Simultaneous temperature and flux controllability for heat equations with memory

Authors:
S. Avdonin and L. Pandolfi

Journal:
Quart. Appl. Math. **71** (2013), 339-368

MSC (2010):
Primary 76A10, 93C05, 47N70

Published electronically:
October 22, 2012

MathSciNet review:
3087427

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Abstract | References | Similar Articles | Additional Information

Abstract: It is known that, in the case of the heat equation with memory, temperature can be controlled to an arbitrary square integrable target provided that the system evolves for a sufficiently long time. The control is the temperature on the boundary. In this paper we consider heat equations with memory (one-dimensional space variable) and we first show that when the control is square integrable, then the flux is square integrable too. Then we prove that both temperature and flux can be simultaneously controlled to a pair of independent targets, both square integrable. This solves a problem first raised by Renardy.

The method of proof relies on moment theory, and one of the contributions of this paper is the identification of -bases and Riesz bases especially suited to heat equations with memory, which so appear to be endowed with a very rich bases structure.

**1.**S. A. Avdonin,*On the question of Riesz bases of exponential functions in 𝐿²*, Vestnik Leningrad. Univ. No. 13 Mat. Meh. Astronom.**Vyp. 3**(1974), 5–12, 154 (Russian, with English summary). MR**0361746****2.**S.A. Avdonin and B.P. Belinskiy,*Exact control of a string under an axial stretching tension*, Discrete and Continuous Dynamical Systems, Expanded Volume (2003), 57-67.**3.**Sergei A. Avdonin and Boris P. Belinskiy,*On the basis properties of the functions arising in the boundary control problem of a string with a variable tension*, Discrete Contin. Dyn. Syst.**suppl.**(2005), 40–49. MR**2192658****4.**Sergei A. Avdonin, Boris P. Belinskiy, and Sergei A. Ivanov,*On controllability of an elastic ring*, Appl. Math. Optim.**60**(2009), no. 1, 71–103. MR**2511787**, 10.1007/s00245-009-9064-2**5.**S. A. Avdonin, B. P. Belinskiy, and L. Pandolfi,*Controllability of a nonhomogeneous string and ring under time dependent tension*, Math. Model. Nat. Phenom.**5**(2010), no. 4, 4–31. MR**2662448**, 10.1051/mmnp/20105401**6.**Sergei A. Avdonin and Sergei A. Ivanov,*Families of exponentials*, Cambridge University Press, Cambridge, 1995. The method of moments in controllability problems for distributed parameter systems; Translated from the Russian and revised by the authors. MR**1366650****7.**S. Avdonin and L. Pandolfi,*Temperature and heat flux dependence/independence for heat equations with memory*, ``Time Delay Systems-Methods, Applications and New Trends,'' 87-101, Lecture Notes in Control and Information Sciences, Springer, 2011.**8.**V. Barbu and M. Iannelli,*Controllability of the heat equation with memory*, Differential Integral Equations**13**(2000), no. 10-12, 1393–1412. MR**1787073****9.**Carlo Cattaneo,*Sulla conduzione del calore*, Atti Sem. Mat. Fis. Univ. Modena**3**(1949), 83–101 (Italian). MR**0032898****10.**Xiaoyu Fu, Jiongmin Yong, and Xu Zhang,*Controllability and observability of a heat equation with hyperbolic memory kernel*, J. Differential Equations**247**(2009), no. 8, 2395–2439. MR**2561284**, 10.1016/j.jde.2009.07.026**11.**Morton E. Gurtin and A. C. Pipkin,*A general theory of heat conduction with finite wave speeds*, Arch. Rational Mech. Anal.**31**(1968), no. 2, 113–126. MR**1553521**, 10.1007/BF00281373**12.**G. Leugering,*Time optimal boundary controllability of a simple linear viscoelastic liquid*, Math. Methods Appl. Sci.**9**(1987), no. 3, 413–430. MR**908599**, 10.1002/mma.1670090130**13.**G. Leugering,*A decomposition method for integro-partial differential equations and applications*, J. Math. Pures Appl. (9)**71**(1992), no. 6, 561–587. MR**1193609****14.**Vilmos Komornik and Paola Loreti,*Fourier series in control theory*, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005. MR**2114325****15.**Paola Loreti and Daniela Sforza,*Reachability problems for a class of integro-differential equations*, J. Differential Equations**248**(2010), no. 7, 1711–1755. MR**2593605**, 10.1016/j.jde.2009.09.016**16.**Luciano Pandolfi,*The controllability of the Gurtin-Pipkin equation: a cosine operator approach*, Appl. Math. Optim.**52**(2005), no. 2, 143–165. MR**2157198**, 10.1007/s00245-005-0819-0**17.**L. Pandolfi,*Controllability of the Gurtin-Pipkin equation.**SISSA, Proceedings of Science, PoS(CSTNA2005)015.***18.**L. Pandolfi,*Riesz systems and controllability of heat equations with memory*, Integral Equations Operator Theory**64**(2009), no. 3, 429–453. MR**2521245**, 10.1007/s00020-009-1682-1**19.**Luciano Pandolfi,*Riesz systems and moment method in the study of viscoelasticity in one space dimension*, Discrete Contin. Dyn. Syst. Ser. B**14**(2010), no. 4, 1487–1510. MR**2679652**, 10.3934/dcdsb.2010.14.1487**20.**Luciano Pandolfi,*Riesz systems, spectral controllability and a source identification problem for heat equations with memory*, Discrete Contin. Dyn. Syst. Ser. S**4**(2011), no. 3, 745–759. MR**2746431**, 10.3934/dcdss.2011.4.745**21.**D. D. Joseph and Luigi Preziosi,*Heat waves*, Rev. Modern Phys.**61**(1989), no. 1, 41–73. MR**977943**, 10.1103/RevModPhys.61.41**22.**Michael Renardy,*Are viscoelastic flows under control or out of control?*, Systems Control Lett.**54**(2005), no. 12, 1183–1193. MR**2175633**, 10.1016/j.sysconle.2005.04.006**23.**A. M. Sedletskiĭ,*Biorthogonal expansions of functions in exponential series on intervals of the real axis*, Uspekhi Mat. Nauk**37**(1982), no. 5(227), 51–95, 248 (Russian). MR**676613****24.**Francesco Giacomo Tricomi,*Differential equations*, Translated by Elizabeth A. McHarg, Hafner Publishing Co., New York, 1961. MR**0138812****25.**Robert M. Young,*An introduction to nonharmonic Fourier series*, 1st ed., Academic Press, Inc., San Diego, CA, 2001. MR**1836633**

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Additional Information

**S. Avdonin**

Affiliation:
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee 37403-2598, USA

Email:
s.avdonin@alaska.edu

**L. Pandolfi**

Affiliation:
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Email:
luciano.pandolfi@polito.it

DOI:
http://dx.doi.org/10.1090/S0033-569X-2012-01287-7

Received by editor(s):
July 27, 2011

Published electronically:
October 22, 2012

Additional Notes:
Supported in part by the National Science Foundation, grant ARC 0724860. This paper was partly written while the first author visited the Dipartimento di Matematica, Politecnico di Torino, as a visiting professor supported by GNAMPA-INDAM

Supported in part by Italian MURST and by the project “Groupement de Recherche en Contrôle des EDP entre la France et l’Italie (CONEDP)”. This paper fits into the research programs of GNAMPA-INDAM

Article copyright:
© Copyright 2012
Brown University