Abstract
Using the Fourier method of separation of variables and a procedure proposed in this paper, namely, reducing integrodifferential equations to systems of ordinary differential equations, the exponential stability of partial functional integro-differential equations is studied. Various tests for the exponential stability are proposed. In contrast to many other known methods our approach does not assume the smallness of integral terms. This allows us to use the method for stabilization of processes described by unstable differential equations by adding controls in the form of integral terms. Finally, using our approach, a phase transition model is analyzed.
Similar content being viewed by others
References
1. N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the theory of functional-differential equations [in Russian]. Nauka, Moscow (1991).
2. T. A. Burton, Volterra integral and differential equations. Academic Press, New York (1983).
3. A. D. Brjuno, Local method of nonlinear analysis of differential equations [in Russian]. Nauka, Moscow (1979).
4. C. Corduneanu, Integral equations and stability of feedback systems. Academic Press, New York (1973).
5. C. Corduneanu, Some problems concerning partial stability. Sympos. Math. 6 (1971), 141–154.
6. C. Corduneanu, Integral equations and applications. Cambridge Univ. Press, Cambridge (1991).
7. B. P. Demidovich, Lectures on the mathematical theory of stability [in Russian]. Nauka, Moscow (1970).
8. A. Domoshnitsky, New concept in the study of differential inequalities. Funct. Differ. Equ. 1 (1993), 52–59.
9. A. Domoshnitsky, About asymptotic and oscillation properties of the Dirichlet problem for delay partial differential equations. Georgian Math. J. 10 (2003), 495–502.
10. A. Domoshnitsky and Ya. Goltser, An approach to study bifurcations and stability of integro-differential equations. Math. Comput. Modelling 36 (2002), 663–678.
11. A. D. Drozdov, Stability of integro-differential equations with periodic operator coefficients. Quart. J. Mech. Appl. Math. 49 (1996), 235–260.
12. A. D. Drozdov, Explicit stability conditions for integro-differential equations with periodic coefficients, Math. Methods Appl. Sci. 21 (1998), 565–588.
13. A. D. Drozdov and M. I. Gil, Stability of linear integro-differential equations with periodic coefficients. Quart. Appl. Math. 54 (1996), 609–624.
14. A. D. Drozdov and V. B. Kolmanovskii, Stability in viscoelasticity. North Holland, Amsterdam (1994).
15. M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity. SIAM Stud. Appl. Math., Philadelphia (1992).
16. J. D. Ferry, Viscoelastic properties of polymers. Wiley, New York (1970).
17. J. M. Golden and G. A. C. Graham, Boundary value problems in linear viscoelasticity. Springer-Verlag, Berlin (1988).
18. Ya. Goltser and A. Domoshnitsky, Bifurcation and stability of integrodifferential equations. Nonlinear Anal. 47 (2001), 953–967.
19. M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31 (1968), 113–126.
20. G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations. Cambridge Univ. Press, Cambridge (1990).
21. J. K. Hale and V. Lunel, Introduction to functional differential equations. Springer-Verlag, New York (1993).
22. T. Hara and R. Miyazaki, Equivalent condition for stability of Volterra integro-differential equations. J. Math. Anal. Appl. 174 (1993), 298–316.
23. M. I. Imanaliev et. al, Integral equations. Differ. Uravn. 18 (1982), 2050–2069.
24. D. D. Joseph and L. Preziosi, Heat waves. Rev. Modern Phys. 61 (1989) 41–73.
25. A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis [in Russian]. Nauka, Moscow (1972).
26. M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii, and V. Ya. Stezenko, Approximate methods for solving operator equations [in Russian]. Nauka, Moscow (1969).
27. N. N. Krasovskii, Stability of motion. Stanford Univ. Press, Stanford (1963).
28. A. Novick-Cohen, Conserved phase-field equations with memory. In: Curvature flows and related topics (A. Damlamian, J. Spruck, and A. Visintin, eds.), GAKUTO Int. Ser. Math. Sci. Appl. 5, Gakkotosho, Tokyo (1995), pp. 179–197.
29. A. Ponosov, A. Shindiapin, and J. Miguel, The W-transform links delay and ordinary differential equations. Funct. Differ. Equ. 9 (2002), 437–470.
30. L. S. Pontryagin, Ordinary differential equations [in Russian]. Nauka, Moscow (1982).
31. J. Pruss, Evolutionary integral equations and applications. Monogr. Math. 87 (1993).
32. M. Renardy, W. J. Hrusa, and J. A. Nohel, Mathematical problems in viscoelasticity. Longman, New York (1987).
33. H. G. Rotstein, S. Brandon, A. Novick-Cohen, and A. Nepomnyashchy, Phase field equations with memory: the hyperbolic case. SIAM J. Appl. Math. 62 (2001), 264–282.
34. N. Roushe, P. Haberts, and M. Laloy, Stability theory by Lyapunov's direct method. Springer-Verlag, New York (1977).
35. V. V. Rumyantsev and A. S. Oziraner, Stability and stabilization of motion with respect to part of variables [in Russian]. Nauka, Moscow (1987).
36. G. E. Shilov, Mathematical analysis. Specialized course [in Russian]. Fizmatgiz, Moscow (1965).
37. M. M. Vainberg and V. A. Trenogin, Branching theory of solutions of nonlinear equations [in Russian]. Nauka, Moscow (1969).
38. J. H. Wu, Theory and applications of partial functional differential equations. Springer-Verlag, New York (1996).
Author information
Authors and Affiliations
Corresponding authors
Additional information
2000 Mathematics Subject Classification. 34K15, 35B05.
This research was supported by the program KAMEA of the Ministry of Absorption of the State of Israel.
Rights and permissions
About this article
Cite this article
Agarwal, R., Domoshnitsky, A. & Goltser, Y. Stability of Partial Functional Integro-Differential Equations. J Dyn Control Syst 12, 1–31 (2006). https://doi.org/10.1007/s10450-006-9681-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10450-006-9681-x