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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.

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[20]

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

[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.

[22]

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

[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.

[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).

[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.

[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.

[27]

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

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[20]

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

[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.

[22]

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

[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.

[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).

[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.

[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.

[27]

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

[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.

[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.

[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.

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