CONJECTURES

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CONJECTURES

Lin’s Conjecture 1: A multivariable n-D system is output feedback stabilizable if and only if it admits a double coprime factorization (DCF).

Background: This conjecture was first posed in [(a) Z. Lin, “Feedback stabilizability of MIMO n-D linear systems”, Multidimensional Systems and Signal Processing, vol. 9, no. 2, pp. 149--172, 1998], and reformulated later in [(b) Z. Lin, “Output feedback stabilizability and stabilization of linear nD systems,” in Multidimensional Signals, Circuits and Systems (K. Galkowski and J. Wood, Eds.), Chapter 4, pp. 59 – 76, Taylor & Francis, UK, 2001].

Solutions:

(I) Partial solutions to this conjecture were presented in [(a) Z. Lin, “Feedback stabilization of MIMO 3-D linear systems”, IEEE Trans. Automatic Control, vol. 44, no. 10, pp. 1950-1955, Oct, 1999; (b) Z. Lin, “Feedback stabilization of MIMO nD linear systems”, IEEE Trans. Automatic Control, vol. 45, no. 12, pp. 2419-2424, Dec. 2000; (c) K. Mori, “Parameterization of stabilizing controllers over commutative rings with application to multidimensional systems,” IEEE Trans. Circuits Syst. I, vol. 49, pp. 743–752, June 2002; (d) K. Mori, “Parameterization of stabilizing controllers with either right- or left-coprime factorization,” IEEE Trans. Automatic Control, vol. 47, no. 10, pp. 1763-1767, Oct, 2002].

(II) Complete solutions to this conjecture have recently been presented in [(e) A. Quadrat, “A lattice approach to analysis and synthesis problems,” Math. Control, Signals and Systems, vol. 18, No. 2, pp. 147-186, May, 2006].

Lin’s Conjecture 2: An n-D polynomial matrix admits a zero prime factorization if and only if its reduced minors are zero coprime.

Background: This conjecture was first posed in [(a) Z. Lin, “Notes on n-D polynomial matrix factorizations”, Multidimensional Systems and Signal Processing, vol. 10, no. 4, pp. 379--393, 1999], and re-posed later in [(b) Z. Lin, N. K. Bose, “A generalization of Serre's conjecture and some related issues”, Linear Algebra and Its Applications, Vol. 338, pp. 125-138, Nov. 2001].

Solutions:

(I) Partial solutions to this conjecture were presented in [(a) Z. Lin, “Notes on n-D polynomial matrix factorizations”, Multidimensional Systems and Signal Processing, vol. 10, no. 4, pp. 379--393, 1999; (b) Z. Lin, “Further results on n-D polynomial matrix factorizations”, Multidimensional Systems and Signal Processing, vol. 12, no. 2, pp. 199-208, April, 2001; (c) H. Park, “Prime factorization of n-D polynomial matrices,” in Proc. IEEE ISCAS, pp. vol. III, pp. 666-669, Bangkok, Thailand, May, 2003].

(II) Complete solutions to this conjecture, using different methods, have recently been presented in [(d) J. F. Pommaret, “Solving Bose conjecture on linear multidimensional systems,” in Proceedings of the European Control Conference, pp. 1853–1855, September 2001; (e) V. Srinivas, “A generalized Serre problem,” J. Algebra, vol. 278, pp. 621-627, Aug. 2004; (f) M. Wang and D. Feng, “On Lin–Bose problem”, Linear Algebra and Its Applications, Vol. 390, pp. 279-285, Oct. 2004].

Lin-Bose’s Conjecture: An n-D polynomial matrix F with content d can be completed into a square n-D polynomial matrix with determinant d if and only if the reduced minors of F are zero coprime.

Background: This conjecture was posed in [Z. Lin, N. K. Bose, “A generalization of Serre's conjecture and some related issues”, Linear Algebra and Its Applications, Vol. 338, pp. 125-138, Nov. 2001].

Solutions: It was proved in [Z. Lin, N. K. Bose, “A generalization of Serre's conjecture and some related issues”, Linear Algebra and Its Applications, Vol. 338, pp. 125-138, Nov. 2001] that Lin-Bose’s Conjecture is equivalent to Lin’s Conjecture 2. Hence, solutions to Lin’s Conjecture 2 are also solutions to Lin-Bose’s Conjecture.

Related Conjectures: Several other apparently different conjectures were also posed in [(a) Z. Lin, “Notes on n-D polynomial matrix factorizations”, Multidimensional Systems and Signal Processing, vol. 10, no. 4, pp. 379--393, 1999; (b) Z. Lin, N. K. Bose, “A generalization of Serre's conjecture and some related issues”, Linear Algebra and Its Applications, Vol. 338, pp. 125-138, Nov. 2001], including n-D polynomial matrices which are not of full rank. All these conjectures were shown to be equivalent to Lin’s Conjecture 2 [Z. Lin, N. K. Bose, “A generalization of Serre's conjecture and some related issues”, Linear Algebra and Its Applications, Vol. 338, pp. 125-138, Nov. 2001].