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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 and Management Optimization, 2011, 7 (2) : 291-315. doi: 10.3934/jimo.2011.7.291
References:
[1]

R. A. Adams, "Sobolev Spaces," Academic Press, Inc. New Cork, 1978.

[2]

J.-P. Aubin, "Approximation of Elliptic Boundary-Value Problems," Pure and Applied Mathematics, Vol. XXVI, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1972.

[3]

Bayer AG, "Provisional Product Information, Rheobay TP AI 3565 and Rheobay TP AI 3566," Bayer AG, Silicones Bisiness Unit, No. AI 12601e, Leverkusen, 1997.

[4]

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

[5]

G. Bossis (Ed), Electrorheological Fluids and Magnetorheological suspensions, in "Proceeding of the Eighth International Conference, Nice, France, July, 2001," World Scientific, Singapore, (2002), 9-13.

[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), 241-271.

[7]

P. Dreyfuss and N. Hungerbühler, Navier-Stokes systems with quasimonotone viscosity tensor, Int. J. Differ. Egu., 9 (2004), 59-79.

[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, Proc. IMechE, Part A, 219 (2005), 639-652.

[9]

A. V. Fursikov, "Optimal Control of Distributed System. Theory and Applications," Translated from the 1999 Russian original by Tamara Rozhkovskaya, Translations of Mathematical Monographs, 187, American Mathematical Society, Providence, RI, 2000.

[10]

R. H. W. Hoppe and W. G. Litvinov, Problems on electrorheological fluid flows, Communication on Pure and Applied Analysis, 3 (2004), 809-848. doi: 10.3934/cpaa.2004.3.809.

[11]

R. H. W. Hoppe, W. G. Litvinov and T. Rahman, Problems on stationary flow of electrorheological fluids in cylindrical coordinate system, SIAM J., Appl. Math., 65 (2005), 1633-1656. doi: 10.1137/S0036139903432883.

[12]

V. C. L. Hutson and J. S. Pym, "Applications of Functional Analysis and Operator Theory," Mathematics in Science and Engineering, 146, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.

[13]

K. Josida, "Functional Analysis," Springer, Berlin, 1965.

[14]

L. V. Kantorovich and G. P.Akilov, "Functional Analysis," Nauka, Moscow, (In Russian), 1977.

[15]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc., Providence, RI, 1968.

[16]

L. D. Landau and E. M. Lifschitz, "Electrodynamics of Continuous Media," Pergamon, Oxford, 1984.

[17]

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

[18]

W. G. Litvinov, "Motion of Nonlinearly Viscous Fluid," Nauka, Moscow, (in Russian), 1982.

[19]

W. G. Litvinov, "Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics," Birkhäuser, Basel, 2000.

[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), 85-113.

[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), 779-803. doi: 10.3934/cpaa.2005.4.779.

[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), 247-277. doi: 10.3934/cpaa.2007.6.247.

[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), 619-640. doi: 10.1093/imamat/hxn008.

[24]

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

[25]

M. Parthasarathy and D. J. Kleingenberg, Electrorheology: mechanisms and models, Material Science and Engineering, R17 (1996), 57-103.

[26]

B. N. Pshenichnyi, "Necessary Optimality Conditions," Nauka, Moscow, (in Russian), 1969.

[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, 384, Kluwer Academic Publishers Group, Dordrecht, 1996.

[28]

Z. P. Shulman and V. I. Kordonskii, "Magnetorheological Effect," Nauka i Technika, Minsk, (in Russian), 1982.

[29]

L. Schwartz, "Analyse Mathématique 1," Hermann, Paris, 1967.

[30]

V. A. Solonnikov, A priori estimates for parabolic equations of the second order, Trudy Mat. Inst. Steklov, (In Russian), 70 (1964), 133-212.

[31]

M. Whittle, R. J. Atkin and W. A. Bullough, Dynamic of a radial electrorheological clutch, Int. J. Mod. Phys., B, 13 (1999), 2119-2126. doi: 10.1142/S0217979299002216.

[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), 61-81. doi: 10.1016/0377-0257(94)01296-T.

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces," Academic Press, Inc. New Cork, 1978.

[2]

J.-P. Aubin, "Approximation of Elliptic Boundary-Value Problems," Pure and Applied Mathematics, Vol. XXVI, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1972.

[3]

Bayer AG, "Provisional Product Information, Rheobay TP AI 3565 and Rheobay TP AI 3566," Bayer AG, Silicones Bisiness Unit, No. AI 12601e, Leverkusen, 1997.

[4]

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

[5]

G. Bossis (Ed), Electrorheological Fluids and Magnetorheological suspensions, in "Proceeding of the Eighth International Conference, Nice, France, July, 2001," World Scientific, Singapore, (2002), 9-13.

[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), 241-271.

[7]

P. Dreyfuss and N. Hungerbühler, Navier-Stokes systems with quasimonotone viscosity tensor, Int. J. Differ. Egu., 9 (2004), 59-79.

[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, Proc. IMechE, Part A, 219 (2005), 639-652.

[9]

A. V. Fursikov, "Optimal Control of Distributed System. Theory and Applications," Translated from the 1999 Russian original by Tamara Rozhkovskaya, Translations of Mathematical Monographs, 187, American Mathematical Society, Providence, RI, 2000.

[10]

R. H. W. Hoppe and W. G. Litvinov, Problems on electrorheological fluid flows, Communication on Pure and Applied Analysis, 3 (2004), 809-848. doi: 10.3934/cpaa.2004.3.809.

[11]

R. H. W. Hoppe, W. G. Litvinov and T. Rahman, Problems on stationary flow of electrorheological fluids in cylindrical coordinate system, SIAM J., Appl. Math., 65 (2005), 1633-1656. doi: 10.1137/S0036139903432883.

[12]

V. C. L. Hutson and J. S. Pym, "Applications of Functional Analysis and Operator Theory," Mathematics in Science and Engineering, 146, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.

[13]

K. Josida, "Functional Analysis," Springer, Berlin, 1965.

[14]

L. V. Kantorovich and G. P.Akilov, "Functional Analysis," Nauka, Moscow, (In Russian), 1977.

[15]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc., Providence, RI, 1968.

[16]

L. D. Landau and E. M. Lifschitz, "Electrodynamics of Continuous Media," Pergamon, Oxford, 1984.

[17]

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

[18]

W. G. Litvinov, "Motion of Nonlinearly Viscous Fluid," Nauka, Moscow, (in Russian), 1982.

[19]

W. G. Litvinov, "Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics," Birkhäuser, Basel, 2000.

[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), 85-113.

[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), 779-803. doi: 10.3934/cpaa.2005.4.779.

[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), 247-277. doi: 10.3934/cpaa.2007.6.247.

[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), 619-640. doi: 10.1093/imamat/hxn008.

[24]

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

[25]

M. Parthasarathy and D. J. Kleingenberg, Electrorheology: mechanisms and models, Material Science and Engineering, R17 (1996), 57-103.

[26]

B. N. Pshenichnyi, "Necessary Optimality Conditions," Nauka, Moscow, (in Russian), 1969.

[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, 384, Kluwer Academic Publishers Group, Dordrecht, 1996.

[28]

Z. P. Shulman and V. I. Kordonskii, "Magnetorheological Effect," Nauka i Technika, Minsk, (in Russian), 1982.

[29]

L. Schwartz, "Analyse Mathématique 1," Hermann, Paris, 1967.

[30]

V. A. Solonnikov, A priori estimates for parabolic equations of the second order, Trudy Mat. Inst. Steklov, (In Russian), 70 (1964), 133-212.

[31]

M. Whittle, R. J. Atkin and W. A. Bullough, Dynamic of a radial electrorheological clutch, Int. J. Mod. Phys., B, 13 (1999), 2119-2126. doi: 10.1142/S0217979299002216.

[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), 61-81. doi: 10.1016/0377-0257(94)01296-T.

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