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Boundary value controllability and observability problems for the wave and heat equation

Published online by Cambridge University Press:  17 February 2009

K.-D. Werner
K.-D. Werner
Affiliation:
Institute of Geometry and Practical Mathematics, Technical University of Aachen, 5100 Aachen, Federal Republic of Germany
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Abstract

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In this paper, we study controllability and observability problems for the wave and heat equation in a spherical region in Rn, where the control enters in the mixed boundary condition. In the main result, we show that all “finite energy” initial states (i.e. (ω0, ν0) ∈ H1(Ω) × L2 (Ω)) can be steered to zero at time T, using a control f ∈ L2 (∂Ω × [0, T]), provided T > 2. On this basis, we use the duality principle to investigate initial observability for the wave equation. Applying the Fourier transform technique, we obtain controllability and observability results for the heat equation.

Type
Research Article
Information
The ANZIAM Journal , Volume 29 , Issue 4 , April 1988 , pp. 461 - 479
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Ahmed, N. U. and Teo, K. L., Optimal control of distributed parameter systems (North Holland, 1981).Google Scholar
[2]Dolecki, S., “Observability for the one-dimensional heat equation”, Studia Math. 48 (1973), 291–305.CrossRefGoogle Scholar
[3]Dolecki, S., “Observability for Regular Processes”, J. Math. Anal. Appl. 58 (1977), 178–188.CrossRefGoogle Scholar
[4]Dolecki, S. and Russell, David L., “A general theory of observation and control”, SIAM J. Control. Optim. 15 (1977), 185–220.CrossRefGoogle Scholar
[5]Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher Transcendental Functions, Vol. 2 (McGraw-Hill, New York, 1953).Google Scholar
[6]Fattorini, H. O. and Russell, David L., “Exact controllability theorems for linear parabolic equations in one space dimension”, Arch. Rational Mech. Anal. 4 (1971), 272–292.CrossRefGoogle Scholar
[7]Fattorini, H. O. and Russell, David L., “Uniform bounds on biorthogonal functions for real exponentials with an application to control theory of parabolic equations”, Quart. Appl. Math. 32 (1974), 45–69.CrossRefGoogle Scholar
[8]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
[9]Graham, K. D. and Russell, David L., “Boundary value control of the wave equation in a spherical region”, SIAM J. Control. Optim. 13 (1975), 174–196.Google Scholar
[10]Lagnese, John, “Control of wave processes with distribution controls supported on a subregion”, SIAM J. Control. Optim. 21 (1983), 68–85.CrossRefGoogle Scholar
[11]Lions, J. L., Optimal control of systems governed by partial differential equations (Springer Verlag, New York, 1971).CrossRefGoogle Scholar
[12]Levinson, N., ‘Gap and density theorems”, Colloquium Publications 26, Amer. Math. Soc. (1940).Google Scholar
[13]Mikhailov, V. P., Partial Differential Equations (English Translation) (Mir Publishers, Moscow, 1978).Google Scholar
[14]Mizel, V. J. and Seidman, T. J., “Observation and prediction for the heat equation II”, J. Math. Anal. Appl. 38 (1972), 149–166.Google Scholar
[15]Russell, David L., “A unified boundary controllability theory for hyperbolic and parabolic differential equations”, Stud. Appl. Math. 52 (1973), 189–211.CrossRefGoogle Scholar
[16]Russell, David L., “Exact boundary value controllability theorems for wave and heat processes in star-complemented regions’, Proc. Conf. of Differential Games and Control Theory, 291–319, Kingston, R. J. (Marcel Dekker, New York, 1974).Google Scholar
[17]Russell, David L., “Nonharmonic Fourier series in the control theory of distributed parameter systems”, J. Math. Anal. Appl. 18 (1967), 542–560.CrossRefGoogle Scholar
[18]Russell, David L., “Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory”, J. Math. Anal. Appl. 40 (1972), 336–368.Google Scholar
[19]Seidman, T. J., “Observation and Prediction for One-Dimensional Diffusion Equations”, J. Math. Anal. Appl. 51 (1975), 165–175.CrossRefGoogle Scholar
[20]Triebel, H., Höhere Analysis (Verlag Harri Deutsch, Frankfurt am Main, 2. Aufiage, 1980).Google Scholar
[21]Werner, K. D., ”, J. Austral. Math. Soc. Ser. B 26 (1984), 92–107.CrossRefGoogle Scholar
[22]Werner, K. D., “Boundary value control problems involving the Bessel differential operator”, J. Austral. Math. Soc. Ser. B 27 (1986), 453–471.CrossRefGoogle Scholar
[23]Wloka, J., Partielle Differentialgleichungen (B. G. Teubner, Stuttgart, 1982).CrossRefGoogle Scholar