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An observation problem for the Bessel differential operator

Published online by Cambridge University Press:  17 February 2009

K.-D. Werner
K.-D. Werner
Affiliation:
School of Mathematics, University of New South Wales, P. O. Box 1, Kensington, N.S.W. 2033, Australia. (Presently on leave from the Department of Mathematics, University of Kassel, University of the State of Hesse, 35 Kassel, West Germany.)
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Abstract

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In this paper, the parabolic partial differential equation ut = urr + (1/r)ur − (v2/r2)u, where v ≥ 0 is a parameter, with Dirichlet, Neumann, and mixed boundary conditions is considered. The final state observability for such problems is investigated.

Type
Research Article
Information
The ANZIAM Journal , Volume 26 , Issue 1 , July 1984 , pp. 92 - 107
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Abramowitz, M. and Stegun, I. A., Handbook of mathematical functions (Dover, New York, 1968).Google Scholar
[2]Ahmed, N. V. and Teo, K. L., Optimal control of distributed parameter systems (North Holland, New York, 1981).Google Scholar
[3]Cooke, R. G., “Gibb's phenomenon in Fourier-Bessel series and integrals”, Proc. London Math. Soc. 27 (1927), 171–192.Google Scholar
[4]Dolecki, S., “Observability for the one-dimensional heat equation”, Studia Math. 48 (1973), 291–305.Google Scholar
[5]Dolecki, S., “Observability for regular processes”, J. Math. Anal. Appl. 58 (1977), 178–188.CrossRefGoogle Scholar
[6]Dolecki, S. and Russel, D. L., “A general theory of observation and control”, SIAM J. Control Optim. 15 (1977), 185–220.Google Scholar
[7]Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher transcendental functions, Vol. II (McGraw-Hill, New York, 1953).Google Scholar
[8]Graham, K. D., “Separation of eigenvalues of the wave equation for the unit ball in RN” Stud. Appl. Math. 52 (1973), 329–343.Google Scholar
[9]Mizel, V. J. and Seidman, T. I., “Observation and prediction for the heat equation”, J. Math. Anal. Appl. 28 (1969), 303–312.CrossRefGoogle Scholar
[10]Mizel, V. J. and Seidmann, T. I., “Observation and prediction for the heat equation II”, J. Math. Anal. Appl. 38 (1972), 149–166.Google Scholar
[11]Moore, C. N., “Summability of developments in Bessel functions”, Trans. A mer. Math. Soc. 10 (1909), 391–435.CrossRefGoogle Scholar
[12]Rolewicz, S., Funktionalanalysis und Steuerungstheorie (Springer, Berlin 1976).Google Scholar
[13]Schafheitlin, P., “Die Nullstellen der Besselschen Funktionen”, J. Reine Angew Math. 122 (1900), 299–321.Google Scholar
[14]Schafheitlin, P., “Uber die Gaussche u. Besselsche Differentialgleichung und eine neue Integral- form der Letzteren”, J. Reine Angewandte Math. 114 (1895), 31–44.Google Scholar
[15]Seidman, T. I., “Observation and prediction for one dimensional diffusion equations”, J. Math. Anal. Appl. 51 (1975), 165–175.Google Scholar
[16]Tolstoy, G. P., Fourier series (Prentice Hall, Englewood Cliffs, N.J., 1976).Google Scholar
[17]Watson, G. N., A treatise on the theory of Bessel functions (Cambridge University Press, 1944).Google Scholar