Well-posed linear systems - a survey with emphasis on conservative systems

Weiss, George ; Staffans, Olof ; Tucsnak, Marius

International Journal of Applied Mathematics and Computer Science, Tome 11 (2001), p. 7-33 / Harvested from The Polish Digital Mathematics Library

Résumé

We survey the literature on well-posed linear systems, which has been an area of rapid development in recent years. We examine the particular subclass of conservative systems and its connections to scattering theory. We study some transformations of well-posed systems, namely duality and time-flow inversion, and their effect on the transfer function and the generating operators. We describe a simple way to generate conservative systems via a second-order differential equation in a Hilbert space. We give results about the stability, controllability and observability of such conservative systems and illustrate these with a system modeling a controlled beam.

Détail

Publié le : 2001-01-01

Zbl 0990.93046

EUDML-ID : urn:eudml:doc:207507

Bibliographie

[000] Adamajan A. and Arov D.Z. (1970): On unitary couplings of semiunitary operators, In: Eleven Papers in Analysis, Vol.95 of American Math. Soc. Transl. - Providence:AMS, pp.75-129. | Zbl 0258.47012

[001] Ammari K., Liu Z. and Tucsnak M. (1999): Decay rates for a beam with pointwise force and moment feedback. - Nancy, preprint. | Zbl 1042.93034

[002] Arov D.Z. and Nudelman M.A. (1996): Passive linear stationary dynamical scattering systems with continuous time.- Int. Eqns. Operat. Theory, Vol.24, pp.1-45. | Zbl 0838.47004

[003] Arov D.Z. (1999): Passive linear systems and scattering theory, In: Dynamical Systems, Control, Coding, Computer Vision (G. Picci and D. Gilliam, Eds.). - Birkhauser, Basel,pp.27-44, | Zbl 0921.93005

[004] Brodskiu i M.S. (1978): Unitary operator colligations and their characteristic functions. - Russian Math. Surveys, Vol.33, No.4, pp.159-191. | Zbl 0415.47007

[005] de Branges L. and Rovnyak J. (1966): Square Summable Power Series. - New York: Holt, Rinehart and Winston. | Zbl 0153.39603

[006] Crawley E.F. and Anderson E.H. (1989): Detailed models forpiezoceramic actuation of beams. - Proc. AIAA Conf., pp.471-476.

[007] Destuynder Ph., Legrain I., Castel L. and Richard N.(1992): Theoretical, numerical and experimental discussion of the use of piezoelectric devices for control-structure interaction. - Eur. J. Mech., ASolids, Vol.11, pp.181-213.

[008] Helton J.W. (1976): Systems with infinite-dimensional state space: The Hilbert space approach. - Proc. IEEE, Vol.64, pp.145-160.

[009] Hille E. and Phillips R.S. (1957): Functional Analysis and Semi-Groups, Rev. Ed. - Providence: AMS. | Zbl 0078.10004

[010] Jaffard S. and Tucsnak M. (1997): Regularity of plate equations with control concentrated in interior curves. - Proc. Roy. Soc. Edinburgh Sect. A, Vol.127, pp.1005-1025. | Zbl 0889.35059

[011] Lax P.D. and Phillips R.S. (1967): Scattering Theory. - New York: Academic Press. | Zbl 0214.12002

[012] Lax P.D. and Phillips R.S. (1973): Scattering theory for dissipative hyperbolic systems. - J. Funct. Anal.,Vol.14, pp.172-235. | Zbl 0295.35069

[013] Livv sic M.S. (173): Operators, Oscillations, Waves (Open Systems). - Transl. Math. Monographs, Vol. 34, Providence: AMS.

[014] Ober R. and Montgomery-Smith S. (1990): Bilinear transformation of infinite-dim-ensional state-space systems and balanced realizations of nonrational transfer functions. - SIAM J. Contr. Optim., Vol.28, pp.438-465. | Zbl 0693.93014

[015] Ober R. and Wu Y. (1996): Infinite-dimensional continuous-time linear systems: stability and structure analysis.- SIAM J. Contr. Optim., Vol.34, pp.757-812. | Zbl 0856.93051

[016] Salamon D. (1987): Infinite dimensional linear systems with unbounded control and observation: A functional analytic approach. - Trans. Amer. Math. Soc., Vol.300, pp.383-431. | Zbl 0623.93040

[017] Salamon D. (1989): Realization theory in Hilbert space. - Math. Syst. Theory, Vol.21, pp.147-164. | Zbl 0668.93018

[018] Staffans O.J. (1997): Quadratic optimal control of stable well-posed linear systems. - Trans. Amer. Math. Soc.,Vol.349, pp.3679-3715. | Zbl 0889.49023

[019] Staffans O.J. (1998a): Coprime factorizations and well-posed linear systems. - SIAM J. Contr. Optim., Vol.36, pp.1268-1292. | Zbl 0919.93040

[020] Staffans O.J. (1998b): On the distributed stable full information H^∞ minimax problem. - Int. J. Robust Nonlin. Contr., Vol.8, pp.1255-1305. | Zbl 0951.93029

[021] Staffans O.J. (1998c): Quadratic optimal control of well-posed linear systems. - SIAM J. Contr. Optim.,Vol.37, pp.131-164. | Zbl 0955.49018

[022] Staffans O.J. (1999): Lax-Phillips scattering and well-posed linear systems. - Proc. 7th IEEE Mediterranean Conf. Control and Systems, Haifa, Israel, published on CD-ROM.

[023] Staffans O.J. (2001): Well-Posed Linear Systems. - Book in preparation.

[024] Staffans O.J. and Weiss G. (2001a): Transfer functions of regular linear systems. Part II: The system operator and the Lax-Phillips semigroup. - London: preprint. | Zbl 0996.93012

[025] Staffans O.J. and Weiss G. (2001b): Transfer functions of regular linear systems. Part III: Inversions and duality. - London: preprint. | Zbl 1052.93032

[026] Sz.-Nagy B. and Foiacs C. (1970): Harmonic Analysis of Operators on Hilbert Space. - Amsterdam and London: North-Holland.

[027] Tucsnak M. and Weiss G. (2001): How to get a conservative well-posed linear system out of thin air. - London: preprint. | Zbl 1125.93383

[028] Weiss G. (1989a): Admissible observation operators for linear semigroups. - Israel J. Math., Vol.65, pp.17-43. | Zbl 0696.47040

[029] Weiss G. (1989b): Admissibility of unbounded control operators. - SIAM J. Contr. Optim., Vol.27, pp.527-545. | Zbl 0685.93043

[030] Weiss G. (1989c): The representation of regular linear systems on Hilbert spaces, In: Control and Optimization of Distributed Parameter Systems (F. Kappel, K. Kunisch, W. Schappacher, Eds.).- Basel: Birkhuser Verlag, pp.401-416.

[031] Weiss G. (1994a): Regular linear systems with feedback. - Math. Contr. Signals Syst., Vol.7, pp.23-57. | Zbl 0819.93034

[032] Weiss G. (1994b): Transfer functions of regular linear systems. Part I: Characterizations of regularity. - Trans. Amer. Math. Soc., Vol.342, pp.827-854. | Zbl 0798.93036

[033] Weiss G. (1999): A powerful generalization of the Carleson measure theorem?, In: Open Problems in Mathematical Systems and Control Theory (V. Blondel, E. Sontag, M. Vidyasagar and J. Willems, Eds.). - London: Springer-Verlag, pp.267-272.

[034] Weiss M. and Weiss G. (1997): Optimal control of stable weakly regular linear systems. - Math. Contr. Signals Syst., Vol.10, pp.287-330. | Zbl 0884.49021

[035] Yamamoto Y. (1981): Realization theory of infinite-dimensionallinear systems, Parts I and II. - Math. Syst. Theory, Vol.15, pp.55-77, pp.169-190.