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

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.
Citation: Sergey Popov, Volker Reitmann. Frequency domain conditions for finite-dimensional projectors and determining observations for the set of amenable solutions. Discrete and 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.

[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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