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March  2017, 6(1): 15-34. doi: 10.3934/eect.2017002

Lumpability of linear evolution Equations in Banach spaces

1. 

Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey

2. 

Max Planck Institute for Mathematics in the Sciences, Inselstra\ss e 22,04103 Leipzig, Germany

Received  March 2016 Revised  September 2016 Published  December 2016

We analyze the lumpability of linear systems on Banach spaces, namely, the possibility of projecting the dynamics by a linear reduction operator onto a smaller state space in which a self-contained dynamical description exists. We obtain conditions for lumpability of dynamics defined by unbounded operators using the theory of strongly continuous semigroups. We also derive results from the dual space point of view using sun dual theory. Furthermore, we connect the theory of lumping to several results from operator factorization. We indicate several applications to particular systems, including delay differential equations.

Citation: Fatihcan M. Atay, Lavinia Roncoroni. Lumpability of linear evolution Equations in Banach spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 15-34. doi: 10.3934/eect.2017002
References:
[1]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators vol. 1184 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922. Google Scholar

[2]

L. Arlotti, A new characterization of B-bounded semigroups with application to implicit evolution equations, Abstract and Applied Analysis, 5 (2000), 227-243. doi: 10.1155/S1085337501000331. Google Scholar

[3]

P. AugerR.B. de la ParraJ.C. PoggialeE. Sánchez and T. Nguyen-Huu, Aggregation of variables and applications to population dynamics, in Structured Population Models in Biology and Epidemiology (eds. P. Magal and S. Ruan), Springer, Berlin, Heidelberg, 1936 (2008), 209-263. doi: 10.1007/978-3-540-78273-5_5. Google Scholar

[4]

J. Banasiak, Generation results for B-bounded semigroups, Annali di Matematica Pura ed Applicata, 175 (1998), 307-326. doi: 10.1007/BF01783690. Google Scholar

[5]

B.A. Barnes, Majorization, range inclusion, and factorization for bounded linear operators, Proc. Amer. Math. Soc., 133 (2005), 155-162 (electronic). doi: 10.1090/S0002-9939-04-07495-7. Google Scholar

[6]

A. Belleni-Morante, B-bounded semigroups and applications, Annali di Matematica Pura ed Applicata, 170 (1996), 359-376. doi: 10.1007/BF01758995. Google Scholar

[7]

A. Belleni-Morante and S. Totaro, The successive reflection method in three dimensional particle transport, Journal of Mathematical Physics, 37 (1996), 2815-2823. doi: 10.1063/1.531541. Google Scholar

[8]

L. BlockJ. Keesling and D. Ledis, Semi-conjugacies and inverse limit spaces, Journal of Difference Equations and Applications, 18 (2012), 627-645. doi: 10.1080/10236198.2011.611803. Google Scholar

[9]

E.M. Bollt and J.D. Skufca, On comparing dynamical systems by defective conjugacy: A symbolic dynamics interpretation of commuter functions, Physica D: Nonlinear Phenomena, 239 (2010), 579-590. doi: 10.1016/j.physd.2009.12.007. Google Scholar

[10]

P. Buchholz, Exact and ordinary lumpability in finite Markov chains, J. Appl. Probab., 31 (1994), 59-75. doi: 10.1017/S0021900200107338. Google Scholar

[11]

P. Coxson, Lumpability and observability of linear systems, Journal of Mathematical Analysis and Applications, 99 (1984), 435-446. doi: 10.1016/0022-247X(84)90224-5. Google Scholar

[12]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory vol. 21 of Texts in Applied Mathematics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6. Google Scholar

[13]

R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17 (1966), 413-415. doi: 10.1090/S0002-9939-1966-0203464-1. Google Scholar

[14]

M.R. Embry, Factorization of operators on Banach space, Proc. Amer. Math. Soc., 38 (1973), 587-590. doi: 10.1090/S0002-9939-1973-0312287-8. Google Scholar

[15]

K. -J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. Google Scholar

[16]

H.O. Fattorini, Some remarks on complete controllability, SIAM J. Control, 4 (1966), 686-694. doi: 10.1137/0304048. Google Scholar

[17]

M. Forough, Majorization, range inclusion, and factorization for unbounded operators on Banach spaces, Linear Algebra Appl., 449 (2014), 60-67. doi: 10.1016/j.laa.2014.02.033. Google Scholar

[18]

S. Goldberg, Unbounded Linear Operators Dover Publications, Inc. , New York, 1985, Theory and applications, Reprint of the 1966 edition. Google Scholar

[19]

L. Gurvits and J. Ledoux, Markov property for a function of a Markov chain: A linear algebra approach, Linear Algebra and its Applications, 404 (2005), 85-117. doi: 10.1016/j.laa.2005.02.007. Google Scholar

[20]

E. Hille, Functional Analysis and Semi-Groups American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, New York, 1948. Google Scholar

[21]

J. Ledoux, On weak lumpability of denumerable Markov chains, Statist. Probab. Lett., 25 (1995), 329-339. doi: 10.1016/0167-7152(94)00238-5. Google Scholar

[22]

G. Li and H. Rabitz, A general analysis of exact lumping in chemical kinetics, Chemical Engineering Science, 44 (1989), 1413-1430. Google Scholar

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[24]

Z. Rózsa and J. Tóth, Exact linear lumping in abstract spaces, in Proceedings of the 7th Colloquium on the Qualitative Theory of Differential Equations vol. 7, Electron. J. Qual. Theory Differ. Equ. , Szeged, 2004, No. 21, 20 pp. (electronic). Google Scholar

[25]

A. TomlinG. LiH. Rabitz and J. Tóth, The effect of lumping and expanding on kinetic differential equations, SIAM J. Appl. Math., 57 (1997), 1531-1556. doi: 10.1137/S0036139995293294. Google Scholar

[26]

R. Triggiani, Extensions of rank conditions for controllability and observability to banach spaces and unbounded operators, SIAM J. Control, 14 (1976), 313-338. doi: 10.1137/0314022. Google Scholar

[27]

R. Triggiani, Controllability and observability in Banach space with bounded operators, SIAM J. Control, 13 (1975), 462-491. doi: 10.1137/0313028. Google Scholar

[28]

J. van Neerven, The Adjoint of a Semigroup of Linear Operators vol. 1529 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0085008. Google Scholar

[29]

J. Wei and J. Kuo, Lumping analysis in monomolecular reaction systems. Analysis of the exactly lumpable system, Industrial & Engineering Chemistry Fundamentals, 8 (1969), 114-123. doi: 10.1021/i160029a019. Google Scholar

[30]

H.J. Zwart, Geometric theory for infinite dimensional systems, Geometric Theory for Infinite Dimensional Systems: Lecture Notes in Control and Information Sciences, 115 (1989), 1-7. doi: 10.1007/BFb0044353. Google Scholar

show all references

References:
[1]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators vol. 1184 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922. Google Scholar

[2]

L. Arlotti, A new characterization of B-bounded semigroups with application to implicit evolution equations, Abstract and Applied Analysis, 5 (2000), 227-243. doi: 10.1155/S1085337501000331. Google Scholar

[3]

P. AugerR.B. de la ParraJ.C. PoggialeE. Sánchez and T. Nguyen-Huu, Aggregation of variables and applications to population dynamics, in Structured Population Models in Biology and Epidemiology (eds. P. Magal and S. Ruan), Springer, Berlin, Heidelberg, 1936 (2008), 209-263. doi: 10.1007/978-3-540-78273-5_5. Google Scholar

[4]

J. Banasiak, Generation results for B-bounded semigroups, Annali di Matematica Pura ed Applicata, 175 (1998), 307-326. doi: 10.1007/BF01783690. Google Scholar

[5]

B.A. Barnes, Majorization, range inclusion, and factorization for bounded linear operators, Proc. Amer. Math. Soc., 133 (2005), 155-162 (electronic). doi: 10.1090/S0002-9939-04-07495-7. Google Scholar

[6]

A. Belleni-Morante, B-bounded semigroups and applications, Annali di Matematica Pura ed Applicata, 170 (1996), 359-376. doi: 10.1007/BF01758995. Google Scholar

[7]

A. Belleni-Morante and S. Totaro, The successive reflection method in three dimensional particle transport, Journal of Mathematical Physics, 37 (1996), 2815-2823. doi: 10.1063/1.531541. Google Scholar

[8]

L. BlockJ. Keesling and D. Ledis, Semi-conjugacies and inverse limit spaces, Journal of Difference Equations and Applications, 18 (2012), 627-645. doi: 10.1080/10236198.2011.611803. Google Scholar

[9]

E.M. Bollt and J.D. Skufca, On comparing dynamical systems by defective conjugacy: A symbolic dynamics interpretation of commuter functions, Physica D: Nonlinear Phenomena, 239 (2010), 579-590. doi: 10.1016/j.physd.2009.12.007. Google Scholar

[10]

P. Buchholz, Exact and ordinary lumpability in finite Markov chains, J. Appl. Probab., 31 (1994), 59-75. doi: 10.1017/S0021900200107338. Google Scholar

[11]

P. Coxson, Lumpability and observability of linear systems, Journal of Mathematical Analysis and Applications, 99 (1984), 435-446. doi: 10.1016/0022-247X(84)90224-5. Google Scholar

[12]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory vol. 21 of Texts in Applied Mathematics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6. Google Scholar

[13]

R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17 (1966), 413-415. doi: 10.1090/S0002-9939-1966-0203464-1. Google Scholar

[14]

M.R. Embry, Factorization of operators on Banach space, Proc. Amer. Math. Soc., 38 (1973), 587-590. doi: 10.1090/S0002-9939-1973-0312287-8. Google Scholar

[15]

K. -J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. Google Scholar

[16]

H.O. Fattorini, Some remarks on complete controllability, SIAM J. Control, 4 (1966), 686-694. doi: 10.1137/0304048. Google Scholar

[17]

M. Forough, Majorization, range inclusion, and factorization for unbounded operators on Banach spaces, Linear Algebra Appl., 449 (2014), 60-67. doi: 10.1016/j.laa.2014.02.033. Google Scholar

[18]

S. Goldberg, Unbounded Linear Operators Dover Publications, Inc. , New York, 1985, Theory and applications, Reprint of the 1966 edition. Google Scholar

[19]

L. Gurvits and J. Ledoux, Markov property for a function of a Markov chain: A linear algebra approach, Linear Algebra and its Applications, 404 (2005), 85-117. doi: 10.1016/j.laa.2005.02.007. Google Scholar

[20]

E. Hille, Functional Analysis and Semi-Groups American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, New York, 1948. Google Scholar

[21]

J. Ledoux, On weak lumpability of denumerable Markov chains, Statist. Probab. Lett., 25 (1995), 329-339. doi: 10.1016/0167-7152(94)00238-5. Google Scholar

[22]

G. Li and H. Rabitz, A general analysis of exact lumping in chemical kinetics, Chemical Engineering Science, 44 (1989), 1413-1430. Google Scholar

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[24]

Z. Rózsa and J. Tóth, Exact linear lumping in abstract spaces, in Proceedings of the 7th Colloquium on the Qualitative Theory of Differential Equations vol. 7, Electron. J. Qual. Theory Differ. Equ. , Szeged, 2004, No. 21, 20 pp. (electronic). Google Scholar

[25]

A. TomlinG. LiH. Rabitz and J. Tóth, The effect of lumping and expanding on kinetic differential equations, SIAM J. Appl. Math., 57 (1997), 1531-1556. doi: 10.1137/S0036139995293294. Google Scholar

[26]

R. Triggiani, Extensions of rank conditions for controllability and observability to banach spaces and unbounded operators, SIAM J. Control, 14 (1976), 313-338. doi: 10.1137/0314022. Google Scholar

[27]

R. Triggiani, Controllability and observability in Banach space with bounded operators, SIAM J. Control, 13 (1975), 462-491. doi: 10.1137/0313028. Google Scholar

[28]

J. van Neerven, The Adjoint of a Semigroup of Linear Operators vol. 1529 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0085008. Google Scholar

[29]

J. Wei and J. Kuo, Lumping analysis in monomolecular reaction systems. Analysis of the exactly lumpable system, Industrial & Engineering Chemistry Fundamentals, 8 (1969), 114-123. doi: 10.1021/i160029a019. Google Scholar

[30]

H.J. Zwart, Geometric theory for infinite dimensional systems, Geometric Theory for Infinite Dimensional Systems: Lecture Notes in Control and Information Sciences, 115 (1989), 1-7. doi: 10.1007/BFb0044353. Google Scholar

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