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New passivity analysis of continuoustime recurrent neural networks with multiple discrete delays
Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions
1.  Institute of Mathematics, University of Augsburg, Universitätsstr. 14, D86159 Augsburg, Germany 
References:
[1] 
R. A. Adams, "Sobolev Spaces," Academic Press, Inc. New Cork, 1978. Google Scholar 
[2] 
J.P. Aubin, "Approximation of Elliptic BoundaryValue Problems," Pure and Applied Mathematics, Vol. XXVI, WileyInterscience [A division of John Wiley & Sons, Inc.], New YorkLondonSydney, 1972. Google Scholar 
[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. Google Scholar 
[4] 
O. V. Besov, V. P. Il'in and S. M. Nikolsky, "Integral Representation of Functions and Embedding Theorems," Nauka, Moscow, (In Russian), 1975. Google Scholar 
[5] 
G. Bossis (Ed), Electrorheological Fluids and Magnetorheological suspensions, in "Proceeding of the Eighth International Conference, Nice, France, July, 2001," World Scientific, Singapore, (2002), 913. Google Scholar 
[6] 
P. Dreyfuss and N. Hungerbühler, Results on a NavierStokes system with applications to electrorheological fluid flow, Int. J. Pure Appl. Math., 14 (2004), 241271. Google Scholar 
[7] 
P. Dreyfuss and N. Hungerbühler, NavierStokes systems with quasimonotone viscosity tensor, Int. J. Differ. Egu., 9 (2004), 5979. Google Scholar 
[8] 
D. J. Ellam, R. J. Atkin and W. A. Bullough, Analysis of a smart clutch with cooling flow using twodimensional Bingham plastic analysis and computational fluid dynamics, J. Power and Energy, Proc. IMechE, Part A, 219 (2005), 639652. Google Scholar 
[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. Google Scholar 
[10] 
R. H. W. Hoppe and W. G. Litvinov, Problems on electrorheological fluid flows, Communication on Pure and Applied Analysis, 3 (2004), 809848. 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., Appl. Math., 65 (2005), 16331656. 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, 146, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1980. Google Scholar 
[13] 
K. Josida, "Functional Analysis," Springer, Berlin, 1965. Google Scholar 
[14] 
L. V. Kantorovich and G. P.Akilov, "Functional Analysis," Nauka, Moscow, (In Russian), 1977. Google Scholar 
[15] 
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc., Providence, RI, 1968. Google Scholar 
[16] 
L. D. Landau and E. M. Lifschitz, "Electrodynamics of Continuous Media," Pergamon, Oxford, 1984. Google Scholar 
[17] 
J.L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," (French) Dunod; GauthierVillars, Paris, 1969. Google Scholar 
[18] 
W. G. Litvinov, "Motion of Nonlinearly Viscous Fluid," Nauka, Moscow, (in Russian), 1982. Google Scholar 
[19] 
W. G. Litvinov, "Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics," Birkhäuser, Basel, 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), 85113. Google Scholar 
[21] 
W. G. Litvinov and R. H. W. Hoppe, Coupled problems on stationary nonisothermal flow of electrorheological fluids, Com. Pure Appl. Anal., 4 (2005), 779803. 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), 247277. 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), 619640. doi: 10.1093/imamat/hxn008. Google Scholar 
[24] 
G. I. Marchuk, V. I. Agoshkov, "Introduction in ProjectiveNet Methods," Nauka, Moscow, (in Russian), 1981. Google Scholar 
[25] 
M. Parthasarathy and D. J. Kleingenberg, Electrorheology: mechanisms and models, Material Science and Engineering, R17 (1996), 57103. Google Scholar 
[26] 
B. N. Pshenichnyi, "Necessary Optimality Conditions," Nauka, Moscow, (in Russian), 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, 384, Kluwer Academic Publishers Group, Dordrecht, 1996. Google Scholar 
[28] 
Z. P. Shulman and V. I. Kordonskii, "Magnetorheological Effect," Nauka i Technika, Minsk, (in Russian), 1982. Google Scholar 
[29] 
L. Schwartz, "Analyse Mathématique 1," Hermann, Paris, 1967. Google Scholar 
[30] 
V. A. Solonnikov, A priori estimates for parabolic equations of the second order, Trudy Mat. Inst. Steklov, (In Russian), 70 (1964), 133212. Google Scholar 
[31] 
M. Whittle, R. J. Atkin and W. A. Bullough, Dynamic of a radial electrorheological clutch, Int. J. Mod. Phys., B, 13 (1999), 21192126. 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. NonNewtonian Fluid Mech., 57 (1995), 6181. doi: 10.1016/03770257(94)01296T. Google Scholar 
show all references
References:
[1] 
R. A. Adams, "Sobolev Spaces," Academic Press, Inc. New Cork, 1978. Google Scholar 
[2] 
J.P. Aubin, "Approximation of Elliptic BoundaryValue Problems," Pure and Applied Mathematics, Vol. XXVI, WileyInterscience [A division of John Wiley & Sons, Inc.], New YorkLondonSydney, 1972. Google Scholar 
[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. Google Scholar 
[4] 
O. V. Besov, V. P. Il'in and S. M. Nikolsky, "Integral Representation of Functions and Embedding Theorems," Nauka, Moscow, (In Russian), 1975. Google Scholar 
[5] 
G. Bossis (Ed), Electrorheological Fluids and Magnetorheological suspensions, in "Proceeding of the Eighth International Conference, Nice, France, July, 2001," World Scientific, Singapore, (2002), 913. Google Scholar 
[6] 
P. Dreyfuss and N. Hungerbühler, Results on a NavierStokes system with applications to electrorheological fluid flow, Int. J. Pure Appl. Math., 14 (2004), 241271. Google Scholar 
[7] 
P. Dreyfuss and N. Hungerbühler, NavierStokes systems with quasimonotone viscosity tensor, Int. J. Differ. Egu., 9 (2004), 5979. Google Scholar 
[8] 
D. J. Ellam, R. J. Atkin and W. A. Bullough, Analysis of a smart clutch with cooling flow using twodimensional Bingham plastic analysis and computational fluid dynamics, J. Power and Energy, Proc. IMechE, Part A, 219 (2005), 639652. Google Scholar 
[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. Google Scholar 
[10] 
R. H. W. Hoppe and W. G. Litvinov, Problems on electrorheological fluid flows, Communication on Pure and Applied Analysis, 3 (2004), 809848. 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., Appl. Math., 65 (2005), 16331656. 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, 146, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1980. Google Scholar 
[13] 
K. Josida, "Functional Analysis," Springer, Berlin, 1965. Google Scholar 
[14] 
L. V. Kantorovich and G. P.Akilov, "Functional Analysis," Nauka, Moscow, (In Russian), 1977. Google Scholar 
[15] 
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc., Providence, RI, 1968. Google Scholar 
[16] 
L. D. Landau and E. M. Lifschitz, "Electrodynamics of Continuous Media," Pergamon, Oxford, 1984. Google Scholar 
[17] 
J.L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," (French) Dunod; GauthierVillars, Paris, 1969. Google Scholar 
[18] 
W. G. Litvinov, "Motion of Nonlinearly Viscous Fluid," Nauka, Moscow, (in Russian), 1982. Google Scholar 
[19] 
W. G. Litvinov, "Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics," Birkhäuser, Basel, 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), 85113. Google Scholar 
[21] 
W. G. Litvinov and R. H. W. Hoppe, Coupled problems on stationary nonisothermal flow of electrorheological fluids, Com. Pure Appl. Anal., 4 (2005), 779803. 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), 247277. 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), 619640. doi: 10.1093/imamat/hxn008. Google Scholar 
[24] 
G. I. Marchuk, V. I. Agoshkov, "Introduction in ProjectiveNet Methods," Nauka, Moscow, (in Russian), 1981. Google Scholar 
[25] 
M. Parthasarathy and D. J. Kleingenberg, Electrorheology: mechanisms and models, Material Science and Engineering, R17 (1996), 57103. Google Scholar 
[26] 
B. N. Pshenichnyi, "Necessary Optimality Conditions," Nauka, Moscow, (in Russian), 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, 384, Kluwer Academic Publishers Group, Dordrecht, 1996. Google Scholar 
[28] 
Z. P. Shulman and V. I. Kordonskii, "Magnetorheological Effect," Nauka i Technika, Minsk, (in Russian), 1982. Google Scholar 
[29] 
L. Schwartz, "Analyse Mathématique 1," Hermann, Paris, 1967. Google Scholar 
[30] 
V. A. Solonnikov, A priori estimates for parabolic equations of the second order, Trudy Mat. Inst. Steklov, (In Russian), 70 (1964), 133212. Google Scholar 
[31] 
M. Whittle, R. J. Atkin and W. A. Bullough, Dynamic of a radial electrorheological clutch, Int. J. Mod. Phys., B, 13 (1999), 21192126. 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. NonNewtonian Fluid Mech., 57 (1995), 6181. doi: 10.1016/03770257(94)01296T. Google Scholar 
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