Partial Internal Control Recovery on 1-D Klein-Gordon Systems


  • Iwan Pranoto Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha No. 10 Bandung 40132



In this exposition, a technique to recover internal control on a distributed parameter system is reported. The system is described by 1-D Klein- Gordon partial differential equation with a time-varying parameter. We would like to recover the internal control applied to the system if we know some limited information about the output. We use a method called sentinel method to recover the internal control. It involves some construction of a linear functional, and we show that this construction relates closely to the exact controllability problem.


Lions, J.L., Sentinels and Stealthy Perturbations. Semi Complete Set of Sentinels, Math. and Numerical Aspects of Wave Propagation Phenomena, by G. Cohen (eds), SIAM Philadelphia, pp. 239-251, 1991.

Lions, J.L., Sentinelles pour Les Systemes Distribues, Masson, Paris, 1992.

Pranoto, Internal Control Recovery on Klein-Gordon Systems, J. Indones. Math. Soc., 10(2), 115-124, 2004.

Pranoto, A Brief Summary on the Control Recovery of Time-Varying K-G Systems, International Journal of Modelling, Identification and Control, 8(1), 2009.

Zuazua, E., Exact Controllability of Semilinear Wave Equations in One Space Dimension, Ann. Inst. H. Poincare Anal. Non Lineaire, 10, 109 - 129, 1993.

Zhang, X. & Zuazua, E., Exact Controllability of The Semi-linear Wave Equations, in Sixty Open Problems in the Mathematics of Systems and Control, ed by V.D. Blondel and A. Megretski, Princeton Univ. Press, 2003.

Zuazua, E., Propagation, Observation, and Control of Waves Approximated by Finite Difference Methods, SIAM Review, 47 (2), 197- 243, 2005.

Zuazua, E., Control and Numerical Approximation of the Wave and Heat Equations, Proceedings of the ICM 2006, Vol. III, "Invited Lectures", European Mathematical Society Publishing House, M. Sanz-Sole et al. eds., pp. 1389-1417, 2006.

Acheli, D., Kernevez, J.P. & Oukaci, F., The Sentinel Method Used in Identification of The Position and Trajectory of A Source of Pollution, Appl. Anal. 70(3-4), 303-319, 1999.

Fliess, M., Join, C. & Sira-Ramirez, H., Non Linear estimation Is Easy, International Journal of Modelling, Identification and Control, 4(1), 12 - 27, 2008.

Olsder, G.J., Mathematical Systems Theory, Delft University Press, 1994.

Wonham, W.M., Linear Multivariable Control: A Geometric Approach, Applications of Mathematics, 10, Springer-Verlag, 1979.

Pranoto, An Equality of the 1-D Klein-Gordon Equation with A Time-Varying Parameter, Journal of Inequality in Pure and Applied Mathematics, Art. 45, 3(3), 2002.

Komornik, V., Exact Controllability and Stabilization: The Multiplier Method, Research in Applied Mathematics, Masson, 1994.

Bardos, C., Lebeau, G. & Rauch, J., Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary, SIAM J. Control and Optimization, 30(5), 1024-1065, 1992.

Lions, J.L., Controllabilite Exacte Perturbations et Stabilisation de Systemes Distribues, Masson, Paris, 1988.

Pranoto, I., Exact Controllability of Klein-Gordon Systems with A Time-varying Parameter, in Topics in Applied and Theoretical Mathematics and Computer Science, edited by V.V. Kluev, et al, World Scientific and Engineering Academy and Society, 76 - 80, 2001.