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Uniform attractor of the non-autonomous discrete Selkov model
Frequency domain conditions for finite-dimensional projectors and determining observations for the set of amenable solutions
1. | Department of Mathematics and Mechanics, Saint Petersburg State University, Saint Petersburg, Russian Federation |
2. | Department of Mathematics and Mechanics, Saint-Petersburg State University, Saint-Petersburg, 198504 |
References:
[1] |
Ju. M. Berezans'kiĭ, "Expansions in Eigenfunctions of Selfadjoint Operators," Naukova Dumka, Kiev; English translation in Translations of Mathematical Monographs, 17, American Mathematical Society, Providence, R.I., 1968. |
[2] |
V. A. Boichenko, G. A. Leonov and V. Reitmann, "Dimension Theory for Ordinary Differential Equations," Teubner, Stuttgart-Leipzig, 2004.
doi: 10.1007/978-3-322-80055-8. |
[3] |
H. Brézis, Problemes unilateraux, J. Math. Pures. Appl., 51 (1972), 1-168. |
[4] |
V. A. Brusin, The Luré equations in Hilbert space and its solvability, (in Russian) Prikl. Math. Mekh., 40 (1976), 947-955. |
[5] |
R. Datko, Extending a theorem of A. M. Liapunov to Hilbert spaces, J. Math. Anal. Appl., 32 (1970), 610-616.
doi: 10.1016/0022-247X(70)90283-0. |
[6] |
G. Duvant and J.-L. Lions, "Inequalities in Mechanics and Physics," Springer-Verlag, Berlin, 1976. |
[7] |
I. N. Ermakov, Y. N. Kalinin and V. Reitmann, Determining modes and almost periodic integrals for cocycles, J. Differential Equations, 47 (2011), 1837-1852. |
[8] |
D. Henry, "Geometric Theory of Semilimear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer, New York, 1981. |
[9] |
F. Flandoli, J. Lasiecka and R. Triggiani, Algebraic Riccati equations with nonsmoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems, Annali di Matematica Pura Applicata, 153 (1988), 307-382.
doi: 10.1007/BF01762397. |
[10] |
C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations, J. Diff. Eq., 73 (1988), 309-353.
doi: 10.1016/0022-0396(88)90110-6. |
[11] |
D. Kalinichenko, V. Reitmann and S. Skopinov, Asymptotic behaviour of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion, in "Proc. $9^{th}$ AIMS Conference on Dynamical Systems, Differential Equations and Applications," Orlando, Florida, USA, 2012. |
[12] |
Y. N. Kalinin and V. Reitmann, Almost periodic solutions in control systems with monotone nonlinearities, Differential Equations and Control Processes, 4 (2012), 40-68. |
[13] |
Y. Kalinin, V. Reitmann and N. Yumaguzin, Asymptotic behavior of Maxwell's equation in one-space dimension with termal effect, Discrete and Cont. Dyn. Sys. Supplement 2011, 2 (2011), 754-762. |
[14] |
H. Kantz and V. Reitmann, "Reconstructing Attractors of Infinite-Dimensional Dynamical Systems from Low-Dimensional Projections," Workshop on Multivaluate Time Series Analysis, IWH Heidelberg, 2004. |
[15] |
O. A. Ladyzhenskaya, On estimates of the fractal dimension and the number of determining modes for invariant sets of dynamical systems, (in Russian) Zapiski Nauchnich Seminarov LOMI, 163 (1987), 105-129.
doi: 10.1007/BF02208714. |
[16] |
A. L. Likhtarnikov, Absolute stability criteria for nonlinear operator equations, (in Russian) Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), 1064-1083. |
[17] |
A. L. Likhtarnikov and V. A. Yakubovich, The frequency theorem for equations of evolutionary type, (in Russian) Siberian Math. J., 17 (1976), 790-803.
doi: 10.1007/BF00966379. |
[18] |
A. L. Likhtarnikov and V. A. Yakubovich, The frequency theorem for continuous one-parameter semigroups, (in Russian) Math. USSR-Izv., 11 (1977), 849-864. |
[19] |
A. L. Likhtarnikov and V. A. Yakubovich, Dichotomy and stability of uncertain nonlinear systems in Hilbert spaces, (in Russian) Algebra and Analysis, 9 (1997), 132-155. |
[20] |
J. Louis and D. Wexler, The Hilbert space regulator problem and operator Riccati equation under stabilizability, Annales de la Societe Scientifique de Bruxelles, 105 (1991), 137-165. |
[21] |
R. V. Manoranjan and H.-M. Yin, On two-phase Stefan problem arising from a microwave heating process, J. Continuous and Discrete Dynamical Systems, 15 (2006), 1155-1168.
doi: 10.3934/dcds.2006.15.1155. |
[22] |
A. Pankov, "Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations," Mathematics and its Applications (Soviet Series), 55, Kluwer Academic Publishers Group, Dordrecht, 1990.
doi: 10.1007/978-94-011-9682-6. |
[23] |
S. Popov, Taken's time delay embedding theorem for dynamical systems on infinite-dimensional manifolds, in "Proc. International Student Conference Science and Progress," St. Petersburg-Peterhof, (2011), 79. |
[24] |
J. C. Robinson, Inertial manifolds and the cone condition, Dyn. Syst. Appl., 2 (1993), 311-330. |
[25] |
J. C. Robinson, "Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[26] |
J. C. Robinson, Taken's embedding theorem for infinite-dimensional dynamical systems, J. Nonlinearity, 18 (2005), 2135-2143.
doi: 10.1088/0951-7715/18/5/013. |
[27] |
G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer, New York, 1990. |
[28] |
R. A. Smith, Orbital stability of ordinary differential equations, J. Differential Equations, 69 (1986), 265-287.
doi: 10.1016/0022-0396(87)90120-3. |
[29] |
R. A. Smith, Convergence theorems for periodic retarded functional differential equations, Proc. London Math. Soc., 60 (1990), 581-608.
doi: 10.1112/plms/s3-60.3.581. |
[30] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, Princeton, New Jersey, 1970. |
[31] |
V. A. Yakubovich, The frequency theorem in control theory, (in Russian) Sibirsk. Matem. Zh., 14 (1973), 384-419.
doi: 10.1007/BF00967952. |
show all references
References:
[1] |
Ju. M. Berezans'kiĭ, "Expansions in Eigenfunctions of Selfadjoint Operators," Naukova Dumka, Kiev; English translation in Translations of Mathematical Monographs, 17, American Mathematical Society, Providence, R.I., 1968. |
[2] |
V. A. Boichenko, G. A. Leonov and V. Reitmann, "Dimension Theory for Ordinary Differential Equations," Teubner, Stuttgart-Leipzig, 2004.
doi: 10.1007/978-3-322-80055-8. |
[3] |
H. Brézis, Problemes unilateraux, J. Math. Pures. Appl., 51 (1972), 1-168. |
[4] |
V. A. Brusin, The Luré equations in Hilbert space and its solvability, (in Russian) Prikl. Math. Mekh., 40 (1976), 947-955. |
[5] |
R. Datko, Extending a theorem of A. M. Liapunov to Hilbert spaces, J. Math. Anal. Appl., 32 (1970), 610-616.
doi: 10.1016/0022-247X(70)90283-0. |
[6] |
G. Duvant and J.-L. Lions, "Inequalities in Mechanics and Physics," Springer-Verlag, Berlin, 1976. |
[7] |
I. N. Ermakov, Y. N. Kalinin and V. Reitmann, Determining modes and almost periodic integrals for cocycles, J. Differential Equations, 47 (2011), 1837-1852. |
[8] |
D. Henry, "Geometric Theory of Semilimear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer, New York, 1981. |
[9] |
F. Flandoli, J. Lasiecka and R. Triggiani, Algebraic Riccati equations with nonsmoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems, Annali di Matematica Pura Applicata, 153 (1988), 307-382.
doi: 10.1007/BF01762397. |
[10] |
C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations, J. Diff. Eq., 73 (1988), 309-353.
doi: 10.1016/0022-0396(88)90110-6. |
[11] |
D. Kalinichenko, V. Reitmann and S. Skopinov, Asymptotic behaviour of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion, in "Proc. $9^{th}$ AIMS Conference on Dynamical Systems, Differential Equations and Applications," Orlando, Florida, USA, 2012. |
[12] |
Y. N. Kalinin and V. Reitmann, Almost periodic solutions in control systems with monotone nonlinearities, Differential Equations and Control Processes, 4 (2012), 40-68. |
[13] |
Y. Kalinin, V. Reitmann and N. Yumaguzin, Asymptotic behavior of Maxwell's equation in one-space dimension with termal effect, Discrete and Cont. Dyn. Sys. Supplement 2011, 2 (2011), 754-762. |
[14] |
H. Kantz and V. Reitmann, "Reconstructing Attractors of Infinite-Dimensional Dynamical Systems from Low-Dimensional Projections," Workshop on Multivaluate Time Series Analysis, IWH Heidelberg, 2004. |
[15] |
O. A. Ladyzhenskaya, On estimates of the fractal dimension and the number of determining modes for invariant sets of dynamical systems, (in Russian) Zapiski Nauchnich Seminarov LOMI, 163 (1987), 105-129.
doi: 10.1007/BF02208714. |
[16] |
A. L. Likhtarnikov, Absolute stability criteria for nonlinear operator equations, (in Russian) Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), 1064-1083. |
[17] |
A. L. Likhtarnikov and V. A. Yakubovich, The frequency theorem for equations of evolutionary type, (in Russian) Siberian Math. J., 17 (1976), 790-803.
doi: 10.1007/BF00966379. |
[18] |
A. L. Likhtarnikov and V. A. Yakubovich, The frequency theorem for continuous one-parameter semigroups, (in Russian) Math. USSR-Izv., 11 (1977), 849-864. |
[19] |
A. L. Likhtarnikov and V. A. Yakubovich, Dichotomy and stability of uncertain nonlinear systems in Hilbert spaces, (in Russian) Algebra and Analysis, 9 (1997), 132-155. |
[20] |
J. Louis and D. Wexler, The Hilbert space regulator problem and operator Riccati equation under stabilizability, Annales de la Societe Scientifique de Bruxelles, 105 (1991), 137-165. |
[21] |
R. V. Manoranjan and H.-M. Yin, On two-phase Stefan problem arising from a microwave heating process, J. Continuous and Discrete Dynamical Systems, 15 (2006), 1155-1168.
doi: 10.3934/dcds.2006.15.1155. |
[22] |
A. Pankov, "Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations," Mathematics and its Applications (Soviet Series), 55, Kluwer Academic Publishers Group, Dordrecht, 1990.
doi: 10.1007/978-94-011-9682-6. |
[23] |
S. Popov, Taken's time delay embedding theorem for dynamical systems on infinite-dimensional manifolds, in "Proc. International Student Conference Science and Progress," St. Petersburg-Peterhof, (2011), 79. |
[24] |
J. C. Robinson, Inertial manifolds and the cone condition, Dyn. Syst. Appl., 2 (1993), 311-330. |
[25] |
J. C. Robinson, "Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[26] |
J. C. Robinson, Taken's embedding theorem for infinite-dimensional dynamical systems, J. Nonlinearity, 18 (2005), 2135-2143.
doi: 10.1088/0951-7715/18/5/013. |
[27] |
G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer, New York, 1990. |
[28] |
R. A. Smith, Orbital stability of ordinary differential equations, J. Differential Equations, 69 (1986), 265-287.
doi: 10.1016/0022-0396(87)90120-3. |
[29] |
R. A. Smith, Convergence theorems for periodic retarded functional differential equations, Proc. London Math. Soc., 60 (1990), 581-608.
doi: 10.1112/plms/s3-60.3.581. |
[30] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, Princeton, New Jersey, 1970. |
[31] |
V. A. Yakubovich, The frequency theorem in control theory, (in Russian) Sibirsk. Matem. Zh., 14 (1973), 384-419.
doi: 10.1007/BF00967952. |
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