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April  2011, 7(2): 291-315. doi: 10.3934/jimo.2011.7.291

Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions

1. 

Institute of Mathematics, University of Augsburg, Universitätsstr. 14, D-86159 Augsburg, Germany

Received  July 2010 Revised  November 2010 Published  April 2011

The operation of the electrorheological clutch is simulated by a nonlinear parabolic equation which describes the motion of electrorheological fluid in the gap between the driving and driven rotors. In this case, the velocity of the driving rotor is prescribed on one part of the boundary. Nonlocal nonlinear boundary condition is given on a part of the boundary, which corresponds to the driven rotor A problem on optimal control of acceleration or braking of the driven rotor is formulated and studied. Functions of time of the angular velocity of the driving rotor and of the voltages are considered to be controls. In the case that the clutch acts as an accelerator, the energy consumed in the acceleration of the driven rotor is minimized under the restriction that at some instant, the angular velocity and the acceleration of the driven rotor are localized within given regions. In the case of braking, the energy production is maximized. The existence of a solution of optimal control problem is proved and necessary optimality conditions are established.
Citation: William G. Litvinov. Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions. Journal of Industrial & Management Optimization, 2011, 7 (2) : 291-315. doi: 10.3934/jimo.2011.7.291
References:
[1]

R. A. Adams, "Sobolev Spaces,", Academic Press, (1978).   Google Scholar

[2]

J.-P. Aubin, "Approximation of Elliptic Boundary-Value Problems,", Pure and Applied Mathematics, XXVI (1972).   Google Scholar

[3]

Bayer AG, "Provisional Product Information, Rheobay TP AI 3565 and Rheobay TP AI 3566,", Bayer AG, (1260).   Google Scholar

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O. V. Besov, V. P. Il'in and S. M. Nikolsky, "Integral Representation of Functions and Embedding Theorems,", Nauka, (1975).   Google Scholar

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G. Bossis (Ed), Electrorheological Fluids and Magnetorheological suspensions,, in, (2002), 9.   Google Scholar

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P. Dreyfuss and N. Hungerbühler, Results on a Navier-Stokes system with applications to electrorheological fluid flow,, Int. J. Pure Appl. Math., 14 (2004), 241.   Google Scholar

[7]

P. Dreyfuss and N. Hungerbühler, Navier-Stokes systems with quasimonotone viscosity tensor,, Int. J. Differ. Egu., 9 (2004), 59.   Google Scholar

[8]

D. J. Ellam, R. J. Atkin and W. A. Bullough, Analysis of a smart clutch with cooling flow using two-dimensional Bingham plastic analysis and computational fluid dynamics,, J. Power and Energy, 219 (2005), 639.   Google Scholar

[9]

A. V. Fursikov, "Optimal Control of Distributed System. Theory and Applications,", Translated from the 1999 Russian original by Tamara Rozhkovskaya, (1999).   Google Scholar

[10]

R. H. W. Hoppe and W. G. Litvinov, Problems on electrorheological fluid flows,, Communication on Pure and Applied Analysis, 3 (2004), 809.  doi: 10.3934/cpaa.2004.3.809.  Google Scholar

[11]

R. H. W. Hoppe, W. G. Litvinov and T. Rahman, Problems on stationary flow of electrorheological fluids in cylindrical coordinate system,, SIAM J., 65 (2005), 1633.  doi: 10.1137/S0036139903432883.  Google Scholar

[12]

V. C. L. Hutson and J. S. Pym, "Applications of Functional Analysis and Operator Theory,", Mathematics in Science and Engineering, (1980).   Google Scholar

[13]

K. Josida, "Functional Analysis,", Springer, (1965).   Google Scholar

[14]

L. V. Kantorovich and G. P.Akilov, "Functional Analysis,", Nauka, (1977).   Google Scholar

[15]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type,", Amer. Math. Soc., (1968).   Google Scholar

[16]

L. D. Landau and E. M. Lifschitz, "Electrodynamics of Continuous Media,", Pergamon, (1984).   Google Scholar

[17]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", (French) Dunod; Gauthier-Villars, (1969).   Google Scholar

[18]

W. G. Litvinov, "Motion of Nonlinearly Viscous Fluid,", Nauka, (1982).   Google Scholar

[19]

W. G. Litvinov, "Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics,", Birkh\, (2000).   Google Scholar

[20]

W. G. Litvinov, A problems on non steady flow of a nonlinear viscous fluid in a deformable pipe,, Methods of Functional Analysis and Topology, 2 (1996), 85.   Google Scholar

[21]

W. G. Litvinov and R. H. W. Hoppe, Coupled problems on stationary non-isothermal flow of electrorheological fluids,, Com. Pure Appl. Anal., 4 (2005), 779.  doi: 10.3934/cpaa.2005.4.779.  Google Scholar

[22]

W. G. Litvinov, Problem on stationary flow of electrorheological fluids at the generalized conditions of slip on the boundary,, Com. Pure Appl. Anal., 6 (2007), 247.  doi: 10.3934/cpaa.2007.6.247.  Google Scholar

[23]

W. G. Litvinov, Dynamics of electrorheological clutch and a problem for nonlinear parabolic equation with nonlocal boundary conditions,, IMA Journal of Applied Mathematics, 73 (2008), 619.  doi: 10.1093/imamat/hxn008.  Google Scholar

[24]

G. I. Marchuk, V. I. Agoshkov, "Introduction in Projective-Net Methods,", Nauka, (1981).   Google Scholar

[25]

M. Parthasarathy and D. J. Kleingenberg, Electrorheology: mechanisms and models,, Material Science and Engineering, R17 (1996), 57.   Google Scholar

[26]

B. N. Pshenichnyi, "Necessary Optimality Conditions,", Nauka, (1969).   Google Scholar

[27]

Y. A. Roitberg, "Elliptic Boundary Value Problems in the Spaces of Distributions,", Translated from the Russian by Peter Malyshev and Dmitry Malyshev. Mathematics and its Applications, (1996).   Google Scholar

[28]

Z. P. Shulman and V. I. Kordonskii, "Magnetorheological Effect,", Nauka i Technika, (1982).   Google Scholar

[29]

L. Schwartz, "Analyse Mathématique 1,", Hermann, (1967).   Google Scholar

[30]

V. A. Solonnikov, A priori estimates for parabolic equations of the second order,, Trudy Mat. Inst. Steklov, 70 (1964), 133.   Google Scholar

[31]

M. Whittle, R. J. Atkin and W. A. Bullough, Dynamic of a radial electrorheological clutch,, Int. J. Mod. Phys., 13 (1999), 2119.  doi: 10.1142/S0217979299002216.  Google Scholar

[32]

M. Whittle, R. J. Atkin and W. A. Bullough, Fluid dynamic limitations on the performance of an electrorheological clutch,, J. Non-Newtonian Fluid Mech., 57 (1995), 61.  doi: 10.1016/0377-0257(94)01296-T.  Google Scholar

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces,", Academic Press, (1978).   Google Scholar

[2]

J.-P. Aubin, "Approximation of Elliptic Boundary-Value Problems,", Pure and Applied Mathematics, XXVI (1972).   Google Scholar

[3]

Bayer AG, "Provisional Product Information, Rheobay TP AI 3565 and Rheobay TP AI 3566,", Bayer AG, (1260).   Google Scholar

[4]

O. V. Besov, V. P. Il'in and S. M. Nikolsky, "Integral Representation of Functions and Embedding Theorems,", Nauka, (1975).   Google Scholar

[5]

G. Bossis (Ed), Electrorheological Fluids and Magnetorheological suspensions,, in, (2002), 9.   Google Scholar

[6]

P. Dreyfuss and N. Hungerbühler, Results on a Navier-Stokes system with applications to electrorheological fluid flow,, Int. J. Pure Appl. Math., 14 (2004), 241.   Google Scholar

[7]

P. Dreyfuss and N. Hungerbühler, Navier-Stokes systems with quasimonotone viscosity tensor,, Int. J. Differ. Egu., 9 (2004), 59.   Google Scholar

[8]

D. J. Ellam, R. J. Atkin and W. A. Bullough, Analysis of a smart clutch with cooling flow using two-dimensional Bingham plastic analysis and computational fluid dynamics,, J. Power and Energy, 219 (2005), 639.   Google Scholar

[9]

A. V. Fursikov, "Optimal Control of Distributed System. Theory and Applications,", Translated from the 1999 Russian original by Tamara Rozhkovskaya, (1999).   Google Scholar

[10]

R. H. W. Hoppe and W. G. Litvinov, Problems on electrorheological fluid flows,, Communication on Pure and Applied Analysis, 3 (2004), 809.  doi: 10.3934/cpaa.2004.3.809.  Google Scholar

[11]

R. H. W. Hoppe, W. G. Litvinov and T. Rahman, Problems on stationary flow of electrorheological fluids in cylindrical coordinate system,, SIAM J., 65 (2005), 1633.  doi: 10.1137/S0036139903432883.  Google Scholar

[12]

V. C. L. Hutson and J. S. Pym, "Applications of Functional Analysis and Operator Theory,", Mathematics in Science and Engineering, (1980).   Google Scholar

[13]

K. Josida, "Functional Analysis,", Springer, (1965).   Google Scholar

[14]

L. V. Kantorovich and G. P.Akilov, "Functional Analysis,", Nauka, (1977).   Google Scholar

[15]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type,", Amer. Math. Soc., (1968).   Google Scholar

[16]

L. D. Landau and E. M. Lifschitz, "Electrodynamics of Continuous Media,", Pergamon, (1984).   Google Scholar

[17]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", (French) Dunod; Gauthier-Villars, (1969).   Google Scholar

[18]

W. G. Litvinov, "Motion of Nonlinearly Viscous Fluid,", Nauka, (1982).   Google Scholar

[19]

W. G. Litvinov, "Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics,", Birkh\, (2000).   Google Scholar

[20]

W. G. Litvinov, A problems on non steady flow of a nonlinear viscous fluid in a deformable pipe,, Methods of Functional Analysis and Topology, 2 (1996), 85.   Google Scholar

[21]

W. G. Litvinov and R. H. W. Hoppe, Coupled problems on stationary non-isothermal flow of electrorheological fluids,, Com. Pure Appl. Anal., 4 (2005), 779.  doi: 10.3934/cpaa.2005.4.779.  Google Scholar

[22]

W. G. Litvinov, Problem on stationary flow of electrorheological fluids at the generalized conditions of slip on the boundary,, Com. Pure Appl. Anal., 6 (2007), 247.  doi: 10.3934/cpaa.2007.6.247.  Google Scholar

[23]

W. G. Litvinov, Dynamics of electrorheological clutch and a problem for nonlinear parabolic equation with nonlocal boundary conditions,, IMA Journal of Applied Mathematics, 73 (2008), 619.  doi: 10.1093/imamat/hxn008.  Google Scholar

[24]

G. I. Marchuk, V. I. Agoshkov, "Introduction in Projective-Net Methods,", Nauka, (1981).   Google Scholar

[25]

M. Parthasarathy and D. J. Kleingenberg, Electrorheology: mechanisms and models,, Material Science and Engineering, R17 (1996), 57.   Google Scholar

[26]

B. N. Pshenichnyi, "Necessary Optimality Conditions,", Nauka, (1969).   Google Scholar

[27]

Y. A. Roitberg, "Elliptic Boundary Value Problems in the Spaces of Distributions,", Translated from the Russian by Peter Malyshev and Dmitry Malyshev. Mathematics and its Applications, (1996).   Google Scholar

[28]

Z. P. Shulman and V. I. Kordonskii, "Magnetorheological Effect,", Nauka i Technika, (1982).   Google Scholar

[29]

L. Schwartz, "Analyse Mathématique 1,", Hermann, (1967).   Google Scholar

[30]

V. A. Solonnikov, A priori estimates for parabolic equations of the second order,, Trudy Mat. Inst. Steklov, 70 (1964), 133.   Google Scholar

[31]

M. Whittle, R. J. Atkin and W. A. Bullough, Dynamic of a radial electrorheological clutch,, Int. J. Mod. Phys., 13 (1999), 2119.  doi: 10.1142/S0217979299002216.  Google Scholar

[32]

M. Whittle, R. J. Atkin and W. A. Bullough, Fluid dynamic limitations on the performance of an electrorheological clutch,, J. Non-Newtonian Fluid Mech., 57 (1995), 61.  doi: 10.1016/0377-0257(94)01296-T.  Google Scholar

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