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Stability analysis of set trajectories for families of impulsive equations. (English) Zbl 1417.34117
Authors’ abstract: “In this paper, for a family of impulsive equations, a heterogeneous matrix-valued Lyapunov-like function is considered, the comparison principle is formulated, and stability conditions for the set of stationary solutions are established. In addition, for a class of impulsive equations with uncertain parameters the monotone iterative technique for constructing a set of solutions is adapted.”
In Section 6 ‘Notes and comments’, the authors emphasize the contribution of the following references: [the first author, Russ. Acad. Sci., Dokl., Math. 50, No. 2, 1 (1994; Zbl 0864.34007); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 338, No. 6, 728–730 (1994)], [M. Z. Djordjevic, Large Scale Syst. 5, 255–262 (1983; Zbl 0538.93048)], [J. Vasundhara Devi and A. S. Vatsala, Nonlinear Stud. 11, No. 4, 639–658 (2004; Zbl 1080.34004)] and [G. S. Ladde et al., Monotone iterative techniques for nonlinear differential equations. Boston-London: Pitman (Advanced Publishing Program); New York: John Wiley & Sons, Inc. (1985; Zbl 0658.35003)].
MSC:
34D20 Stability of solutions to ordinary differential equations
34A37 Ordinary differential equations with impulses
34A45 Theoretical approximation of solutions to ordinary differential equations
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[1] Martynyuk, AA, Stability of motion. The role of multicomponent Liapunov’s functions, (2007), Cambridge: Cambridge Scientific Publishers, Cambridge
[2] Martynyuk, AA, Elements of the stability theory of motions for hybrid systems (Review), Prikl Mekh, 51, 3-66, (2015)
[3] Bainov, DD; Simeonov, PS, Systems with impulsive effect: stability, theory and applications, (1989), Chichester: Ellis Horwood, Chichester
[4] Lakshmikantham, V.; Bhaskar, TG; Vasundhara Devi, J., Theory of set differential equations in metric spaces, (2006), Cambridge: Cambridge Scientific, Cambridge · Zbl 1156.34003
[5] Akhmetov, MU; Zafer, A., Stability of zero solution of impulsive differential equations by the Lyapunov second method, J Math Anal Appl, 248, 69-82, (2000) · Zbl 0965.34007
[6] Martynyuk-Chernienko, YuA, On the stability of motion in impulsive systems with uncertain parameter values, Dokl Akad Nauk, 395, 1-4, (2004)
[7] Milman, VD; Myshkis, AD, On the stability of motion in the presence of impulses, Siberian Math J, 1, 233-237, (1960)
[8] Lakshmikantham, V.; Bainov, DD; Simeonov, PS, Theory of impulse differential equations, (1989), Singapore: World Scientific, Singapore
[9] Stamova, IM, Stability analysis of impulsive functional differential equations, (2009), Berlin: Walter de Gruyter, Berlin · Zbl 1189.34001
[10] Stamova, IM; Stamov, GT, Applied impulsive mathematical models, CMS books in mathematics, (2016), Cham: Springer, Cham
[11] Martynyuk, AA, Lyapunov matrix functions and stability with respect to two measures of impulsive systems, Dokl Akad Nauk, 338, 728-730, (1994)
[12] Djordjevic, MZ, Stability analysis of large scale systems whose subsystems may be unstable, Large Scale Syst, 5, 255-262, (1983) · Zbl 0538.93048
[13] Vasundhara Devi, J.; Vatsala, AS, Monotone iterative technique for impulsive and set differential equations, Nonlinear Stud, 11, 639-658, (2004) · Zbl 1080.34004
[14] Ladde, GS; Lakshmikantham, V.; Vatsala, AS, Monotone iterative techniques for nonlinear differential equations, (1985), Boston (MA): Pitman, Boston (MA)
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