Hostname: page-component-f554764f5-68cz6 Total loading time: 0 Render date: 2025-04-14T17:40:15.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Differentiability of solutions of systems with impulse effect with respect to initial data and parameter

Published online by Cambridge University Press:  20 January 2009

P. S. Simeonov  and
D. D. Bainov
P. S. Simeonov
Affiliation:
Plovdiv University“Paissii Hilendarski”PlovdivBulgaria
D. D. Bainov
Affiliation:
Plovdiv University“Paissii Hilendarski”PlovdivBulgaria
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is characteristic for the development of many processes that in certain moments they change their state in jumps. Systems with impulse effect provide an adequate mathematical model of such processes. The investigation of these systems begins with the paper of Millman and Myshkis [7] and afterwards the number of publications dedicated to this problem rapidly increases.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 31 , Issue 3 , October 1988 , pp. 353 - 368
Copyright
Copyright © Edinburgh Mathematical Society 1988

References

REFERENCES

1.Alekseev, V. M., An estimate for perturbations of solutions of ordinary differential equations, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (1961), 28–36.Google Scholar
2.Batnov, D. D. and Simeonov, P. S., Integral and differential inequalities for a class of piecewise continuous functions, Czechoslovak Mat. J. 36 (111) (1986), 417–426 (in Russian).Google Scholar
3.Bibikov, J. N., A General Course of Ordinary Differential Equations (Publ. House of Leningrad State University, 1981 (in Russian)).Google Scholar
4.Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations (McGraw-Hill, New York, Toronto, London, 1955).Google Scholar
5.Dishliev, A. B. and Bainov, D. D., Conditions for the absence of the phenomenon “beating” for systems of impulse differential equations, Bull. Inst. Math. Acad. Sinica 13 (1985), 237–256.Google Scholar
6.Lord, M. E. and Mitchell, A. R., A new approach to the method of nonlinear variation of parameters, J. Appl. Math. Comp., to appear.Google Scholar
7.Millman, V. D. and Myshkis, A. D., On the stability of motion in the presence of impulses, Siberian Math. J. 1 (1960), 233–237 (in Russian).Google Scholar
8.Myshkis, A. D. and Samoilenko, A. M., Systems with impulses at prescribed moments of the time, Matn. Sb. 74 (1967), 202–208 (in Russian).Google Scholar
9.Pandit, S. G. and Deo, S. G., Differential Systems Involving Impulses (Springer-Verlag, 1982).CrossRefGoogle Scholar
10.Pavlidis, T., Stability of systems described by differential equations containing impulses, IEEE Trans. 12 (1967), 43–45.Google Scholar
11.Raghavendra, V. and Rama Mohana Rao, M., On stability of differential systems with respect to impulsive perturbations, J. Math. Anal Appl. 48 (1974), 515–526.CrossRefGoogle Scholar
12.Samoilenko, A. M., Bainov, D. D. and Milusheva, S. D., On the application of the averaging method for systems of integro-differential equations of standard type with discontinuous right-hand side, Acta Math. Acad. Sci. Hung. 29 (1977), 31–48.CrossRefGoogle Scholar
13.Samoilenko, A. M. and Perestiuk, N. A., Stability of the solutions of differential equations with impulse effect, Differ. Uravn. 11 (1977), 1981–1992 (in Russian).Google Scholar
14.Samoilenko, A. M. and Perestiuk, N. A., On the stability of systems with impulse effect, Differ. Uravn. 11 (1981), 1995–2001 (in Russian).Google Scholar
15.Samoilenko, A. M., Perestiuk, N. A. and Ahmetov, M. U., Almost periodic solutions of impulse differential equations, Kiev Inst. Math. Acad. Sci. of Ukr. SSR (1983) (Preprint 83.26, in Russian).Google Scholar
16.Simeonov, P. S. and Bainov, D. D., Asymptotic equivalence of two systems of differential equations with impulse effect, Systems & Control Lett. 3 (1983), 297–301.CrossRefGoogle Scholar
17.Simeonov, P. S. and Bainov, D. D., Stability under persistent disturbances for systems with impulse effect, J. Math. Anal. Appl. 109 (1985), 546–563.CrossRefGoogle Scholar
18.Simeonov, P. S. and Bainov, D. D., The second method of Liapunov for systems with an impulse effect, Tamkang J. Math. 16 (1985), 19–40.Google Scholar
19.Simeonov, P. S. and Bainov, D. D., Stability with respect to part of the variables in systems with impulse effect, J. Math. Anal. Appl. 117 (1986), 247–263.CrossRefGoogle Scholar
20.Sree Hari Rao, V., On boundedness of impulsively perturbed systems, Bull. Austral. Math. Soc. 18 (1978), 237–242.Google Scholar