On stability and robust stability of positive linear volterra equations

SIAM Journal on Control and Optimization

Volumn 47, Issue 2, 2008, Pages 975-996

  Huu Anh Ngoc, Pham ^a   Naito, Toshiki ^a   Shin, Jong Son ^a   Murakami, Satoru ^b  

^a UNIVERSITY OF ELECTRO COMMUNICATIONS   (Japan)

^b OKAYAMA UNIVERSITY OF SCIENCE   (Japan)

Author keywords

Linear Volterra equation; Positive system; Stability radius; Uniformly asymptotic stability

Indexed keywords

ASYMPTOTIC ANALYSIS; ASYMPTOTIC STABILITY; INTEGRAL EQUATIONS; INTEGRODIFFERENTIAL EQUATIONS; SYSTEM STABILITY; TURBULENT FLOW;

AFFINE PERTURBATIONS; LINEAR VOLTERRA EQUATION; POSITIVE SYSTEM; ROBUST STABILITIES; STABILITY RADIUS; STRUCTURED PERTURBATIONS; UNIFORMLY ASYMPTOTIC STABILITY; VOLTERRA EQUATIONS; VOLTERRA INTEGRODIFFERENTIAL EQUATIONS;

LINEAR EQUATIONS;

EID: 55549117815     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/070679740     Document Type: Article

References (48)

1. D. AEYELS and P. DE LEENHEER, Extension of the Perron-Frobenius theorem to homogeneous systems, SIAM J. Control Optim., 41 (2002), pp. 563-582.
2. A. BERMAN and R. J. PLEMMONS, Nonnegative Matrices in Mathematical Sciences, Academic Press, New York, 1979.
3. T. A. BURTON, Volterra Integral and Differential Equations, Math. Sci. Eng. 167, Academic Press, New York, 1983.
4. T. A. BURTON, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Math. Sci. Eng. 178, Academic Press, New York, 1985.
5. L. FARINA and S. RINALDI, Positive Linear Systems: Theory and Applications, John Wiley, New York, 2000.
6. S. GAUBERT and J. GUNAWARDENA, The Perron-Frobenius theorem for homogeneous, monotone functions, Trans. Amer. Math. Soc., 356 (2004), pp. 4931-4950.
7. 1-kernels, J. Differential Equations, 13 (1973), pp. 551-566.
8. W. M. HADDAD and V. CHELLABOINA, Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay, in Advances in Time-Delay Systems, Lect. Notes Comput. Sci. Eng. 38, Springer, Berlin, 2004, pp. 421-435.
9. W. M. HADDAD and V. CHELLABOINA, Stability theory for nonnegative and compartmental dynamical systems with delay, Systems Control Lett., 51 (2004), pp. 355-361.
10. W. M. HADDAD and V. CHELLABOINA, Stability and dissipativity theory for nonnegative dynamical systems: A unified analysis framework for biological and physiological systems, Nonlinear Anal. Real World Appl., 6 (2005), pp. 35-65.
11. Y. HINO and S. MURAKAMI, Stability properties of linear Volterra integro-differential equations in a Banach space, Funkcial. Ekvac., 48 (2005), pp. 367-392.
12. D. HINRICHSEN and A. J. PRITCHARD, Stability radius for structured perturbations and the algebraic Riccati equation, Systems Control Lett., 8 (1986), pp. 105-113.
13. D. HINRICHSEN and A. J. PRITCHARD, Real and complex stability radii: A survey, in Control of Uncertain Systems, Progr. Systems Control Theory 6, D. Hinrichsen and B. Märtensson, eds., Birkhäuser, Basel, 1990, pp. 119-162.
14. D. HINRICHSEN and N. K. SON, μ-analysis and robust stability of positive linear systems, Appl. Math. Comput. Sci., 8 (1998), pp. 253-268.
15. D. HINRICHSEN, N. K. SON, and P. H. A. NGOC, Stability radii of positive higher order difference systems, Systems Control Lett., 49 (2003), pp. 377-388.
16. R. A. HORN and C. R. JOHNSON, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1993.
17. G. JAMES and V. RUMCHEV, Stability of positive linear discrete-time systems, Systems Sci., 30 (2004), pp. 51-67.
18. G. JAMES, S. P. KOSTOVA, and V. G. RUMCHEV, Pole-assignment for a class of positive linear systems, Internat. J. Systems Sci., 32 (2001), pp. 1377-1388.
19. G. S. JORDAN and R. L. WHEELER, Structure of resolvents of Volterra integral and integrodifferential systems, SIAM J. Math. Anal., 11 (1980), pp. 119-132.
20. 1-spaces and applications to stability problems for Volterra equations, Trans. Amer. Math. Soc., 274 (1982), pp. 749-782.
21. F. E. KLOEDEN and A. M. RUBINOV, A generalization of Perron-Frobenius theorem, Nonlinear Anal., 41 (2000), pp. 97-115.
22. M. J. KREIN and M. A. RUTMAN, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl., 26 (1950), pp. 199-325.
23. V. LAKSHMIKANTHAM and M. R. M. RAO, Theory of Integro-differential Equations, Stability Control Theory Methods Appl. 1, Gordon and Breach, Lausanne, 1995
24. P. DE LEENHEER and D. P. AEYELS, Stabilization of positive systems with first integrals, Automatica, 38 (2002), pp. 1583-1589.
25. D. G. LUENBERGER, Introduction to Dynamic Systems, Theory, Models and Applications, John Wiley, New York, 1979.
26. R. K. MILLER, Asymptotic stability properties of Volterra integro-differential systems, J. Differential Equations, 10 (1971), pp. 485-506.
27. R. K. MILLER, Structure of solutions of unstable linear Volterra integro-differential equations, J. Differential Equations, 15 (1974), pp. 129-157.
28. R. NAGEL, ed., One-Parameter Semigroups of Positive Operators, Springer, Berlin, 1986.
29. T. NAITO, J. S. SHIN, S. MURAKAMI, and P. H. A. NGOC, Characterizations of linear Volterra integral equations with nonnegative kernels, J. Math. Anal. Appl., 335 (2007), pp. 298-313.
30. P. H. A. NGOC, Strong stability radii of positive linear time-delay systems, Internat. J. Robust Nonlinear Control, 15 (2005), pp. 459-472.
31. P. H. A. NGOC, A Perron-Frobenius for a class of positive quasi-polynomial matrices, Appl. Math. Lett., 19 (2006), pp. 747-751.
32. P. H. A. NGOC and B. S. LEE, A characterization of spectral abscissa and Perron-Frobenius theorem of positive linear functional differential equations, IMA J. Math. Control Inform., 23 (2006), pp. 259-268.
33. P. H. A. NGOC and N. K. SON, Stability radii of positive linear difference equations under affne parameter perturbations, Appl. Math. Comput., 134 (2003), pp. 577-594.
34. P. H. A. NGOC and N. K. SON, Stability radii of linear systems under multi-perturbations, Numer. Funct. Anal. Optim., 25 (2004), pp. 221-238.
35. P. H. A. NGOC and N. K. SON, Stability radii of positive linear functional differential equations under multi-perturbations, SIAM J. Control Optim., 43 (2005), pp. 2278-2295.
36. P. H. A. NGOC, B. S. LEE, and N. K. SON, Perron-Frobenius theorem for positive polynomial matrices, Vietnam J. Math., 32 (2004), pp. 475-481.
37. P. H. A. NGOC, T. NAITO, and J. S. SHIN, Characterizations of positive linear functional differential equations, Funkcial. Ekvac., 50 (2007), pp. 1-17.
38. P. H. A. NGOC, T. NAITO, and J. S. SHIN, On stability of a class of positive linear functional difference equations, Math. Control Signals Systems, 19 (2007), pp. 361-382.
39. A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983.
40. L. QIU, B. BERNHARDSSON, A. RANTZER, E. J. DAVISON, P. M. YOUNG, and J. C. DOYLE, A formula for computation of the real stability radius, Automatica, 31 (1995), pp. 879-890.
41. W. RUDIN, Real and Complex Analysis, McGraw-Hill, New York, 1987.
42. S. M. RUMP, Theorems of Perron-Frobenius type for matrices without sign restrictions, Linear Algebra Appl., 266 (1997), pp. 1-42.
43. B. SHAFAI, J. CHEN, and M. KOTHANDARAMAN, Explicit formulas for stability radii of nonnegative and Metzlerian matrices, IEEE Trans. Automat. Control, 42 (1997), pp. 265-269.
44. N. K. SON and D. HINRICHSEN, Robust stability of positive continuous time systems, Numer. Funct. Anal. Optim., 17 (1996), pp. 649-659.
45. N. K. SON and P. H. A. NGOC, Robust stability of positive linear time delay systems under affne parameter perturbations, Acta Math. Vietnam., 24 (1999), pp. 353-372.
46. N. K. SON and P. H. A. NGOC, Robust stability of linear functional differential equations, Adv. Stud. Contemp. Math., 3 (2001), pp. 43-59.
47. B. ZHANG, Asymptotic stability criteria and integrability properties of the resolvent of Volterra and functional equations, Funkcial. Ekvac., 40 (1997), pp. 335-351.
48. B. ZHANG, Necessary and suffcient conditions for stability in Volterra equations of nonconvolution type, Dynam. Systems Appl., 14 (2005), pp. 525-549.