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June  2011, 4(3): 745-759. doi: 10.3934/dcdss.2011.4.745

## 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

Received  April 2009 Revised  November 2009 Published  November 2010

In this paper we show that recent results on a Riesz basis associated to a heat equation with memory can be used in order to solve a source identification problem.
Citation: Luciano Pandolfi. Riesz systems, spectral controllability and a source identification problem for heat equations with memory. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 745-759. doi: 10.3934/dcdss.2011.4.745
##### 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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##### 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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