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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

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 & 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. Google Scholar

[2]

Isaak Chepkwony, Analysis and Control Theory of some Cochlea Models, Ph.D. thesis, Department of Mathematics, Iowa State University, Ames, IA, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. Lighthill, Energy flow in the cochlea, J. Fluid Mech., 106 (1981), 149-213. doi: 10.1017/S0022112081001560.  Google Scholar

[10]

R. D. Luce, Sound and Hearing. A Conceptual Introduction, Lawrence Erlbaum Assoc. Inc., Publishers, Hillsdale, New Jersey, 1993. Google Scholar

[11]

G. A. Manley and R. R. Fay, Active Processes and Otoacoustic Emissions in Hearing, Springer Science & Business Media 30, 2007. Google Scholar

[12]

J. Nečas, Les Méthodes Directes en théorie des équations Elliptiques. Paris: Masson, 1967. Google Scholar

[13]

S. T. Neely, Mathematical modeling of cochlear mechanics, J. Acoust. Soc. Am., 78 (1985), 345-352. doi: 10.1121/1.392497.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

G. von Békésy, Experiments in Hearing McGraw-Hill Inc., New York, 1960. Google Scholar

[2]

Isaak Chepkwony, Analysis and Control Theory of some Cochlea Models, Ph.D. thesis, Department of Mathematics, Iowa State University, Ames, IA, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. Lighthill, Energy flow in the cochlea, J. Fluid Mech., 106 (1981), 149-213. doi: 10.1017/S0022112081001560.  Google Scholar

[10]

R. D. Luce, Sound and Hearing. A Conceptual Introduction, Lawrence Erlbaum Assoc. Inc., Publishers, Hillsdale, New Jersey, 1993. Google Scholar

[11]

G. A. Manley and R. R. Fay, Active Processes and Otoacoustic Emissions in Hearing, Springer Science & Business Media 30, 2007. Google Scholar

[12]

J. Nečas, Les Méthodes Directes en théorie des équations Elliptiques. Paris: Masson, 1967. Google Scholar

[13]

S. T. Neely, Mathematical modeling of cochlear mechanics, J. Acoust. Soc. Am., 78 (1985), 345-352. doi: 10.1121/1.392497.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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