Time-variant Darlington synthesis and induced realizations

Derk Pik

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 6, page 1331-1360
  • ISSN: 1641-876X

Abstract

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For a block lower triangular contraction T, necessary and sufficient conditions are given in order that there exist block lower triangular contractions T_{1,1}, T_{2,1} and T_{2,2} such that T_{1,1} T U_T = [ ] T_{2,1} T_{2,2} is unitary. For the case when T^*_{1,1} and T_{2,2} have dense ranges, all such embeddings are described. Each unitary embedding of UT induces a contractive realization of T , and various properties of this realization are characterized in terms of the unitary embedding.

How to cite

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Pik, Derk. "Time-variant Darlington synthesis and induced realizations." International Journal of Applied Mathematics and Computer Science 11.6 (2001): 1331-1360. <http://eudml.org/doc/207558>.

@article{Pik2001,
abstract = {For a block lower triangular contraction T, necessary and sufficient conditions are given in order that there exist block lower triangular contractions T\_\{1,1\}, T\_\{2,1\} and T\_\{2,2\} such that T\_\{1,1\} T U\_T = [ ] T\_\{2,1\} T\_\{2,2\} is unitary. For the case when T^*\_\{1,1\} and T\_\{2,2\} have dense ranges, all such embeddings are described. Each unitary embedding of UT induces a contractive realization of T , and various properties of this realization are characterized in terms of the unitary embedding.},
author = {Pik, Derk},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {time-variant linear systems; contractive linear systems; Darlington synthesis; unitary contraction; nonstationary case; lower triangular contractions},
language = {eng},
number = {6},
pages = {1331-1360},
title = {Time-variant Darlington synthesis and induced realizations},
url = {http://eudml.org/doc/207558},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Pik, Derk
TI - Time-variant Darlington synthesis and induced realizations
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 6
SP - 1331
EP - 1360
AB - For a block lower triangular contraction T, necessary and sufficient conditions are given in order that there exist block lower triangular contractions T_{1,1}, T_{2,1} and T_{2,2} such that T_{1,1} T U_T = [ ] T_{2,1} T_{2,2} is unitary. For the case when T^*_{1,1} and T_{2,2} have dense ranges, all such embeddings are described. Each unitary embedding of UT induces a contractive realization of T , and various properties of this realization are characterized in terms of the unitary embedding.
LA - eng
KW - time-variant linear systems; contractive linear systems; Darlington synthesis; unitary contraction; nonstationary case; lower triangular contractions
UR - http://eudml.org/doc/207558
ER -

References

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  1. Arov D.Z. (1971): Darlington’s method for dissipative systems. — Dokl. Akad. Nauk SSSR, Vol.201, No.3, pp.559–562; English translation: Soviet Physics Doklady, Vol.16, No.11, pp.954–956. Zbl0246.47035
  2. Arov D.Z. (1979): Stable dissipative linear stationary dynamical scattering systems. — J. Oper. Theory, Vol.2, No.1, pp.95–126 (in Russian). Zbl0461.47005
  3. Arov D.Z. (1985): Linear stationary passive systems with losses. — 2-nd doctoral dissertation, Phys. Math. Sci., Institute of Mathematics, Akad. Nauk Ukr. SSR, Kiev. 
  4. Arov D.Z., Kaashoek M.A. and Pik D.R. (1998): Optimal time-variant systems and factorization of operators, I: Minimal and optimal systems. — Int. Eqns. Oper. Theory, Vol.31, No.4, pp.389–420. Zbl0927.47006
  5. Arov D.Z., Kaashoek M.A. and Pik D.R. (2000): Optimal time variant systems and factorization of operators, II: Factorization. — J. Oper. Theory, Vol.43, No.2, pp.263–294. Zbl0992.47008
  6. Belevitch V. (1968): Classical Network Theory. — San Francisco: Holden-Day. Zbl0172.20404
  7. Constantinescu T. (1995): Schur Parameters, Factorizations and Dilation Problems. — Basel: Birkhäuser. 
  8. Dewilde P. (1971): Roomy scattering matrix synthesis. — Tech. Rep., Dept. of Mathematics, Univ. of California, Berkeley. 
  9. Dewilde P. (1999): Generalized Darlington Synthesis. — IEEE Trans. Circ. Syst., 1: Fund. Theory Appl., Vol.46, No.1, pp.41–58. Zbl0991.93040
  10. Dewilde P. and Van der Veen A. (1998): Time-Varying Systems and Computations. — Boston: Kluwer Academic Publishers. Zbl0937.93002
  11. Douglas R.G. and William Helton J. (1973): Inner dilations of analytic matrix functions and Darlington synthesis. — Acta Sci. Math. (Szeged), Vol.34, pp.61–67. 1360 Zbl0267.30032
  12. Douglas R.G., Shapiro A.L., and Shields A.L. (1970): Cyclic vectors and invariant subspaces for the backward shift. — Ann. Inst. Fourier (Grenoble), Vol.20, No.1, pp.37–76. Zbl0186.45302
  13. Foias C., Frazho A.E., Gohberg I. and Kaashoek, M.A. (1998); Metric Constrained Interpolation, Commutant Lifting and Systems. — Basel: Birkhäuser. 
  14. Gohberg I., Kaashoek M.A. and Lerer L. (1992): Minimality and realization of discrete time- variant systems, In: Time-variant Systems and Interpolation (I. Gohberg, Ed.). — Basel: Birkhäuser, pp.261–296. Zbl0747.93054
  15. Halanay A. and Ionescu V. (1994): Time-Varying Discrete Linear Systems, Input-Output Operators, Riccati Equations, Disturbance Attenuation. — Basel: Birkhäuser. Zbl0799.93035
  16. Kaashoek M.A. and Pik D.R. (1998): Factorization of lower triangular unitary operators with finite Kronecker index into elementary factors, In: Recent Progress in Operator Theory, International Workshop on Operator Theory and Applications, IWOTA 95, Regensburg, Germany, July 31–August 4, 1995 (I. Gohberg, R. Mennicken, C. Tretter, Eds.). — Basel: Birkhäuser, pp.183–217. Zbl0906.47010
  17. Pik D.R. (1999): Block lower triangular operators and optimal contractive systems. — Ph.D. Thesis, Vrije Universiteit Amsterdam, Amsterdam. 

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