Abstract
The spectral or canonical factorization of matrix- or operator-valued function F defined on the imaginary axis is defined as F = Y *X, where Y ±1,X ±1 are H− (bounded and holomorphic on Rez > 0), or, more generally, Y ±1,X ±1 belong to some weighted strong H2 space.
It is well known that the invertibility of the corresponding Toeplitz operator P + FP 0+ is necessary for this factorization to exist, where P + : L2 → H2 is the orthogonal projection. When F is positive, this condition is also sufficient for the factors to be H∞. In the general (indefinite) case, this is not so. However, if F is smooth enough, then the H∞ canonical factorization does exist even in the indefinite case; we give a solution assuming that F is the Fourier transform of a measure with no singular continuous part.
If the (Popov function determined by the) transfer function of a control system has a canonical factorization, then a well-posed optimal state feedback exists for the corresponding control problem. Conversely, a well-posed optimal state feedback determines a canonical factorization of the transfer function. We generalize this to unstable systems, i.e., to transfer functions that are holomorphic and bounded on some right half-plane | Rez > r.
Then we show that if the generalized Popov Toeplitz operator is uniformly positive, then the canonical factorization exists (the stable case is well known). However, the results on the regularity of the factors and in the nonpositive case remain very few — we explain them and the remaining open problems.
The first author was supported by Magnus Ehrnrooth Foundation. The second author was supported in part by NSF Grant DMS-0456625.
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References
Albrecht Böttcher, Yuri I. Karlovich, and Ilya M. Spitkovsky, Convolution operators and factorization of almost periodic matrix functions, Operator Theory: Advances and Applications, vol. 131, Birkhäuser Verlag, Basel, 2002.
R.G. Babadzhanyan and V.S. Rabinovich, Factorization of almost periodic operator functions, Differential and integral equations and complex analysis (Russian), Kalmytsk. Gos. Univ., Elista, 1986, pp. 13–22, i.
Frank M. Callier and Charles A. Desoer, Stabilization, tracking and disturbance rejection in multivariable convolution systems, Ann. Soc. Sci. Bruxelles Sér. I 94 (1980), no. 1, 7–51.
Kevin F. Clancey and Israel Gohberg, Factorization of matrix functions and singular integral operators, Operator Theory: Advances and Applications, vol. 3, Birkhäuser-Verlag, Basel Boston Berlin, 1981.
Frank M. Callier and Joseph J. Winkin, The spectral factorization problem for multivariable distributed parameter systems, Integral Equations Operator Theory 34 (1999), no. 3, 270–292.
Ruth F. Curtain and George Weiss, Exponential stabilization of well-posed systems by colocated feedback, SIAM J. Control Optim. (2006).
Ruth F. Curtain and Hans Zwart, An introduction to infinite-dimensional linear systems theory, Springer-Verlag, New York, 1995.
Franco Flandoli, Irena Lasiecka, and Roberto Triggiani, Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems, Ann. Mat. Pura Appl. 153 (1988), 307–382.
Israel Gohberg, Marinus A. Kaashoek, and Ilya M. Spitkovsky, An overview of matrix factorization theory and operator applications, Factorization and integrable systems (Faro, 2000), Oper. Theory Adv. Appl., vol. 141, Birkhäuser, Basel, 2003, pp. 1–102.
Israel C. Gohberg and Yuri Laiterer, A criterion for factorization of operator-functions with respect to a contour, Soviet Math. Dokl. 14 (1973), 425–429.
_____, The factorization of operator-functions relative to a contour. III. Factorization in algebras. (Russian), Math. Nachr. 55 (1973), 33–61.
Gustaf Gripenberg, Stability of Volterra equations with measure kernels in Banach spaces, J. Math. Anal. Appl. 178 (1993), 156–164.
Vlad Ionescu, Cristian Oară, and Martin Weiss, Generalized Riccati theory and robust control. a Popov function approach, John Wiley, New York, London, 1999.
Georgii S. Litvinchuk and Ilia M. Spitkovskii, Factorization of measurable matrix functions, Operator Theory: Advances and Applications, vol. 25, Birkhäuser Verlag, Basel, 1987, Translated from the Russian by Bernd Luderer, with a foreword by Bernd Silbermann.
Irena Lasiecka and Roberto Triggiani, Control theory for partial differential equations: continuous and approximation theories. I abstract parabolic systems, Encyclopedia of Mathematics and its Applications, vol. 74, Cambridge University Press, Cambridge, 2000.
Kalle M. Mikkola, Infinite-dimensional linear systems, optimal control and algebraic Riccati equations, Doctoral dissertation, technical report A452, Institute of Mathematics, Helsinki University of Technology, Espoo, Finland, 2002, www.math.hut.fi/~kmikkola/research/.
_____, Characterization of transfer functions of Pritchard-Salamon or other realizations with a bounded input or output operator, Integral Equations Operator Theory 54 (2006), no. 3, 427–440.
_____, State-feedback stabilization of well-posed linear systems, Integral Equations Operator Theory 55 (2006), no. 2, 249–271.
_____, Weakly coprime factorization and state-feedback stabilization of discrete-time systems, submitted, 2007.
_____, Indefinite optimal control and well-posed minimizing control, in preparation, 2007.
_____, Fourier multipliers for L 2 functions with values in nonseparable Hilbert spaces and operator-valued Hp boundary functions, Ann. Acad. Sci. Fenn.Math. 33 (2008), no. 1, to appear.
Marvin Rosenblum and James Rovnyak, Hardy classes and operator theory, Oxford University Press, New York, Oxford, 1985.
Dietmar Salamon, Realization theory in Hilbert space, Math. Systems Theory 21 (1989), 147–164.
Olof J. Staffans, Quadratic optimal control of stable well-posed linear systems, Trans. Amer. Math. Soc. 349 (1997), 3679–3715.
_____, Feedback representations of critical controls for well-posed linear systems, Internat. J. Robust Nonlinear Control 8 (1998), 1189–1217.
_____, Well-Posed Linear Systems, Encyclopedia Math. Appl., vol. 103, Cambridge University Press, Cambridge, 2005.
Bert van Keulen, H ∞-control for distributed parameter systems: A state space approach, Birkhäuser Verlag, Basel Boston Berlin, 1993.
Mathukumalli Vidyasagar, Hans Schneider, and Bruce A. Francis, Algebraic and topological aspects of feedback stabilization, IEEE Trans. Autom. Control 27 (1982), no. 5, 880–894.
Joseph Winkin, Spectral factorization and feedback control for infinite-dimensional control systems, Doctoral dissertation, Facultées Universitaires Notre-Dame de la Paix à Namur, 1989.
Martin Weiss and George Weiss, Optimal control of stable weakly regular linear systems, Math. Control Signals Systems 10 (1997), 287–330.
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Mikkola, K.M., Spitkovsky, I.M. (2008). Spectral Factorization, Unstable Canonical Factorization, and Open Factorization Problems in Control Theory. In: Bastos, M.A., Lebre, A.B., Speck, FO., Gohberg, I. (eds) Operator Algebras, Operator Theory and Applications. Operator Theory: Advances and Applications, vol 181. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8684-9_16
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