## Stability of infinite-dimensional sampled-data systems

HTML articles powered by AMS MathViewer

- by Hartmut Logemann, Richard Rebarber and Stuart Townley PDF
- Trans. Amer. Math. Soc.
**355**(2003), 3301-3328 Request permission

## Abstract:

Suppose that a static-state feedback stabilizes a continuous-time linear infinite-dimensional control system. We consider the following question: if we construct a sampled-data controller by applying an idealized sample-and-hold process to a continuous-time stabilizing feedback, will this sampled-data controller stabilize the system for all sufficiently small sampling times? Here the state space $X$ and the control space $U$ are Hilbert spaces, the system is of the form $\dot x(t) = Ax(t) + Bu(t)$, where $A$ is the generator of a strongly continuous semigroup on $X$, and the continuous time feedback is $u(t) = Fx(t)$. The answer to the above question is known to be “yes” if $X$ and $U$ are finite-dimensional spaces. In the infinite-dimensional case, if $F$ is not compact, then it is easy to find counterexamples. Therefore, we restrict attention to compact feedback. We show that the answer to the above question is “yes”, if $B$ is a bounded operator from $U$ into $X$. Moreover, if $B$ is unbounded, we show that the answer “yes” remains correct, provided that the semigroup generated by $A$ is analytic. We use the theory developed for static-state feedback to obtain analogous results for dynamic-output feedback control.## References

- Tongwen Chen and Bruce A. Francis,
*Input-output stability of sampled-data systems*, IEEE Trans. Automat. Control**36**(1991), no. 1, 50–58. MR**1084245**, DOI 10.1109/9.62267 - Tongwen Chen and Bruce Francis,
*Optimal sampled-data control systems*, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1996. Reprint of the 1995 original. MR**1410060** - Ruth F. Curtain and Hans Zwart,
*An introduction to infinite-dimensional linear systems theory*, Texts in Applied Mathematics, vol. 21, Springer-Verlag, New York, 1995. MR**1351248**, DOI 10.1007/978-1-4612-4224-6 - Nelson Dunford and Jacob T. Schwartz,
*Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space*, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR**0188745** - Klaus-Jochen Engel and Rainer Nagel,
*One-parameter semigroups for linear evolution equations*, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR**1721989** - G. F. Franklin, J. D. Powell and M. Workman,
*Digital Control of Dynamic Systems*, 3rd edition, Addison Wesley, Menlo Park, 1998. - R. E. Kalman, Y. C. Ho, and K. S. Narendra,
*Controllability of linear dynamical systems*, Contributions to Differential Equations**1**(1963), 189–213. MR**155070** - Tosio Kato,
*Perturbation theory for linear operators*, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR**1335452**, DOI 10.1007/978-3-642-66282-9 - Hartmut Logemann,
*Stability and stabilizability of linear infinite-dimensional discrete-time systems*, IMA J. Math. Control Inform.**9**(1992), no. 3, 255–263. MR**1203451**, DOI 10.1093/imamci/9.3.255 - I. Lasiecka and R. Triggiani,
*Stabilization and structural assignment of Dirichlet boundary feedback parabolic equations*, SIAM J. Control Optim.**21**(1983), no. 5, 766–803. MR**711000**, DOI 10.1137/0321047 - I. Lasiecka and R. Triggiani,
*The regulator problem for parabolic equations with Dirichlet boundary control. I. Riccati’s feedback synthesis and regularity of optimal solution*, Appl. Math. Optim.**16**(1987), no. 2, 147–168. MR**894809**, DOI 10.1007/BF01442189 - I. Lasiecka and R. Triggiani,
*The regulator problem for parabolic equations with Dirichlet boundary control. I. Riccati’s feedback synthesis and regularity of optimal solution*, Appl. Math. Optim.**16**(1987), no. 2, 147–168. MR**894809**, DOI 10.1007/BF01442189 - Arnaud Denjoy,
*Sur certaines séries de Taylor admettant leur cercle de convergence comme coupure essentielle*, C. R. Acad. Sci. Paris**209**(1939), 373–374 (French). MR**50** - A. Pazy,
*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR**710486**, DOI 10.1007/978-1-4612-5561-1 - R. Rebarber and S. Townley,
*Stabilization of distributed parameter systems by piecewise polynomial control*, IEEE Trans. Automat. Control**42**(1997), no. 9, 1254–1257. MR**1470563**, DOI 10.1109/9.623087 - Richard Rebarber and Stuart Townley,
*Generalized sampled data feedback control of distributed parameter systems*, Systems Control Lett.**34**(1998), no. 5, 229–240. MR**1639017**, DOI 10.1016/S0167-6911(98)00011-5 - R. Rebarber and S. Townley, Non-robustness of closed-loop stability for infinite-dimensional systems under sample and hold,
*IEEE Trans. Automat. Control***47**(2002), pp. 1381-1385. - I. G. Rosen and C. Wang,
*On stabilizability and sampling for infinite-dimensional systems*, IEEE Trans. Automat. Control**37**(1992), no. 10, 1653–1656. MR**1188781**, DOI 10.1109/9.256405 - Eduardo D. Sontag,
*Mathematical control theory*, 2nd ed., Texts in Applied Mathematics, vol. 6, Springer-Verlag, New York, 1998. Deterministic finite-dimensional systems. MR**1640001**, DOI 10.1007/978-1-4612-0577-7 - Tzyh Jong Tarn, John R. Zavgren Jr., and Xiaoming Zeng,
*Stabilization of infinite-dimensional systems with periodic feedback gains and sampled output*, Automatica J. IFAC**24**(1988), no. 1, 95–99. MR**927837**, DOI 10.1016/0005-1098(88)90012-X - George Weiss,
*Admissibility of unbounded control operators*, SIAM J. Control Optim.**27**(1989), no. 3, 527–545. MR**993285**, DOI 10.1137/0327028 - Yutaka Yamamoto,
*A function space approach to sampled data control systems and tracking problems*, IEEE Trans. Automat. Control**39**(1994), no. 4, 703–713. MR**1276768**, DOI 10.1109/9.286247 - Leo F. Epstein,
*A function related to the series for $e^{e^x}$*, J. Math. Phys. Mass. Inst. Tech.**18**(1939), 153–173. MR**58**, DOI 10.1002/sapm1939181153

## Additional Information

**Hartmut Logemann**- Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
- Email: hl@maths.bath.ac.uk
**Richard Rebarber**- Affiliation: Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0323
- Email: rrebarbe@math.unl.edu
**Stuart Townley**- Affiliation: School of Mathematical Sciences, University of Exeter, Exeter, EX4 4QE, United Kingdom
- Email: townley@maths.ex.ac.uk
- Received by editor(s): December 21, 2000
- Received by editor(s) in revised form: February 21, 2002
- Published electronically: April 25, 2003
- Additional Notes: This work was supported by NATO (Grant CRG 950179) and by the National Science Foundation (Grant DMS-9623392).
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**355**(2003), 3301-3328 - MSC (2000): Primary 34G10, 47A55, 47D06, 93C25, 93C57, 93D15
- DOI: https://doi.org/10.1090/S0002-9947-03-03142-8
- MathSciNet review: 1974689