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

International Journal of Applied Mathematics and Computer Science (2012)

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

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topZbigniew 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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- Emirsajłow, Z. (2005). Implemented Semigroup in Analysis of Infinite-Dimensional Sylvester and Lyapunov equations, Technical University of Szczecin Press, Szczecin, (in Polish).
- 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.
- 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
- Engel, J. and Nagel, R. (2000). One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, NY. Zbl0952.47036
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- Kuehnemund, F. (2001). Bi-continuous Semigroups on Spaces with Two Topologies: Theory and Applications, Ph.D. thesis, Eberhard-Karls-Univerität Tuebingen, Tuebingen.
- 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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