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

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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

Abstract
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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 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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