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

Controllability of a basic cochlea model

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.
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., (1960). Google Scholar

[2]

Isaak Chepkwony, Analysis and Control Theory of some Cochlea Models,, Ph.D. thesis, (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. 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. 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). Google Scholar

[6]

J. B. Keller and J. C. Neu, Asymptotic analysis of a viscous cochlear model,, J. Acoust. Soc. Amer., 77 (1985), 2107. 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. 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. doi: 10.1137/0148009. Google Scholar

[9]

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

[10]

R. D. Luce, Sound and Hearing. A Conceptual Introduction,, Lawrence Erlbaum Assoc. Inc., (1993). Google Scholar

[11]

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

show all references

References:
[1]

G. von Békésy, Experiments in Hearing, McGraw-Hill Inc., (1960). Google Scholar

[2]

Isaak Chepkwony, Analysis and Control Theory of some Cochlea Models,, Ph.D. thesis, (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. 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. 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). Google Scholar

[6]

J. B. Keller and J. C. Neu, Asymptotic analysis of a viscous cochlear model,, J. Acoust. Soc. Amer., 77 (1985), 2107. 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. 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. doi: 10.1137/0148009. Google Scholar

[9]

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

[10]

R. D. Luce, Sound and Hearing. A Conceptual Introduction,, Lawrence Erlbaum Assoc. Inc., (1993). Google Scholar

[11]

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

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