Abstract
For an operator-valued function θ in the Schur class a new geometric proof, using state space considerations only, of the construction of a minimal and optimal realization is given. A minimal and optimal realization also appears as a restricted shift realization where the state space is the completion of the range of the associated Hankel operator in the de Branges-Rovnyak norm associated with θ. It is also shown that minimal and optimal, and minimal and star-optimal realizations of a rational matrix function in the Schur class are intimately connected to the extremal positive solutions of the associated Kalman-Yakubobich-Popov operator inequality.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
T. Ando,De Branges spaces and analytic operator functions, Lecture notes of the division of Applied Mathematics Research Institute of Applied Electricity, Hokkaido University, Sapporo, Japan, 1990.
M.A. Arbib, P.L. Falb, R.E. Kalman,Topics in Mathematical System Theory, McGraw-Hill, New York, 1969.
D.Z. Arov, Scattering theory with dissipation of energy,Dokl. Akad. Nauk SSSR,216 (4) (1974), 713–716; English translation with addenda:Sov. Math. Dokl.,15 (1974), 848–854.
D.Z. Arov, Passive linear stationary dynamic systems,Sibirsk. Mat. Zhurn.,20 (2) (1979), 211–228 (Russian), English translation:Siberian Math. J.,20 (1979), 149–162.
D.Z. Arov, Stable dissipative linear stationary dynamical scattering systems,Journal of Operator Theory,1 (1979), 95–126 (Russian).
D.Z. Arov,Linear stationary passive systems with losses, doctoral dissertation, phys. math. sciences, Odessa, 1985.
D.Z. Arov, M.A. Kaashoek, D.R. Pik, Optimal time-varying systems and factorization of operators, I: minimal and optimal systems,to appear.
D.Z. Arov, M.A. Kaashoek, D.R. Pik, Optimal time-varying systems and factorization of operators, II: factorization of operators,to appear.
D.Z. Arov, M.A. Nudelman, Passive linear stationary dynamical scattering systems with continuous time,Integral Equations Operator Theory,24 (1996), 1–45.
J. Ball, I. Cohen, De Branges-Rovnyak operator models and systems theory: a survey, in:Topics in Matrix and Operator Theory (eds. H. Bart, I. Gohberg, M. A. Kaashoek) Operator Theory: Advances and Applications50, Birkhäuser Verlag, Basel, 1989, pp. 93–136.
H. Bart, I. Gohberg, M.A. Kaashoek,Minimal Factorization of Matrix and Operator Functions, Operator Theory: Advances and Applications1, Birkhäuser Verlag, Basel, 1979.
S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan,Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, Vol. 15, SIAM, Philadelphia, 1994.
L. de Branges, Unitary linear systems whose transfer functions are Riemann mapping functions, in:Operator theory and systems, Proceedings Workshop Amsterdam, Operator Theory: Advances and Applications19, Birkhäuser Verlag, Basel, 1986, pp. 105–124.
L. de Branges, J. Rovnyak,Square Summable Power Series, Holt, Rinehart and Winston, New York, 1966.
L. de Branges, J. Rovnyak, Appendix on square summable power series, Canonical models in quantum scattering theory, in:Perturbation Theory and its Applications in Quantum Mechanics, (ed. C.H. Wilcox), New York, 1966, pp. 295–392.
M.S. Brodskii, Unitary operator colligations and their characteristic functions,Uspekhi Math. Nauk,33 (4) (1978), 141–168.
I. Gohberg, S. Goldberg, M.A. Kaashoek,Classes of linear operators Vol. II, Operator Theory: Advances and Applications63, Birkhäuser Verlag, Basel, 1993.
T. Kailath, Norbert Wiener and the development of mathematical engineering, in:Communications, computation, control and signal processing, (eds. A. Paulraj, V. Roychowdhury, C.D. Schaper), Kluwer Academic Publishers, Norwell, Massachusetts, 1997, pp. 35–64.
P. Lancaster, L. Rodman,Algebraic Riccati Equations, Clarendon Press, Oxford, 1995.
M.A. Nudelman, Optimal passive systems and semi boundedness of quadratic functionals,Sibirsk. Mat. Zhurn.,33 (1) (1992), 78–86 Russian.
V.M. Popov,Hyperstability of Control Systems, Springer Verlag, Berlin, 1973.
A.C.M. Ran, R. Vreugdenhil, Existence and comparison theorems for algebraic Riccati equations for continuous- and discrete-time systems,Linear Algebra and its Applications,99 (1988), 63–83.
B. Szökefalvi-Nagy, C. Foias,Analyse harmonique des operateurs de l'espace de Hilbert, Akademiai Kiado, Budapest, 1967.
J.C. Willems, Dissipative dynamical systems,Archive for rational mechanics and analysis,45 (5) (1972), 321–393.
K. Zhou, J.C. Doyle, K. Glover,Robust and Optimal Control, Prentice Hall, New Jersey, 1996.
Author information
Authors and Affiliations
Additional information
The first author thanks the International Association for the promotion of co-operation with scientists from the New Independent States of the former Soviet Union for its support (under project INTAS 93-249), and the Vrije Universiteit, Amsterdam, for its hospitality
Rights and permissions
About this article
Cite this article
Arov, D.Z., Kaashoek, M.A. Minimal and optimal linear discrete time-invariant dissipative scattering systems. Integr equ oper theory 29, 127–154 (1997). https://doi.org/10.1007/BF01191426
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01191426