# Infinite-dimensional Sylvester equations: Basic theory and application to observer design

• Volume: 22, Issue: 2, page 245-257
• ISSN: 1641-876X

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## Abstract

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This paper develops a mathematical framework for the infinite-dimensional Sylvester equation both in the differential and the algebraic form. It uses the implemented semigroup concept as the main mathematical tool. This concept may be found in the literature on evolution equations occurring in mathematics and physics and is rather unknown in systems and control theories. But it is just systems and control theory where Sylvester equations widely appear, and for this reason we intend to give a mathematically rigorous introduction to the subject which is tailored to researchers and postgraduate students working on systems and control. This goal motivates the assumptions under which the results are developed. As an important example of applications we study the problem of designing an asymptotic state observer for a linear infinitedimensional control system with a bounded input operator and an unbounded output operator.

## How to cite

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Zbigniew Emirsajłow. "Infinite-dimensional Sylvester equations: Basic theory and application to observer design." International Journal of Applied Mathematics and Computer Science 22.2 (2012): 245-257. <http://eudml.org/doc/208105>.

@article{ZbigniewEmirsajłow2012,
abstract = {This paper develops a mathematical framework for the infinite-dimensional Sylvester equation both in the differential and the algebraic form. It uses the implemented semigroup concept as the main mathematical tool. This concept may be found in the literature on evolution equations occurring in mathematics and physics and is rather unknown in systems and control theories. But it is just systems and control theory where Sylvester equations widely appear, and for this reason we intend to give a mathematically rigorous introduction to the subject which is tailored to researchers and postgraduate students working on systems and control. This goal motivates the assumptions under which the results are developed. As an important example of applications we study the problem of designing an asymptotic state observer for a linear infinitedimensional control system with a bounded input operator and an unbounded output operator.},
author = {Zbigniew Emirsajłow},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {infinite-dimensional Sylvester equation; implemented semigroup; state observer design},
language = {eng},
number = {2},
pages = {245-257},
title = {Infinite-dimensional Sylvester equations: Basic theory and application to observer design},
url = {http://eudml.org/doc/208105},
volume = {22},
year = {2012},
}

TY - JOUR
AU - Zbigniew Emirsajłow
TI - Infinite-dimensional Sylvester equations: Basic theory and application to observer design
JO - International Journal of Applied Mathematics and Computer Science
PY - 2012
VL - 22
IS - 2
SP - 245
EP - 257
AB - This paper develops a mathematical framework for the infinite-dimensional Sylvester equation both in the differential and the algebraic form. It uses the implemented semigroup concept as the main mathematical tool. This concept may be found in the literature on evolution equations occurring in mathematics and physics and is rather unknown in systems and control theories. But it is just systems and control theory where Sylvester equations widely appear, and for this reason we intend to give a mathematically rigorous introduction to the subject which is tailored to researchers and postgraduate students working on systems and control. This goal motivates the assumptions under which the results are developed. As an important example of applications we study the problem of designing an asymptotic state observer for a linear infinitedimensional control system with a bounded input operator and an unbounded output operator.
LA - eng
KW - infinite-dimensional Sylvester equation; implemented semigroup; state observer design
UR - http://eudml.org/doc/208105
ER -

## References

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1. Alber, J. (2001). On implemented semigroups, Semigroup Forum 63(3): 371-386. Zbl1041.47028
2. Arendt, W., Raebiger, F. and Sourour, A. (1994). Spectral properties of the operator equation ax + xb = y, Quarterly Journal of Mathematics 45(2): 133-149. Zbl0826.47013
3. Chen, C.-T. (1984). Linear Systems Theory and Design, Holt, Rinehart and Winston, New York, NY.
4. Emirsajłow, Z. (2005). Implemented Semigroup in Analysis of Infinite-Dimensional Sylvester and Lyapunov equations, Technical University of Szczecin Press, Szczecin, (in Polish).
5. Emirsajłow, Z. and Townley, S. (2000). From PDEs with a boundary control to the abstract state equation with an unbounded input operator, European Journal of Control 7(1): 1-23.
6. Emirsajłow, Z. and Townley, S. (2005). On application of the implemented semigroup to a problem arising in optimal control, International Journal of Control 78(4): 298-310. Zbl1213.34074
7. Engel, J. and Nagel, R. (2000). One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, NY. Zbl0952.47036
8. Gajic, Z. and Qureshi, M. (2000). Lyapunov Matrix Equation in Systems Stability and Control, Academic Press, San Diego, CA.
9. Kuehnemund, F. (2001). Bi-continuous Semigroups on Spaces with Two Topologies: Theory and Applications, Ph.D. thesis, Eberhard-Karls-Univerität Tuebingen, Tuebingen.
10. Phong, V. Q. (1991). The operator equation ax − xb = c with unbounded operators a and b and related Cauchy problems, Mathematische Zeitschrift 208(7): 567-588. Zbl0726.47029

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