Abstract
In this paper we attempt to describe a circle of ideas which makes possible a unified exposition of a large part of linear algezbra, operator theory and both finite and infinite dimensional multivariable linear systems. As the title suggests the unifying conzcepts will be those of functional models, module theory and various factorizations of polynomial and analytic matrix valued functions. The full exposition of the ideas presented here will be the theme of a forthcoming monograph [20].
Partially supported by National Science Foundation under Grant NSF ENG77-28444.
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References
Baras, J. S., and Brockett, R. W.: 1975, H 2 functions and infinite dimensional realization theory, SIAM J. Control, 13, pp. 221–241.
Beurling, A.: 1949, On two -problems concerning linear transformations in Eilbert space, Acta Math. 81, pp. 239–255.
deBranges, L., and Rovnyak, J.: 1964, The existence of invariant subspaces, Bull. Amer. Math. Soc., 70, pp. 718–721.
Brockett, R. W.: 1977, The geometry of the set of controlzlable linear systems, Reserach Report of Automatic Control Laboratory, Nagoya University, v. 24, pp. 1–7.
Carleson, L.: 1962, Interpolation by bounded analytic functions and the corona problem, Ann. of Math., (2), 76, pp. 547–559.
Dewilde, P.: 1976, Input-output description of roomy sysztems, SIAM J. Control, 14, pp. 712–736.
Emre, E.: Nonsingular factors of polynomial matrices and (A3B)-invariant and reachability subspaces, Memorandum C0S0R 78-19, Eindhoven University of Technology.
Emre, E., and Hautus, M. L. J.: A polynomial characterizaztion of (A,B)-invariant and reachability subspaces, Memoranzdum C0S0R 78-19, Eindhoven University of Technology.
Fuhrmann, P. A.: 1968, On the corona theorem and its applizcations to spectral problems in Hilbert space, Trans. Amer. Math. Soc., 132, pp. 55–66.
Fuhrmann, P. A.: 1968, A functional calculus in Hilbert space based on operator valued analytic functions, Israel J. Math. 6, pp. 267–278.
Fuhrmann, P. A.: 1975, Realization theory in Hilbert space for a class of transfer functions, J. Funct. Anal. 18, pp. 338–349.
Fuhrmann, P. A.: 1975, On generalized Eankel operators induced by sums and products, Israel J. Math., 21, pp. 279–295.
Fuhrmann, P. A.: 1976, On series and parallel coupling of infinite dimensional linear systems, SIAM J. Control, 14, p. 339–358.
Fuhrmann, P. A.: 1976, Algebraic system theory; An analystrs point of view, J. Franklin Inst., 301, pp. 521–540.
Fuhrmann, P. A.: 1977, On strict system equivalence and similarity, Int. J. Control, 25, pp. 5–10.
Fuhrmann, P. A.: 1978, Simulation of linear systems and factorization of matrix polynomials, Int. J. Control, 28, pp. 689–705.
Fuhrmann, P. A.: Linear feedback via polynomial models, Int. J. Control, to appear.
Fuhrmann, P. A., and Willems, J. C.: The factorization indices for rational matrix functions, to appear.
Fuhrmann, P. A., and Willems, J. C.: A study of (A,B)- invariant subspaces by polynomial models, to appear.
Fuhrmann, P. A.: 1979, Linear systems and operators in Hilbert space, McGraw-Hi11.
Gohberg, I. C., and Krein, M. G.: 1960, Systems of integral equations on a half line with kernels depending on the difference of arguments, English translation, A.M.S. Transzlations (2), 14, pp. 217–287.
Halmos, P. R.: 1961, Shifts on Hilbert spaces, J. Reine Angew. Math. 208, pp. 102–112.
Hautus, M. L. J., and Heymann, M.: 1973, Feedback-an algezbraic approach, SIAM J. Control and Optimization, 16, pp. 83–105.
Helson, H.: 1964, Lectures on invariant subspaces, Academic Press, New York.
Helton, J. W.: 1974, Discrete time systems, operator models and scattering theory, J. Funct. Anal., 16, pp. 15–38.
Helton, J. W.: 1976, Systems with infinite dimensional state space: The Eilbert space approach, Proc. of the IEEE, 60, pp. 145–160.
Lax, P. D.: 1959, Translation invariant subspaces, Acta Math. 101, pp. 163–178.
Lax, P. D., and Phillips, R. S.: 1967, Scattering theory, Academic Press, New York.
Li vsic, M. S. c 1960, On a class of linear operators in Eilzbert space, Amer. Math. Soc. Transl. (2), 13, pp. 61–83.
Morse, A. S.: Systems invariants under feedback and cascade control, in Mathematical Systems Theory, Lecture Notes in Economics and Mathematical Systems, v. 131, Springer, pp. 61–74.
Munzner, H. F., and Prätzel-Wolters, D.: 1978, Minimalbasen polynomialer Modulen und Brunovsky-Transformationen, Mathematik Arbeitspapiere, Nr. 12, Univ. Poremen.
Rosenbrock, H. H.: 1970, State space and multivariable theory, J. Wiley, New York.
Rota, G. E.: 1960, On models for linear operators, Comm. Pure Appl. Math., 13, pp. 469–472. 00
Sarason, D.: 1967, Generalized interpolation in E, Trans. Amer. Math. Soc., 127, pp. 179–203.
Sz.-Nagy, B., and Foias, C.: 1970, Harmonic analysis of operators on Eilbert space, North Holland, Amsterdam.
Wolovich, W. A.: 1974, Linear multivariable systems, Springer.
Wonham, W. M.: 1974, Linear multivariable control, Springer.
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Fuhrmann, P.A. (1980). Functional Models, Factorizations and Linear Systems. In: Byrnes, C.I., Martin, C.F. (eds) Geometrical Methods for the Theory of Linear Systems. Nato Advanced Study Institutes Series, vol 62. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9082-1_7
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DOI: https://doi.org/10.1007/978-94-009-9082-1_7
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