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 - A, 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, 17 (1968).   Google Scholar

[2]

V. A. Boichenko, G. A. Leonov and V. Reitmann, "Dimension Theory for Ordinary Differential Equations,", Teubner, (2004).  doi: 10.1007/978-3-322-80055-8.  Google Scholar

[3]

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

[4]

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

[5]

R. Datko, Extending a theorem of A. M. Liapunov to Hilbert spaces,, J. Math. Anal. Appl., 32 (1970), 610.  doi: 10.1016/0022-247X(70)90283-0.  Google Scholar

[6]

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

[8]

D. Henry, "Geometric Theory of Semilimear Parabolic Equations,", Lecture Notes in Mathematics, 840 (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.  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.  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, (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.   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.   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, (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.  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.   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.  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.   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.   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.   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.  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 (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, (2011).   Google Scholar

[24]

J. C. Robinson, Inertial manifolds and the cone condition,, Dyn. Syst. Appl., 2 (1993), 311.   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, (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.  doi: 10.1088/0951-7715/18/5/013.  Google Scholar

[27]

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

[28]

R. A. Smith, Orbital stability of ordinary differential equations,, J. Differential Equations, 69 (1986), 265.  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.  doi: 10.1112/plms/s3-60.3.581.  Google Scholar

[30]

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

[31]

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

show all references

References:
[1]

Ju. M. Berezans'kiĭ, "Expansions in Eigenfunctions of Selfadjoint Operators,", Naukova Dumka, 17 (1968).   Google Scholar

[2]

V. A. Boichenko, G. A. Leonov and V. Reitmann, "Dimension Theory for Ordinary Differential Equations,", Teubner, (2004).  doi: 10.1007/978-3-322-80055-8.  Google Scholar

[3]

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

[4]

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

[5]

R. Datko, Extending a theorem of A. M. Liapunov to Hilbert spaces,, J. Math. Anal. Appl., 32 (1970), 610.  doi: 10.1016/0022-247X(70)90283-0.  Google Scholar

[6]

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

[8]

D. Henry, "Geometric Theory of Semilimear Parabolic Equations,", Lecture Notes in Mathematics, 840 (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.  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.  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, (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.   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.   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, (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.  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.   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.  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.   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.   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.   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.  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 (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, (2011).   Google Scholar

[24]

J. C. Robinson, Inertial manifolds and the cone condition,, Dyn. Syst. Appl., 2 (1993), 311.   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, (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.  doi: 10.1088/0951-7715/18/5/013.  Google Scholar

[27]

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

[28]

R. A. Smith, Orbital stability of ordinary differential equations,, J. Differential Equations, 69 (1986), 265.  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.  doi: 10.1112/plms/s3-60.3.581.  Google Scholar

[30]

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

[31]

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

[1]

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
[2]

Hongyu Cheng, Shimin Wang. Response solutions to harmonic oscillators beyond multi–dimensional brjuno frequency. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020222
[3]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105
[4]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073
[5]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079
[6]

Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096
[7]

Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126
[8]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[9]

Anton A. Kutsenko. Isomorphism between one-dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270
[10]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351
[11]

Aisling McGlinchey, Oliver Mason. Observations on the bias of nonnegative mechanisms for differential privacy. Foundations of Data Science, 2020, 2 (4) : 429-442. doi: 10.3934/fods.2020020
[12]

Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations. Electronic Research Archive, 2021, 29 (1) : 1819-1839. doi: 10.3934/era.2020093
[13]

Divine Wanduku. Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021005
[14]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240
[15]

Philippe G. Ciarlet, Liliana Gratie, Cristinel Mardare. Intrinsic methods in elasticity: a mathematical survey. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 133-164. doi: 10.3934/dcds.2009.23.133
[16]

Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020280
[17]

Xiaofeng Ren, David Shoup. The impact of the domain boundary on an inhibitory system: Interior discs and boundary half discs. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3957-3979. doi: 10.3934/dcds.2020048
[18]

Wenrui Hao, King-Yeung Lam, Yuan Lou. Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 367-400. doi: 10.3934/dcdsb.2020283
[19]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115
[20]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences