Source localization and sensor placement in environmental monitoring

Khapalov, A.

International Journal of Applied Mathematics and Computer Science, Tome 20 (2010) no. 3, pp. 445-458.

Résumé

In this paper we discuss two closely related problems arising in environmental monitoring. The first is the source localization problem linked to the question How can one find an unknown 'contamination source'? The second is an associated sensor placement problem: Where should we place sensors that are capable of providing the necessary 'adequate data' for that? Our approach is based on some concepts and ideas developed in mathematical control theory of partial differential equations.

Détail

Keywords: source localization, sensor placement, environmental monitoring, controllability, identifiability, parabolic system
Mots-clés : lokalizacja źródła, położenie sensora, monitoring środowiska, sterowalność, identyfikowalność, równanie paraboliczne

Bibliographie

[1] Afifi, L., El Jai, A. and Merry, M. (2000). Detection and sources reconstruction in a tube, International Journal of Systems Science 31(2): 149-159.

[2] Afifi, L., El Jai, A. and Merry, M. (2001). Regional detection and reconstruction of unknown internal or boundary sources, International Journal of Applied Mathematics and Computer Science 11(2): 319-348.

[3] Alvez, C., Silvestre, A.L., Takahashi, T. and Tuscnak M. (2009). Solving inverse source problems using observability (with), SIAMJournal of Control and Optimization 48(3): 1632-1659.

[4] Butkovskii, A.G. and Pustylnikov, A.M. (1987). Mobile Control of Distributed Parameter Systems, Ellis Horwood, Chichester.

[5] Dagan, G. (1987). Theory of solute transport by groundwater, Annual Reviews of Fluid Mechanics 19: 183-215.

[6] Dagan, G. (1989). Flow and Transport in Porous Formations, Springer-Verlag, Heidelberg, p. 465.

[7] Demetriou, M. (2009). Centralized and decentralized policies for the containment of moving source in 2D diffusion processes using sensor/actuator network, 2009 American Control Conference, St. Louis, MO, USA, pp. 127-132.

[8] Devooght, J. and Smidts, O.F. (1996). Transport of radionuclides in stochastic media: 1. The quasi asymptotic approximation, Annals of Nuclear Energy 23(6): 499-516.

[9] Dolecki, Sz. (1973). Observation for the one-dimensional heat equation, Studia Mathematica 48: 291-305.

[10] Dolecki, Sz. and Russell, D.L. (1977). A general theory of observation and control, SIAM Journal of Control and Optimization 15: 185-219.

[11] Fattorini, H.O. and Russell, D.L. (1974). Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quarterly of Applied Mathematics 43: 45-69.

[12] Friedman, A. (1964). Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ.

[13] Isakov, V. (2009). Inverse Problems for Partial Differential Equations, Springer, New York, NY, p. 360

[14] Khapalov, A.Y. (1994a). L∞-exact observability of the heat equation with scanning pointwise sensor, SIAM Journal Control and Optimization 32(4): 1037-1051.

[15] Khapalov, A.Y. (1994b). Localization of unknown sources for parabolic systems on the basis of available observations, International Journal of System Sciences 25(8): 1305-1322.

[16] Khapalov, A.Y. (1996). On unique continuation of the solutions of the parabolic equation from a curve, Control and Cybernetics 25(1): 451-463.

[17] Khapalov, A.Y. (1998). Exact null-controllability for the semilinear heat equation with mobile controls of degenerate support, Technical Report Series 98-2, Mathematics Department, Washington State University, Pullman, WA.

[18] Khapalov, A.Y. (2001). Mobile point controls versus locally distributed ones for the controllability of the semilinear parabolic equation, SIAM Journal of Control and Optimization 40(7): 231-252.

[19] Khapalov, A.Y. (2010). Controllability of Partial Differential Equations Governed by Multiplicative Controls, Lecture Notes in Mathematics, Vol. 1995, Springer, Heidelberg/Dordrecht/London/New York, NY.

[20] Komornik, V. and Yamamoto, M. (2002). Upper and lower estimates in determining point sources in a wave equation, Inverse Problems 18(2): 319-329.

[21] Komornik, V. and Yamamoto, M. (2005). Estimation of point sources and applications to inverse problems, Inverse Problems 21(6): 2051-2070.

[22] Kurzhanski, A.B. and Khapalov, A.Y. (1990).Mathematical problem motivated by environmental monitoring, Proceedings of the 11th IFAC World Congress, Tallinn, Estonia, p. 5.

[23] Ladyzhenskaja, O.H., Solonnikov, V.A. and Ural'ceva, N.N. (1968). Linear and Quasi-linear Equations of Parabolic Type, AMS, Providence, RI.

[24] Matthes, J., Gruell, L. and Keller, H. (2005). Source localization by spatially distributed electronic noses for advection and diffusion, IEEE Transactions on Signal Processing 53(5): 1711-1719.

[25] Nicaise, S. and Zair, O. (2004). Determination of point sources in vibrating beams by boundary measurements: Identifiability, stability, and reconstruction results, Electronic Journal of Differential Equations (20): 1-17.

[26] PAGIS (1989). Performance Assessment of Geological Isolation Systems for Radioactive Waste-Summary, Nuclear Science and Technology, Joint Research Centre, EUR 11775, 11776 EN.

[27] Puel, J.-P. and Yamamoto, M. (1995). Applications de la controlabilite exacte à quelques problèmes inverses hyperboliques, Computer Rendus de l'Académie des Sciences Serie 1, Mathématique 320(10): 1171-1176.

[28] Saut, J.-C. and Scheurer, B. (1987). Unique continuation for some evolution equations, Journal of Differential Equations 66(1): 118-139.

[29] Seinfeld, J.H. (1986), Atmospheric Chemistry and Physics of Pollution, Wiley Interscience, Somerset, NJ.

[30] Shukla, J.B., Hallam T. G. and Capasso V. (Eds.) (1987). Mathematical Modelling of Environmental and Ecological Systems, Elsevier, Amsterdam.

[31] Sivergina, I.F., Polis, M.P. and Kolmanovsky, I. (2003). Source identification for parabolic equations, Mathematics of Control, Signals, and Systems 16(2-3): 141-157.

[32] Tzanos, P. and Zefran, M. (2006). Stability analysis of information based control for biochemical source localization, 2006 IEEE International Conference on Robotics and Automation, Orlando, FL, USA, pp. 3116-3121.

[33] Uciński, D. (2000). Optimal sensor location for parameter estimation of distributed processes, International Journal of Control 73(13): 1235-1248.

[34] Uciński, D. (2005). Optimal Measurement Methods for Distributed Parameter System Identification, CRC Press, Boca Raton, FL.

[35] Wang, Y. H. (2007). Global uniqueness and stability for an inverse plate problem, Journal of Optimization Theory and Applications 132(1): 161-173.

[36] Williams,M.M.R. (1992). A new model for describing the transport of radionuclides through fractured rocks, Annals of Nuclear Energy 19(10-12): 791-824.

[37] Williams,M.M.R. (1993). Radionuclides transport through fractured rocks, Annals of Nuclear Energy 20(3): 279-297.

[38] Yamamoto, M. (1995). Stability, reconstruction formula and regularizationfor an inverse source hyperbolic problem by a control method, Inverse Problems 11(2): 481-496.