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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. 
[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. 
[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), 913. 
[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. 
[7] 
P. Dreyfuss and N. Hungerbühler, NavierStokes systems with quasimonotone viscosity tensor, Int. J. Differ. Egu., 9 (2004), 5979. 
[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. 
[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), 809848. 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), 16331656. 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 YorkLondon, 1980. 
[13]  
[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; GauthierVillars, 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), 85113. 
[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. 
[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. 
[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. 
[24] 
G. I. Marchuk, V. I. Agoshkov, "Introduction in ProjectiveNet Methods," Nauka, Moscow, (in Russian), 1981. 
[25] 
M. Parthasarathy and D. J. Kleingenberg, Electrorheology: mechanisms and models, Material Science and Engineering, R17 (1996), 57103. 
[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), 133212. 
[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. 
[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. 
show all references
References:
[1] 
R. A. Adams, "Sobolev Spaces," Academic Press, Inc. New Cork, 1978. 
[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. 
[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), 913. 
[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. 
[7] 
P. Dreyfuss and N. Hungerbühler, NavierStokes systems with quasimonotone viscosity tensor, Int. J. Differ. Egu., 9 (2004), 5979. 
[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. 
[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), 809848. 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), 16331656. 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 YorkLondon, 1980. 
[13]  
[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; GauthierVillars, 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), 85113. 
[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. 
[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. 
[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. 
[24] 
G. I. Marchuk, V. I. Agoshkov, "Introduction in ProjectiveNet Methods," Nauka, Moscow, (in Russian), 1981. 
[25] 
M. Parthasarathy and D. J. Kleingenberg, Electrorheology: mechanisms and models, Material Science and Engineering, R17 (1996), 57103. 
[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), 133212. 
[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. 
[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. 
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