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Differential equations with positive evolutions and some applications

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  1. H. Abou-Kandil, G. Freiling, V. Ionescu, and G. Jank. Matrix Ricati Equations in Control and Systems Theory. Birkhäuser, Basel, 2003.

    Book  Google Scholar 

  2. B. D. O. Anderson and J. B. Moore. New results in linear system stability. SIAM J. Control, 7:398–413, 1969.

    Article  MathSciNet  MATH  Google Scholar 

  3. W. Arendt, editor. One-parameter Semigroups of Positive Operators, volume 1184 of Lecture Notes in Mathematics. Springer, 1986.

  4. W. Arendt. Resolvent positive operators. Proc. London Math. Soc., 54(3):321–349, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Azevedo-Perdicoulis and G. Jank. Linear quadratic Nash games on positive linear systems. to appear in European Journal of Control, 11: 2005.

  6. T. Ba§ar and G. J. Olsder. Dynamic Noncooperative Game Theory. Academic Press, London, 1995.

    Google Scholar 

  7. E.A. Barbashin and N.N. Krasovskii. Introduction to the Theory of Stability. Volters Noorthoff, Groningen, 1970.

  8. A. Berman, M. Neumann, and R. J. Stern. Nonnegative Matrices in Dynamic Systems. John Wiley & Sons, New York, 1989.

    MATH  Google Scholar 

  9. A. Berman and R. J. Plemmons. Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics. SIAM, 1994.

  10. B. van der Broek. Uncertainty in differential games. PhD thesis, Univ. Tilburg, The Netherlands, 2001.

  11. T. Damm. Rational Matrix Equations in Stochastic Control. Number 297 in Lecture Notes in Control and Information Sciences. Springer, 2004.

  12. T. Damm and D. Hinrichsen. Newton’s method for concave operators with resolvent positive derivatives in ordered Banach spaces. Linear Algebra Appl., 363:43–64, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  13. V. Dragan, G. Freiling, A. Hochhaus, and T. Morozan. A class of nonlinear differential equations on the space of symmetric matrices. Electron. J. Differ. Equ., (96):48 pages, 2003.

  14. V. Dragan and T. Morozan. Exponential stability for discrete time linear equations defined by positive operators. Integral Equations and Operator Theory, published electronically 2005 (s00020-005-1371-7).

  15. V. Dragan and T. Morozan. Stochastic observability and applications. IMA J. Math. Control Inf., 21(3):323–344, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  16. V. Dragan and T. Morozan. Observability and detectability of a class of discrete time stochastic linear systems. IMA J. Math. Control Inf., (to appear in 2006).

  17. J. Engwerda. LQ Dynamic Optimization and Differential Games. John Wiley & Sons, New York, 2005.

    Google Scholar 

  18. L. Eisner. Quasimonotonie und Ungleichungen in halbgeordneten Räumen. Linear Algebra Appl., 8:249–261, 1974.

    Article  Google Scholar 

  19. A. Fischer, D. Hinrichsen, and N. K. Son. Stability radii of Metzler operators. Vietnam J. Math, 26:147–163, 1998.

    MathSciNet  MATH  Google Scholar 

  20. G. Freiling and A. Hochhaus. Properties of the solutions of rational matrix difference equations. Comput. Math. Appl., 45(6-9): 1137–1154, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  21. G. Freiling, G. Jank, and H. Abou-Kandil. Generalized Riccati difference and differential equations. Linear Algebra Appl., 241-243:291–303, 1996.

    Article  MathSciNet  Google Scholar 

  22. Chun-Hua Guo. Nonsymmetric algebraic Riccati equations and Wiener-Hopf factorization of M-matrices. SIAM J. Matrix. Anal Appl., 23(1):225–242, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Halanay. Differential Equations: Stability, Oscillations, Time Lags. Academic Press, New-York, London, 1966.

    MATH  Google Scholar 

  24. T. Kaczorek. Externally and internally positive time-varying linear systems. Int. J. Appl. Math. Comput. Sci., 11(4):957–964, 2001.

    MathSciNet  MATH  Google Scholar 

  25. T. Kaczorek. Positive 1D and 2D Systems. Springer, 2002.

  26. R.E. Kaiman. Contributions on the theory of optimal control. Bol. Soc. Mat. Mex., II. Ser. 5:102–119, 1960.

    Google Scholar 

  27. M. A. Krasnosel’skij, Je. A. Lifshits, and A. V. Sobolev. Positive Linear Systems — The Method of Positive Operators, volume 5 of Sigma Series in Applied Mathematics. Heldermann Verlag, Berlin, 1989.

    Google Scholar 

  28. H. Kwakernaak and R. Sivan. Linear Optimal Control Systems. John Wiley & Sons, New York, 1972.

    MATH  Google Scholar 

  29. H. Schneider. Positive operators and an inertia theorem. Numer. Math., 7:11–17, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  30. H. Schneider and M. Vidyasagar. Cross-positive matrices. SIAM J. Numer. Anal, 7(4):508–519, 1970.

    Article  MathSciNet  MATH  Google Scholar 

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Dragan, V., Damm, T., Freiling, G. et al. Differential equations with positive evolutions and some applications. Results. Math. 48, 206–236 (2005).

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