Nonlinear Interpolation Theory in H∞
Recently there has been an interesting cross-fertilisation between the areas of mathematical interpolation theory, operator theory and control theory which has led to new results in all areas. In particular the Sarason-Sz. Nagy Foias commutant lifting theorem provides a very broad framework for generalized Nevalinna-Pick interpolation which in turn gives a solution to the H∞-optimal weighted sensitivity problem in control theory. In this paper we give a generalization of this “commutant lifting theorem” for certain classes of nonlinear analytic operators and discuss some possible applications.
KeywordsHankel Operator Interpolation Theory Fading Memory Feedback Compensator Commutant Lift
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- V. M. Adamjan, D. Z. Arov, and M. G. Krein, “Infinite block Hankel matrices and related extension problems”, AMS Translations 111 (1978), pp. 133–156.Google Scholar
- J. A. Ball and A. C. M. Ran, “Optimal Hankel norm model reductions and Wiener-Hopf factorizations I: the canonical case”, SIAM J. Control and Opt., to appear.Google Scholar
- C. Foias, “Contractive intertwining dilations and waves in layered media”, Proceedings of International Congress of Mathematicians, Helsinki (1978), vol. 2, pp. 605–613.Google Scholar
- C. Foias and A. Tannenbaum, “On the Nehari problem for a certain class of L∞-functions appearing in control theory”, J. Functional Analysis, to appear.Google Scholar
- B. A. Francis, A Course in H ∞ Control Theory, Lecture Notes in Control and Inf. Sci., Springer-Verlag, to appear, 1986.Google Scholar
- E. Hille and R. S. Phillips, Functional Analysis and Semigroups, AMS Colloquium Publications, vol. XXXI, Providence, Rhode Island, 1957.Google Scholar
- B. Sz. -Nagy and C. Foias, “Dilation des commutants d’operateurs”, C.R. Acad. Sci. Paris, série A, 265 (1968), pp. 493–495.Google Scholar
- A. Tannenbaum, Invariance and System Theory: Algebraic and Geometric Aspects, Lecture Notes in Mathematics, vol. 845, Springer-Verlag, New York, 1981.Google Scholar
- J. A. Bail, C. Foias, J. W. Helton, and A. Tannenbaum, “On a local nonlinear commutant lifting theorem”, Technical Report, Department of Electrical Engineering, University of Minnesota, October 1986, submitted for publication.Google Scholar