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Existence of optimal output feedback control law for a class of uncertain infinite dimensional stochastic systems: A direct approach
Memory relaxation of type III thermoelastic extensible beams and Berger plates
1. | Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, Via Bonardi 9, Milano 20133 |
2. | Dipartimento di Matematica “Francesco Brioschi”, Politecnico di Milano, Via Bonardi 9, Milano 20133 |
References:
[1] |
H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. |
[2] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992. |
[3] |
F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discrete Cont. Dyn. Systems, 22 (2008), 557-586.
doi: 10.3934/dcds.2008.22.557. |
[4] |
V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273. |
[5] |
C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), p.154301.
doi: 10.1103/PhysRevLett.94.154301. |
[6] |
I. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," Acta, Kharkov, 2002. |
[7] |
I. Chueshov and I. Lasiecka, Attractors and long-time behavior of von Karman thermoelastic plates, Appl. Math. Optim., 58 (2008), 195-241.
doi: 10.1007/s00245-007-9031-8. |
[8] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. |
[9] |
M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720. |
[10] |
M. Coti Zelati, F. Dell'Oro and V. Pata, Energy decay of type III linear thermoelastic plates with memory, submitted. |
[11] |
M. Coti Zelati, V. Pata and R. Quintanilla, Regular global attractors of type III thermoelastic extensible beams, Chin. Ann. Math. Series B, 31 (2010), 619-630.
doi: 10.1007/s11401-010-0605-4. |
[12] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. |
[13] |
G. Fichera, Is the Fourier theory of heat propagation paradoxical?, Rend. Circ. Mat. Palermo, 41 (1992), 5-28. |
[14] |
S. Gatti, A. Miranville, V. Pata and S. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation, Rocky Mountain J. Math., 38 (2008), 1117-1138.
doi: 10.1216/RMJ-2008-38-4-1117. |
[15] |
S. Gatti, A. Miranville, V. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 329-366.
doi: 10.1017/S0308210509000365. |
[16] |
C. Giorgi, M. G. Naso, V. Pata and M. Potomkin, Global attractors for the extensible thermoelastic beam system, J. Diff. Eqs, 246 (2009), 3496-3517.
doi: 10.1016/j.jde.2009.02.020. |
[17] |
C. Giorgi, V. Pata and E. Vuk, On the extensible viscoelastic beam, Nonlinearity, 21 (2008), 713-733.
doi: 10.1088/0951-7715/21/4/004. |
[18] |
A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264.
doi: 10.1080/01495739208946136. |
[19] |
A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208.
doi: 10.1007/BF00044969. |
[20] |
A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. I. Classical continuum physics, Proc. Roy. Soc. London A, 448 (1995), 335-356.
doi: 10.1098/rspa.1995.0020. |
[21] |
A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. II. Generalized continua, Proc. Roy. Soc. London A, 448 (1995), 357-377.
doi: 10.1098/rspa.1995.0021. |
[22] |
A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. III. Mixtures of interacting continua, Proc. Roy. Soc. London A, 448 (1995), 379-388.
doi: 10.1098/rspa.1995.0022. |
[23] |
M. Grasselli and M. Squassina, Exponential stability and singular limit for a linear thermoelastic plate with memory effects, Adv. Math. Sci. Appl., 16 (2006), 15-31. |
[24] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, 1988. |
[25] |
A. Haraux, "Syst\`emes Dynamiques Dissipatifs et Applications," Coll. RMA no.17, Masson, Paris, 1991. |
[26] |
O. A. Ladyzhenskaya, Finding minimal global attractors for the Navier-Stokes equations and other partial differential equations, Russian Math. Surveys, 42 (1987), 27-73.
doi: 10.1070/RM1987v042n06ABEH001503. |
[27] |
V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., 64 (2006), 499-513. |
[28] |
V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.
doi: 10.1007/s00032-009-0098-3. |
[29] |
V. Pata, Uniform estimates of Gronwall type, J. Math. Anal. Appl., 373 (2011), 264-270.
doi: 10.1016/j.jmaa.2010.07.006. |
[30] |
V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529. |
[31] |
M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192. |
[32] | |
[33] |
R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1997. |
[34] |
S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. |
show all references
References:
[1] |
H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. |
[2] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992. |
[3] |
F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discrete Cont. Dyn. Systems, 22 (2008), 557-586.
doi: 10.3934/dcds.2008.22.557. |
[4] |
V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273. |
[5] |
C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), p.154301.
doi: 10.1103/PhysRevLett.94.154301. |
[6] |
I. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," Acta, Kharkov, 2002. |
[7] |
I. Chueshov and I. Lasiecka, Attractors and long-time behavior of von Karman thermoelastic plates, Appl. Math. Optim., 58 (2008), 195-241.
doi: 10.1007/s00245-007-9031-8. |
[8] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. |
[9] |
M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720. |
[10] |
M. Coti Zelati, F. Dell'Oro and V. Pata, Energy decay of type III linear thermoelastic plates with memory, submitted. |
[11] |
M. Coti Zelati, V. Pata and R. Quintanilla, Regular global attractors of type III thermoelastic extensible beams, Chin. Ann. Math. Series B, 31 (2010), 619-630.
doi: 10.1007/s11401-010-0605-4. |
[12] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. |
[13] |
G. Fichera, Is the Fourier theory of heat propagation paradoxical?, Rend. Circ. Mat. Palermo, 41 (1992), 5-28. |
[14] |
S. Gatti, A. Miranville, V. Pata and S. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation, Rocky Mountain J. Math., 38 (2008), 1117-1138.
doi: 10.1216/RMJ-2008-38-4-1117. |
[15] |
S. Gatti, A. Miranville, V. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 329-366.
doi: 10.1017/S0308210509000365. |
[16] |
C. Giorgi, M. G. Naso, V. Pata and M. Potomkin, Global attractors for the extensible thermoelastic beam system, J. Diff. Eqs, 246 (2009), 3496-3517.
doi: 10.1016/j.jde.2009.02.020. |
[17] |
C. Giorgi, V. Pata and E. Vuk, On the extensible viscoelastic beam, Nonlinearity, 21 (2008), 713-733.
doi: 10.1088/0951-7715/21/4/004. |
[18] |
A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264.
doi: 10.1080/01495739208946136. |
[19] |
A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208.
doi: 10.1007/BF00044969. |
[20] |
A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. I. Classical continuum physics, Proc. Roy. Soc. London A, 448 (1995), 335-356.
doi: 10.1098/rspa.1995.0020. |
[21] |
A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. II. Generalized continua, Proc. Roy. Soc. London A, 448 (1995), 357-377.
doi: 10.1098/rspa.1995.0021. |
[22] |
A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. III. Mixtures of interacting continua, Proc. Roy. Soc. London A, 448 (1995), 379-388.
doi: 10.1098/rspa.1995.0022. |
[23] |
M. Grasselli and M. Squassina, Exponential stability and singular limit for a linear thermoelastic plate with memory effects, Adv. Math. Sci. Appl., 16 (2006), 15-31. |
[24] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, 1988. |
[25] |
A. Haraux, "Syst\`emes Dynamiques Dissipatifs et Applications," Coll. RMA no.17, Masson, Paris, 1991. |
[26] |
O. A. Ladyzhenskaya, Finding minimal global attractors for the Navier-Stokes equations and other partial differential equations, Russian Math. Surveys, 42 (1987), 27-73.
doi: 10.1070/RM1987v042n06ABEH001503. |
[27] |
V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., 64 (2006), 499-513. |
[28] |
V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.
doi: 10.1007/s00032-009-0098-3. |
[29] |
V. Pata, Uniform estimates of Gronwall type, J. Math. Anal. Appl., 373 (2011), 264-270.
doi: 10.1016/j.jmaa.2010.07.006. |
[30] |
V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529. |
[31] |
M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192. |
[32] | |
[33] |
R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1997. |
[34] |
S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. |
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