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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

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