American Institute of Mathematical Sciences

Advanced
January  2014, 34(1): 249-267. doi: 10.3934/dcds.2014.34.249

Frequency domain conditions for finite-dimensional projectors and determining observations for the set of amenable solutions

Sergey Popov 1, and  Volker Reitmann 2,

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

Received  October 2012 Revised  April 2013 Published  June 2013

Frequency domain conditions for the existence of finite-dimensional projectors and determining observations for the set of amenable solutions of semi-dynamical systems in Hilbert spaces are derived. Evolutionary variational equations are considered as control systems in a rigged Hilbert space structure. As an example we investigate a coupled system of Maxwell's equations and the heat equation in one-space dimension. We show the controllability of the linear part and the frequency domain conditions for this example.
Keywords: determining observations, Finite-dimensional projections, frequency domain methods..
Mathematics Subject Classification: Primary: 93C80, 93C20; Secondary: 35K20, 35L2.
Citation: Sergey Popov, Volker Reitmann. Frequency domain conditions for finite-dimensional projectors and determining observations for the set of amenable solutions. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 249-267. doi: 10.3934/dcds.2014.34.249
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.  Google Scholar

[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.  Google Scholar

[3]

H. Brézis, Problemes unilateraux, J. Math. Pures. Appl., 51 (1972), 1-168.  Google Scholar

[4]

V. A. Brusin, The Luré equations in Hilbert space and its solvability, (in Russian) Prikl. Math. Mekh., 40 (1976), 947-955.  Google Scholar

[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.  Google Scholar

[6]

G. Duvant and J.-L. Lions, "Inequalities in Mechanics and Physics," Springer-Verlag, Berlin, 1976.  Google Scholar

[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.  Google Scholar

[8]

D. Henry, "Geometric Theory of Semilimear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer, New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[16]

A. L. Likhtarnikov, Absolute stability criteria for nonlinear operator equations, (in Russian) Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), 1064-1083.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[24]

J. C. Robinson, Inertial manifolds and the cone condition, Dyn. Syst. Appl., 2 (1993), 311-330.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer, New York, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, Princeton, New Jersey, 1970.  Google Scholar

[31]

V. A. Yakubovich, The frequency theorem in control theory, (in Russian) Sibirsk. Matem. Zh., 14 (1973), 384-419. doi: 10.1007/BF00967952.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

H. Brézis, Problemes unilateraux, J. Math. Pures. Appl., 51 (1972), 1-168.  Google Scholar

[4]

V. A. Brusin, The Luré equations in Hilbert space and its solvability, (in Russian) Prikl. Math. Mekh., 40 (1976), 947-955.  Google Scholar

[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.  Google Scholar

[6]

G. Duvant and J.-L. Lions, "Inequalities in Mechanics and Physics," Springer-Verlag, Berlin, 1976.  Google Scholar

[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.  Google Scholar

[8]

D. Henry, "Geometric Theory of Semilimear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer, New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[16]

A. L. Likhtarnikov, Absolute stability criteria for nonlinear operator equations, (in Russian) Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), 1064-1083.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[24]

J. C. Robinson, Inertial manifolds and the cone condition, Dyn. Syst. Appl., 2 (1993), 311-330.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer, New York, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, Princeton, New Jersey, 1970.  Google Scholar

[31]

V. A. Yakubovich, The frequency theorem in control theory, (in Russian) Sibirsk. Matem. Zh., 14 (1973), 384-419. doi: 10.1007/BF00967952.  Google Scholar

[1]

Xing Wang, Chang-Qi Tao, Guo-Ji Tang. Differential optimization in finite-dimensional spaces. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1495-1505. doi: 10.3934/jimo.2016.12.1495
[2]

Paolo Maria Mariano. Line defect evolution in finite-dimensional manifolds. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 575-596. doi: 10.3934/dcdsb.2012.17.575
[3]

A. Jiménez-Casas, Mario Castro, Justine Yassapan. Finite-dimensional behavior in a thermosyphon with a viscoelastic fluid. Conference Publications, 2013, 2013 (special) : 375-384. doi: 10.3934/proc.2013.2013.375
[4]

Barbara Panicucci, Massimo Pappalardo, Mauro Passacantando. On finite-dimensional generalized variational inequalities. Journal of Industrial & Management Optimization, 2006, 2 (1) : 43-53. doi: 10.3934/jimo.2006.2.43
[5]

Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029
[6]

Michael L. Frankel, Victor Roytburd. A Finite-dimensional attractor for a nonequilibrium Stefan problem with heat losses. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 35-62. doi: 10.3934/dcds.2005.13.35
[7]

Luigi C. Berselli, Franco Flandoli. Remarks on determining projections for stochastic dissipative equations. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 197-214. doi: 10.3934/dcds.1999.5.197
[8]

Igor Chueshov, Alexander V. Rezounenko. Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1685-1704. doi: 10.3934/cpaa.2015.14.1685
[9]

Laurence Cherfils, Hussein Fakih, Alain Miranville. Finite-dimensional attractors for the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard equation in image inpainting. Inverse Problems & Imaging, 2015, 9 (1) : 105-125. doi: 10.3934/ipi.2015.9.105
[10]

Brahim Alouini. Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1781-1801. doi: 10.3934/cpaa.2015.14.1781
[11]

Haijuan Hu, Jacques Froment, Baoyan Wang, Xiequan Fan. Spatial-Frequency domain nonlocal total variation for image denoising. Inverse Problems & Imaging, 2020, 14 (6) : 1157-1184. doi: 10.3934/ipi.2020059
[12]

Jamal Mrazgua, El Houssaine Tissir, Mohamed Ouahi. Frequency domain $ H_{\infty} $ control design for active suspension systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021036
[13]

Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden. Minimum free energy in the frequency domain for a heat conductor with memory. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 793-816. doi: 10.3934/dcdsb.2010.14.793
[14]

Bopeng Rao, Xu Zhang. Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2789-2809. doi: 10.3934/cpaa.2021119
[15]

Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems & Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163
[16]

Hongyu Cheng, Shimin Wang. Response solutions to harmonic oscillators beyond multi–dimensional Brjuno frequency. Communications on Pure & Applied Analysis, 2021, 20 (2) : 467-494. doi: 10.3934/cpaa.2020222
[17]

Jinye Shen, Xian-Ming Gu. Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021086
[18]

George Avalos, Roberto Triggiani. Rational decay rates for a PDE heat--structure interaction: A frequency domain approach. Evolution Equations & Control Theory, 2013, 2 (2) : 233-253. doi: 10.3934/eect.2013.2.233
[19]

Abderrahim Azouani, Edriss S. Titi. Feedback control of nonlinear dissipative systems by finite determining parameters - A reaction-diffusion paradigm. Evolution Equations & Control Theory, 2014, 3 (4) : 579-594. doi: 10.3934/eect.2014.3.579
[20]

Evelyn Lunasin, Edriss S. Titi. Finite determining parameters feedback control for distributed nonlinear dissipative systems -a computational study. Evolution Equations & Control Theory, 2017, 6 (4) : 535-557. doi: 10.3934/eect.2017027

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy