Simultaneous temperature and flux controllability for heat equations with memory

Avdonin, S.; Pandolfi, L.

doi:10.1090/S0033-569X-2012-01287-7

Authors: S. Avdonin and L. Pandolfi
Journal: Quart. Appl. Math. 71 (2013), 339-368
MSC (2010): Primary 76A10, 93C05, 47N70
DOI: https://doi.org/10.1090/S0033-569X-2012-01287-7
Published electronically: October 22, 2012
MathSciNet review: 3087427
Full-text PDF Free Access

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 $\mathcal {L}$-bases and Riesz bases especially suited to heat equations with memory, which so appear to be endowed with a very rich bases structure.

References

S. A. Avdonin, On the question of Riesz bases of exponential functions in $L^{2}$, Vestnik Leningrad. Univ. No. 13 Mat. Meh. Astronom. Vyp. 3 (1974), 5–12, 154 (Russian, with English summary). MR 0361746

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.

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

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, DOI https://doi.org/10.1007/s00245-009-9064-2

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, DOI https://doi.org/10.1051/mmnp/20105401

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

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.

V. Barbu and M. Iannelli, Controllability of the heat equation with memory, Differential Integral Equations 13 (2000), no. 10-12, 1393–1412. MR 1787073

Carlo Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3 (1949), 83–101 (Italian). MR 0032898

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, DOI https://doi.org/10.1016/j.jde.2009.07.026

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, DOI https://doi.org/10.1007/BF00281373

G. Leugering, Time optimal boundary controllability of a simple linear viscoelastic liquid, Math. Methods Appl. Sci. 9 (1987), no. 3, 413–430. MR 908599, DOI https://doi.org/10.1002/mma.1670090130

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

Vilmos Komornik and Paola Loreti, Fourier series in control theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005. MR 2114325

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, DOI https://doi.org/10.1016/j.jde.2009.09.016

Luciano Pandolfi, The controllability of the Gurtin-Pipkin equation: a cosine operator approach, Appl. Math. Optim. 52 (2005), no. 2, 143–165. MR 2157198, DOI https://doi.org/10.1007/s00245-005-0819-0

L. Pandolfi, Controllability of the Gurtin-Pipkin equation. SISSA, Proceedings of Science, PoS(CSTNA2005)015.

L. Pandolfi, Riesz systems and controllability of heat equations with memory, Integral Equations Operator Theory 64 (2009), no. 3, 429–453. MR 2521245, DOI https://doi.org/10.1007/s00020-009-1682-1

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, DOI https://doi.org/10.3934/dcdsb.2010.14.1487

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, DOI https://doi.org/10.3934/dcdss.2011.4.745

D. D. Joseph and Luigi Preziosi, Heat waves, Rev. Modern Phys. 61 (1989), no. 1, 41–73. MR 977943, DOI https://doi.org/10.1103/RevModPhys.61.41

Michael Renardy, Are viscoelastic flows under control or out of control?, Systems Control Lett. 54 (2005), no. 12, 1183–1193. MR 2175633, DOI https://doi.org/10.1016/j.sysconle.2005.04.006

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

Francesco Giacomo Tricomi, Differential equations, Hafner Publishing Co., New York, 1961. Translated by Elizabeth A. McHarg. MR 0138812

Robert M. Young, An introduction to nonharmonic Fourier series, 1st ed., Academic Press, Inc., San Diego, CA, 2001. MR 1836633