] |
C. Cavaterra, A. Lorenzi and M. Yamamoto, A stability result via Carleman estimates for an inverse source problem related to a hyperbolic integro-differential equation, Comp. Applied Math., 25 (2006), 229-250. |
| [3] |
D. D. Ang, R. Gorenflo, V. K. Le and D. D. Trong, "Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction," Lecture Notes in Mathematics, 1792, Springer-Verlag, Berlin, 2002. |
| [4] |
C. Alves, A. N. Silvestre, T. Takhahashi and M. Tuksnak, Solving inverse source problems using observability. Application to the Euler-Bernoulli plate equation, SIAM J. Control Optim., 48 (2009), 1632-1659. |
| [5] |
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. |
| [6] |
V. Barbu and M. Iannelli, Controllability of the heat equation with memory, Diff. Integral Eq., 13 (2000), 1393-1412. |
| [7] |
F. Fagnani and L. Pandolfi, A singular perturbation approach to a recursive deconvolution problem, SIAM J. Control Optim., 40 (2002), 1384-1405. |
| [8] |
A. Favini and L. Pandolfi, Multiscale Lavrentiev method for systems of Volterra equations of the first kind, J. Inverse Ill-Posed Probl., 16 (2008), 221-238. |
| [9] |
X. Fu, J. Yong and X. Zhang, Controllability and observability of a heat equation with hyperbolic memory kernel, J. Differential Equations, 247 (2009), 2395-2439. |
| [10] |
I. C. Gohberg and M. G. Krejn, "Introduction á la Thèorie des Opèrateurs Linèairs Non Auto-Adjoints dans un Espace Hilbertien" (French) [Linear non selfadjoint operators in a Hilbert space], Dunod, Paris, 1971. |
| [11] |
M. Grasselli and M. Yamamoto, Identifying a spatial body force in linear elastodynamic via traction measurements, SIAM J. Contr. Optim., 36 (1998), 1190-1206. |
| [12] |
S. Ivanov and L. Pandolfi, Heat equation with memory: Lack of controllability to rest, J. Math. Anal. Appl., 355 (2009), 1-11. |
| [13] |
I. Lasieska and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems," Encyclopedia of Mathematics and its Applications, 74, Cambridge University Press, Cambridge, 2000. |
| [14] |
G. Leugering, Time optimal boundary controllability of a simple linear viscoelastic liquid, Math. Methods in the Appl. Sci., 9 (1987), 413-430. |
| [15] |
J. L. Lions, "Contrôlabilitè Exacte Perturbations et Stabilization de Systémes Distribuès" (French) [Exact controllability, perturbation and stabilization of distributed systems], Masson, Paris, 1988. |
| [16] |
L. Pandolfi, The controllability of the Gurtin-Pipkin equation: A cosine operator approach, Applied Mathematics and Optimization, 52 (2005), 143-165. |
| [17] |
L. Pandolfi, Riesz system and the controllability of heat equations with memory, Integral Eq. Oper. Theory, 64 (2009), 429-453. |
| [18] |
L. Pandolfi, Riesz basis and moment method in the study of heat equations with memory in one space dimension, Discrete Continuous Dynamical Systems Ser. B, 14 (2010), 1487-1510. |
| [19] |
J.-P. Puel and M. Yamamoto, Generic well-posedness in a multidimensional hyperbolic inverse problem, J. Inverse Ill-Posed Probl., 5 (1997), 55-83. |
| [20] |
M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse problems, 11 (1995), 481-496. |
| [21] |
R. M. Young, "An Introduction to Nonharmonic Fourier Series," Academic Press, New York, 1980. |
show all references
References:
| [1] |
A. Belleni-Morante, An integro-differential equation arising from the theory of heat conduction in rigid materials with memory, Boll. Unione Mat. Ital. B (5), 15 (1978), 470-482. |
| [2] |
C. Cavaterra, A. Lorenzi and M. Yamamoto, A stability result via Carleman estimates for an inverse source problem related to a hyperbolic integro-differential equation, Comp. Applied Math., 25 (2006), 229-250. |
| [3] |
D. D. Ang, R. Gorenflo, V. K. Le and D. D. Trong, "Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction," Lecture Notes in Mathematics, 1792, Springer-Verlag, Berlin, 2002. |
| [4] |
C. Alves, A. N. Silvestre, T. Takhahashi and M. Tuksnak, Solving inverse source problems using observability. Application to the Euler-Bernoulli plate equation, SIAM J. Control Optim., 48 (2009), 1632-1659. |
| [5] |
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. |
| [6] |
V. Barbu and M. Iannelli, Controllability of the heat equation with memory, Diff. Integral Eq., 13 (2000), 1393-1412. |
| [7] |
F. Fagnani and L. Pandolfi, A singular perturbation approach to a recursive deconvolution problem, SIAM J. Control Optim., 40 (2002), 1384-1405. |
| [8] |
A. Favini and L. Pandolfi, Multiscale Lavrentiev method for systems of Volterra equations of the first kind, J. Inverse Ill-Posed Probl., 16 (2008), 221-238. |
| [9] |
X. Fu, J. Yong and X. Zhang, Controllability and observability of a heat equation with hyperbolic memory kernel, J. Differential Equations, 247 (2009), 2395-2439. |
| [10] |
I. C. Gohberg and M. G. Krejn, "Introduction á la Thèorie des Opèrateurs Linèairs Non Auto-Adjoints dans un Espace Hilbertien" (French) [Linear non selfadjoint operators in a Hilbert space], Dunod, Paris, 1971. |
| [11] |
M. Grasselli and M. Yamamoto, Identifying a spatial body force in linear elastodynamic via traction measurements, SIAM J. Contr. Optim., 36 (1998), 1190-1206. |
| [12] |
S. Ivanov and L. Pandolfi, Heat equation with memory: Lack of controllability to rest, J. Math. Anal. Appl., 355 (2009), 1-11. |
| [13] |
I. Lasieska and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems," Encyclopedia of Mathematics and its Applications, 74, Cambridge University Press, Cambridge, 2000. |
| [14] |
G. Leugering, Time optimal boundary controllability of a simple linear viscoelastic liquid, Math. Methods in the Appl. Sci., 9 (1987), 413-430. |
| [15] |
J. L. Lions, "Contrôlabilitè Exacte Perturbations et Stabilization de Systémes Distribuès" (French) [Exact controllability, perturbation and stabilization of distributed systems], Masson, Paris, 1988. |
| [16] |
L. Pandolfi, The controllability of the Gurtin-Pipkin equation: A cosine operator approach, Applied Mathematics and Optimization, 52 (2005), 143-165. |
| [17] |
L. Pandolfi, Riesz system and the controllability of heat equations with memory, Integral Eq. Oper. Theory, 64 (2009), 429-453. |
| [18] |
L. Pandolfi, Riesz basis and moment method in the study of heat equations with memory in one space dimension, Discrete Continuous Dynamical Systems Ser. B, 14 (2010), 1487-1510. |
| [19] |
J.-P. Puel and M. Yamamoto, Generic well-posedness in a multidimensional hyperbolic inverse problem, J. Inverse Ill-Posed Probl., 5 (1997), 55-83. |
| [20] |
M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse problems, 11 (1995), 481-496. |
| [21] |
R. M. Young, "An Introduction to Nonharmonic Fourier Series," Academic Press, New York, 1980. |
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