## Transfer functions of regular linear systems Part II: The system operator and the Lax–Phillips semigroup

HTML articles powered by AMS MathViewer

- by Olof Staffans and George Weiss
- Trans. Amer. Math. Soc.
**354**(2002), 3229-3262 - DOI: https://doi.org/10.1090/S0002-9947-02-02976-8
- Published electronically: April 3, 2002
- PDF | Request permission

## Abstract:

This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as “Part I”. We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by $\dot x=Ax+Bu$, $y=Cx+Du$ would be the $s$-dependent matrix $S_\Sigma (s)= \left [ {}^{A-sI}_{\ \; C} {\ } ^{B}_{D} \right ]$. In the general case, $S_\Sigma (s)$ is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks $A-sI$ and $B$, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of $S_\Sigma (s)$ where the right lower block is the feedthrough operator of the system. Using $S_\Sigma (0)$, we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the “initial time” is $-\infty$. We also introduce the Lax-Phillips semigroup $\boldsymbol {\mathfrak {T}}$ induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index ${\omega }\in {\mathbb R}$ which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of $A$ and also the points where $S_\Sigma (s)$ is not invertible, in terms of the spectrum of the generator of $\boldsymbol {\mathfrak {T}}$ (for various values of ${\omega }$). The system $\Sigma$ is dissipative if and only if $\boldsymbol {\mathfrak {T}}$ (with index zero) is a contraction semigroup.## References

- D. Z. Arov and M. A. Nudelman,
*Passive linear stationary dynamical scattering systems with continuous time*, Integral Equations Operator Theory**24**(1996), no. 1, 1–45. MR**1366539**, DOI 10.1007/BF01195483 - George Avalos, Irena Lasiecka, and Richard Rebarber,
*Lack of time-delay robustness for stabilization of a structural acoustics model*, SIAM J. Control Optim.**37**(1999), no. 5, 1394–1418. MR**1710226**, DOI 10.1137/S0363012997331135 - C. J. Everett Jr.,
*Annihilator ideals and representation iteration for abstract rings*, Duke Math. J.**5**(1939), 623–627. MR**13** - Klaus-J. Engel,
*On the characterization of admissible control- and observation operators*, Systems Control Lett.**34**(1998), no. 4, 225–227. MR**1637265**, DOI 10.1016/S0167-6911(98)00019-X - Piotr Grabowski and Frank M. Callier,
*Admissible observation operators. Semigroup criteria of admissibility*, Integral Equations Operator Theory**25**(1996), no. 2, 182–198. MR**1388679**, DOI 10.1007/BF01308629 - J. William Helton,
*Systems with infinite-dimensional state space: the Hilbert space approach*, Proc. IEEE**64**(1976), no. 1, 145–160. Recent trends in system theory. MR**0416694** - Saunders MacLane and O. F. G. Schilling,
*Infinite number fields with Noether ideal theories*, Amer. J. Math.**61**(1939), 771–782. MR**19**, DOI 10.2307/2371335 - D. Hinrichsen and A. J. Pritchard,
*Robust stability of linear evolution operators on Banach spaces*, SIAM J. Control Optim.**32**(1994), no. 6, 1503–1541. MR**1297095**, DOI 10.1137/S0363012992230404 - B. Jacob and J.R. Partington. The Weiss conjecture on admissibility of observation operators for contraction semigroups.
*Integral Equations and Operator Theory*, 40:231–243, 2001. - B. Jacob and H. Zwart. Realization of inner functions. Preprint, Twente, 1998.
- V. Katsnelson and G. Weiss,
*A counterexample in Hardy spaces with an application to systems theory*, Z. Anal. Anwendungen**14**(1995), no. 4, 705–730. MR**1376574**, DOI 10.4171/ZAA/648 - Peter D. Lax and Ralph S. Phillips,
*Scattering theory*, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR**0217440** - P. D. Lax and R. S. Phillips,
*Scattering theory for dissipative hyperbolic systems*, J. Functional Analysis**14**(1973), 172–235. MR**0353016**, DOI 10.1016/0022-1236(73)90049-9 - Hartmut Logemann, Richard Rebarber, and George Weiss,
*Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop*, SIAM J. Control Optim.**34**(1996), no. 2, 572–600. MR**1377713**, DOI 10.1137/S0363012993250700 - H. Logemann and E. P. Ryan,
*Time-varying and adaptive integral control of infinite-dimensional regular linear systems with input nonlinearities*, SIAM J. Control Optim.**38**(2000), no. 4, 1120–1144. MR**1760063**, DOI 10.1137/S0363012998339228 - H. Logemann, E. P. Ryan, and S. Townley,
*Integral control of infinite-dimensional linear systems subject to input saturation*, SIAM J. Control Optim.**36**(1998), no. 6, 1940–1961. MR**1638027**, DOI 10.1137/S0363012996314142 - H. Logemann, E.P. Ryan, and S. Townley. Integral control of linear systems with actuator nonlinearities: lower bounds for the maximal regulating gain.
*IEEE Trans. Autom. Control*, 44:1315–1319, 1999. - Hartmut Logemann and Stuart Townley,
*Discrete-time low-gain control of uncertain infinite-dimensional systems*, IEEE Trans. Automat. Control**42**(1997), no. 1, 22–37. MR**1439362**, DOI 10.1109/9.553685 - Hartmut Logemann and Stuart Townley,
*Low-gain control of uncertain regular linear systems*, SIAM J. Control Optim.**35**(1997), no. 1, 78–116. MR**1430284**, DOI 10.1137/S0363012994275920 - K. A. Morris,
*Justification of input-output methods for systems with unbounded control and observation*, IEEE Trans. Automat. Control**44**(1999), no. 1, 81–85. MR**1665308**, DOI 10.1109/9.739075 - Raimund Ober and Stephen Montgomery-Smith,
*Bilinear transformation of infinite-dimensional state-space systems and balanced realizations of nonrational transfer functions*, SIAM J. Control Optim.**28**(1990), no. 2, 438–465. MR**1040469**, DOI 10.1137/0328024 - Raimund J. Ober and Yuanyin Wu,
*Infinite-dimensional continuous-time linear systems: stability and structure analysis*, SIAM J. Control Optim.**34**(1996), no. 3, 757–812. MR**1384953**, DOI 10.1137/S0363012993245318 - Raymond E. A. C. Paley and Norbert Wiener,
*Fourier transforms in the complex domain*, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR**1451142**, DOI 10.1090/coll/019 - 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 - Richard Rebarber,
*Conditions for the equivalence of internal and external stability for distributed parameter systems*, IEEE Trans. Automat. Control**38**(1993), no. 6, 994–998. MR**1227215**, DOI 10.1109/9.222318 - Richard Rebarber,
*Exponential stability of coupled beams with dissipative joints: a frequency domain approach*, SIAM J. Control Optim.**33**(1995), no. 1, 1–28. MR**1311658**, DOI 10.1137/S0363012992240321 - R. Rebarber and S. Townley,
*Robustness and continuity of the spectrum for uncertain distributed parameter systems*, Automatica J. IFAC**31**(1995), no. 11, 1533–1546. MR**1359350**, DOI 10.1016/0005-1098(95)00088-E - Dietmar Salamon,
*Infinite-dimensional linear systems with unbounded control and observation: a functional analytic approach*, Trans. Amer. Math. Soc.**300**(1987), no. 2, 383–431. MR**876460**, DOI 10.1090/S0002-9947-1987-0876460-7 - Dietmar Salamon,
*Realization theory in Hilbert space*, Math. Systems Theory**21**(1989), no. 3, 147–164. MR**977021**, DOI 10.1007/BF02088011 - Olof J. Staffans,
*Quadratic optimal control of stable well-posed linear systems*, Trans. Amer. Math. Soc.**349**(1997), no. 9, 3679–3715. MR**1407712**, DOI 10.1090/S0002-9947-97-01863-1 - Olof J. Staffans,
*Coprime factorizations and well-posed linear systems*, SIAM J. Control Optim.**36**(1998), no. 4, 1268–1292. MR**1618041**, DOI 10.1137/S0363012995285417 - Olof J. Staffans,
*Quadratic optimal control of well-posed linear systems*, SIAM J. Control Optim.**37**(1999), no. 1, 131–164. MR**1645436**, DOI 10.1137/S0363012996314257 - Olof J. Staffans,
*Feedback representations of critical controls for well-posed linear systems*, Internat. J. Robust Nonlinear Control**8**(1998), no. 14, 1189–1217. MR**1658797**, DOI 10.1002/(SICI)1099-1239(19981215)8:14<1189::AID-RNC385>3.3.CO;2-W - Olof J. Staffans,
*On the distributed stable full information $H^\infty$ minimax problem*, Internat. J. Robust Nonlinear Control**8**(1998), no. 15, 1255–1305. MR**1658961**, DOI 10.1002/(SICI)1099-1239(19981230)8:15<1255::AID-RNC386>3.0.CO;2-P - O.J. Staffans. Lax–Phillips scattering and well-posed linear systems. In
*Proceedings of the 7th IEEE Mediterranean Conference on Control and Systems*, CD-ROM, Haifa, Israel, July 28–30, 1999. - O.J. Staffans.
*Well-Posed Linear Systems*. Book manuscript, 2002. - E. G. F. Thomas,
*Vector-valued integration with applications to the operator-valued $H^\infty$ space*, IMA J. Math. Control Inform.**14**(1997), no. 2, 109–136. Distributed parameter systems: analysis, synthesis and applications, Part 2. MR**1470030**, DOI 10.1093/imamci/14.2.109 - George Weiss,
*Admissibility of unbounded control operators*, SIAM J. Control Optim.**27**(1989), no. 3, 527–545. MR**993285**, DOI 10.1137/0327028 - George Weiss,
*Admissible observation operators for linear semigroups*, Israel J. Math.**65**(1989), no. 1, 17–43. MR**994732**, DOI 10.1007/BF02788172 - George Weiss,
*The representation of regular linear systems on Hilbert spaces*, Control and estimation of distributed parameter systems (Vorau, 1988) Internat. Ser. Numer. Math., vol. 91, Birkhäuser, Basel, 1989, pp. 401–416. MR**1033074** - Takao Hinamoto, Takashi Hamanaka, and Sadao Maekawa,
*Lyapunov stability of two-dimensional digital filters*, Electron. Comm. Japan Part III Fund. Electron. Sci.**72**(1989), no. 8, 83–90. MR**1041701**, DOI 10.1002/ecjc.4430720809 - George Weiss,
*Regular linear systems with feedback*, Math. Control Signals Systems**7**(1994), no. 1, 23–57. MR**1359020**, DOI 10.1007/BF01211484 - Vincent D. Blondel, Eduardo D. Sontag, M. Vidyasagar, and Jan C. Willems (eds.),
*Open problems in mathematical systems and control theory*, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1999. MR**1727924**, DOI 10.1007/978-1-4471-0807-8 - George Weiss and Ruth F. Curtain,
*Dynamic stabilization of regular linear systems*, IEEE Trans. Automat. Control**42**(1997), no. 1, 4–21. MR**1439361**, DOI 10.1109/9.553684 - Martin Weiss and George Weiss,
*Optimal control of stable weakly regular linear systems*, Math. Control Signals Systems**10**(1997), no. 4, 287–330. MR**1486727**, DOI 10.1007/BF01211550 - G. Weiss and M. Häfele. Repetitive control of MIMO systems using $H^\infty$ design.
*Automatica*, 35:1185–1199, 1999. - Olga Taussky,
*An algebraic property of Laplace’s differential equation*, Quart. J. Math. Oxford Ser.**10**(1939), 99–103. MR**83**, DOI 10.1093/qmath/os-10.1.99 - K. Zhou, J. Doyle and K. Glover.
*Robust and Optimal Control*. Prentice-Hall, Upper Saddle River, 1996.

## Bibliographic Information

**Olof Staffans**- Affiliation: Department of Mathematics, Åbo Akademi University, FIN-20500 Åbo, Finland
- Email: Olof.Staffans@abo.fi
**George Weiss**- Affiliation: Department of Electrical & Electronic Engineering, Imperial College of Science & Technology, Exhibition Road, London SW7 2BT, United Kingdom
- Email: G.Weiss@ic.ac.uk
- Received by editor(s): February 23, 2001
- Received by editor(s) in revised form: November 16, 2001
- Published electronically: April 3, 2002
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**354**(2002), 3229-3262 - MSC (2000): Primary 93C25; Secondary 34L25, 37L99, 47D06
- DOI: https://doi.org/10.1090/S0002-9947-02-02976-8
- MathSciNet review: 1897398