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Boundary value control problems involving the bessel differential operator

Published online by Cambridge University Press:  17 February 2009

K.-D. Werner
K.-D. Werner
Affiliation:
This paper was written during the author's stay at the School of Mathematics, University of New South Wales, N.S.W., 2033, Australia. (Present address: Institute of Geometry and Practical Mathematics, Technical University of Aachen, Templergraben 55, 5100 Aachen, Fed. Rep. of Germany)
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Abstract

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In this paper, we consider the hyperbolic partial differential equation wrr = wrr + 1/r wr − ν2 /r2w, where v ≥ 1/2 or ν = 0 is aprameter, with the Dirichlet, Neumann and mixed boundary conditions. The boundary controllability for such problems is investigated. The main resutl is that all “finite energy” intial states can be steered to the zero state in time T, using a control f ∈ L2 (0, T), provided T > 2. Furthermore, necessary conditions for controllability are also presented.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 27 , Issue 4 , April 1986 , pp. 453 - 472
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Ahmed, N. U. and Teo, K. L., Optimal control of distributed parameter systems (North-Holland, New York, 1981).Google Scholar
[2]Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher transcendental functions, Vol. 2 (McGraw-Hill, New York, 1953).Google Scholar
[3]Graham, K. D., “Separation of eigenvalues of the wave equation for the unit ball in RN” Stud. Appl. Math. 52 (1973), 329–344.CrossRefGoogle Scholar
[4]Graham, K. D. and Russell, D. L., “Boundary value control of the wave equation in a spherical region”, SIAM J. Control Optin. 13 (1975), 174–196.CrossRefGoogle Scholar
[5]Lagnese, J., “Control of wave processes with distributed controls supported on a subregion”, SIAM J. Control Optim. 21 (1983), 868–85.CrossRefGoogle Scholar
[6]Levinson, N., “Gap and density theorems”, Colloquium Publications, Vol. 26 (Amer. Math. Soc., New York, 1940).Google Scholar
[7]Lions, J. L., Optimal control of systems governed by partial differential equations (Springer Verlag, New York, 1971).CrossRefGoogle Scholar
[8]Lions, J. L. and Magenes, E., Problémes aux limites non homogènes et applications, Vol. 1 (Dunod, Paris, 1968).Google Scholar
[9]Littmann, W., “Boundary control theory for hyperbolic and parabolic differential equations with constant coefficients”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 567–580.Google Scholar
[10]Moore, C. N., “Summability of developments in Bessel functions”, Trans. Amer. Math. Soc. 10 (1909), 391–435.CrossRefGoogle Scholar
[11]Russell, D. L., “Nonharmonic Fourier series in the control theory of distributed parameter systems”, J. Math. Anal. Appl. 18 (1967), 542–560.CrossRefGoogle Scholar
[12]Russell, D. L., “Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory”, J. Math. Anal. Appl. 40 (1972), 336–368.CrossRefGoogle Scholar
[13]Russell, D. L., “A unified boundary controllability theory for hyperbolic and parabolic differential equations”, Stud. Appl. Math. Vol. 52 (1973), 189–211.CrossRefGoogle Scholar
[14]Russell, D. L., “Boundary value control of the higher dimensional wave equation”, SIAM J. Control Optim. 9 (1971), 29–42.CrossRefGoogle Scholar
[15]Russell, D. L., “Boundary value control of the higher dimensional wave equation, Part II”, SIAM J. Control Optim. 9 (1971),, 401–419.CrossRefGoogle Scholar
[16]Russell, D. L., “Exact boundary value controllability theorems for wave and heat processes in star-complemented regions”, Proc. conference on differential games and control theory (Kingston, R. I., 1973), (Marcel Dekker, New York, 1973).Google Scholar
[17]Schafheitlin, P., “Die Nullstellen der Besselschen Funktionen”, J. Reine A ngew. Math. 122 (1900), 299–321.Google Scholar
[18]Schafheitlin, P., “Über die Gaussche u. Bessel Differentialgleichung und eine neue Integral form der Letzteren”, J. Reine Angew. Math. 114 (1895), 31–44.Google Scholar
[19]Triebel, H., Höhere Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, 1972).Google Scholar
[20]Werner, K. D., “An observation problem for the Bessel differential operator”, J. Austral. Math. Soc. Ser. B. 26 (1984), 92–107.CrossRefGoogle Scholar
[21]Werner, K. D., “Boundary value control and observation problems for the wave and heat equation in a spherical region”, (unpublished).Google Scholar
[22]Wloka, J., Grundräume und verallgemeinerte Funktionen, Lecture Notes in Math., Vol. 82 (Springer, Berlin, 1969).Google Scholar