We extend the classical Perron-Frobenius theorem for positive quasipolynomial matrices associated with homogeneous difference equations. Finally, the result obtained is applied to derive necessary and sufficient conditions for the stability of positive system.
C. E. Avellar and J. K. Hale, “On the zeros of exponential polynomials,” Journal of Mathematical Analysis and Applications, vol. 73, no. 2, pp. 434–452, 1980.
View at: Publisher Site | Google Scholar | MathSciNetJ. K. Hale, Functional Differential Equations, vol. 3 of Applied Mathematical Sciences, Springer, Berlin, Germany, 1971.
View at: MathSciNetJ. K. Hale and S. M. Verduyn Lunel, “Strong stabilization of neutral functional differential equations,” IMA Journal of Mathematical Control and Information, vol. 19, no. 1-2, pp. 5–23, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetW. Michiels and T. Vyhlídal, “An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type,” Automatica, vol. 41, no. 6, pp. 991–998, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetD. Aeyels and P. De Leenheer, “Extension of the Perron-Frobenius theorem to homogeneous systems,” SIAM Journal on Control and Optimization, vol. 41, no. 2, pp. 563–582, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetP. H. A. Ngoc, “A Perron-Frobenius theorem for a class of positive quasi-polynomial matrices,” Applied Mathematics Letters, vol. 19, no. 8, pp. 747–751, 2006.
View at: Google Scholar | MathSciNetS. M. Rump, “Theorems of Perron-Frobenius type for matrices without sign restrictions,” Linear Algebra and Its Applications, vol. 266, pp. 1–42, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. M. Rump, “Perron-Frobenius theory for complex matrices,” Linear Algebra and Its Applications, vol. 363, pp. 251–273, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetA. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, NY, USA, 1979.
View at: MathSciNetO. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H.-O. Walther, Delay Equations, Functional, Complex, and Nonlinear Analysis, Springer, New York, NY, USA, 1995.
View at: Zentralblatt MATH | MathSciNet