Controllability of a basic cochlea model

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December 2016, 5(4): 475-487. doi: 10.3934/eect.2016015

Controllability of a basic cochlea model

Scott W. Hansen ^1,

1.	Department of Mathematics, Iowa State University, Ames, IA 50011, United States

Received August 2016 Revised September 2016 Published October 2016

Abstract
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Two variations of a basic model for a cochlea are described which consist of a basilar membrane coupled with a linear potential fluid. The basilar membrane is modeled as an array of oscillators which may or may not include longitudinal elasticity. The fluid is assumed to be a linear potential fluid described by Laplace's equation in a domain that surrounds the basilar membrane. The problem of controllability of the system is considered with control active on a portion of the basilar membrane. Approximate controllability is proved for both models and moreover lack of exact controllability is shown to hold when longitudinal stiffness is not included.

Keywords: Controllability, cochlea model, distributed control, fluid structure interaction, potential fluid..

Mathematics Subject Classification: Primary: 93B05, 93C20; Secondary: 74F10, 74K1.

Citation: Scott W. Hansen. Controllability of a basic cochlea model. Evolution Equations and Control Theory, 2016, 5 (4) : 475-487. doi: 10.3934/eect.2016015

References:

[1]	G. von Békésy, Experiments in Hearing McGraw-Hill Inc., New York, 1960.
[2]	Isaak Chepkwony, Analysis and Control Theory of some Cochlea Models, Ph.D. thesis, Department of Mathematics, Iowa State University, Ames, IA, 2006.
[3]	S. W. Hansen, Exact controllability of an elastic membrane coupled with a potential fluid, Int. J. Appl. Math. Comput. Sci., 11 (2001), 1231-1248.
[4]	S. W. Hansen and A. Lyashenko, Exact controllability of a beam in an incompressible inviscid fluid, Disc. Cont. Dyn. Syst., 3 (1997), 59-78.
[5]	H. L. F. von Helmoltz, On the sensations of tone as a physiological basis for the theory of music, (Translation by A. J. Ellis of Die Lehre von den Tonempfindungen als physiologiche Grundlage für die Theorie der Musik: Verlag von Fr. Vieweg u. Sohn. 4th ed., 1877; originally published 1863) Dover, New York, 1954.
[6]	J. B. Keller and J. C. Neu, Asymptotic analysis of a viscous cochlear model, J. Acoust. Soc. Amer., 77 (1985), 2107-2110. doi: 10.1121/1.391735.
[7]	R. J. Leveque, C. S. Peskin and P. D. Lax, Solution of a two-dimensional cochlea model using transform techniques, SIAM J. Applied Math., 45 (1988), 450-464. doi: 10.1137/0145026.
[8]	R. J. Leveque, C. S. Peskin and P. D. Lax, Solution of a two-dimensional cochlea model with fluid viscosity, SIAM J. Applied Math., 48 (1988), 191-213. doi: 10.1137/0148009.
[9]	J. Lighthill, Energy flow in the cochlea, J. Fluid Mech., 106 (1981), 149-213. doi: 10.1017/S0022112081001560.
[10]	R. D. Luce, Sound and Hearing. A Conceptual Introduction, Lawrence Erlbaum Assoc. Inc., Publishers, Hillsdale, New Jersey, 1993.
[11]	G. A. Manley and R. R. Fay, Active Processes and Otoacoustic Emissions in Hearing, Springer Science & Business Media 30, 2007.
[12]	J. Nečas, Les Méthodes Directes en théorie des équations Elliptiques. Paris: Masson, 1967.
[13]	S. T. Neely, Mathematical modeling of cochlear mechanics, J. Acoust. Soc. Am., 78 (1985), 345-352. doi: 10.1121/1.392497.
[14]	S. T. Neely and D. O. Kim, An active cochlear model showing sharp tuning and high sensitivity, Hearing Research, 9 (1983) 123-130. doi: 10.1016/0378-5955(83)90022-9.
[15]	A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[16]	O. F. Ranke, Theory of operation of cochlear: A contribution to the hydrodynamics of the cochlear, J. Acoust. Soc. Am., 22 (1950), 772-777.
[17]	W. S. Rhode, Observations of the vibration of the Basilar Membrane in squirrel monkeys using the Mössbauer technique, Journal of the Acoustical Society of America, 49 (1971), 1218-1231.
[18]	J. Xin, Dispersive instability and its minimization in time-domain computation of steady-state responses of cochlea models, J. Acoust. Soc. Am., 115 (2004), 2173-2177.

show all references

References:

[1]	G. von Békésy, Experiments in Hearing McGraw-Hill Inc., New York, 1960.
[2]	Isaak Chepkwony, Analysis and Control Theory of some Cochlea Models, Ph.D. thesis, Department of Mathematics, Iowa State University, Ames, IA, 2006.
[3]	S. W. Hansen, Exact controllability of an elastic membrane coupled with a potential fluid, Int. J. Appl. Math. Comput. Sci., 11 (2001), 1231-1248.
[4]	S. W. Hansen and A. Lyashenko, Exact controllability of a beam in an incompressible inviscid fluid, Disc. Cont. Dyn. Syst., 3 (1997), 59-78.
[5]	H. L. F. von Helmoltz, On the sensations of tone as a physiological basis for the theory of music, (Translation by A. J. Ellis of Die Lehre von den Tonempfindungen als physiologiche Grundlage für die Theorie der Musik: Verlag von Fr. Vieweg u. Sohn. 4th ed., 1877; originally published 1863) Dover, New York, 1954.
[6]	J. B. Keller and J. C. Neu, Asymptotic analysis of a viscous cochlear model, J. Acoust. Soc. Amer., 77 (1985), 2107-2110. doi: 10.1121/1.391735.
[7]	R. J. Leveque, C. S. Peskin and P. D. Lax, Solution of a two-dimensional cochlea model using transform techniques, SIAM J. Applied Math., 45 (1988), 450-464. doi: 10.1137/0145026.
[8]	R. J. Leveque, C. S. Peskin and P. D. Lax, Solution of a two-dimensional cochlea model with fluid viscosity, SIAM J. Applied Math., 48 (1988), 191-213. doi: 10.1137/0148009.
[9]	J. Lighthill, Energy flow in the cochlea, J. Fluid Mech., 106 (1981), 149-213. doi: 10.1017/S0022112081001560.
[10]	R. D. Luce, Sound and Hearing. A Conceptual Introduction, Lawrence Erlbaum Assoc. Inc., Publishers, Hillsdale, New Jersey, 1993.
[11]	G. A. Manley and R. R. Fay, Active Processes and Otoacoustic Emissions in Hearing, Springer Science & Business Media 30, 2007.
[12]	J. Nečas, Les Méthodes Directes en théorie des équations Elliptiques. Paris: Masson, 1967.
[13]	S. T. Neely, Mathematical modeling of cochlear mechanics, J. Acoust. Soc. Am., 78 (1985), 345-352. doi: 10.1121/1.392497.
[14]	S. T. Neely and D. O. Kim, An active cochlear model showing sharp tuning and high sensitivity, Hearing Research, 9 (1983) 123-130. doi: 10.1016/0378-5955(83)90022-9.
[15]	A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[16]	O. F. Ranke, Theory of operation of cochlear: A contribution to the hydrodynamics of the cochlear, J. Acoust. Soc. Am., 22 (1950), 772-777.
[17]	W. S. Rhode, Observations of the vibration of the Basilar Membrane in squirrel monkeys using the Mössbauer technique, Journal of the Acoustical Society of America, 49 (1971), 1218-1231.
[18]	J. Xin, Dispersive instability and its minimization in time-domain computation of steady-state responses of cochlea models, J. Acoust. Soc. Am., 115 (2004), 2173-2177.

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