Lyapunov functions and comparison theorems for control systems with impulsive actions
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 1, pp. 51-59
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We extend the results of E. L. Tonkov and E. A. Panasenko to differential equations and control systems with impulsive actions. In terms of Lyapunov functions and the Clarke derivative we obtain comparison theorems for systems with impulsive effect. We consider the set $\mathfrak M\doteq\bigl\{(t,x)\in[t_0,+\infty)\times\mathbb R^n\colon x\in M(t)\bigr\}$, defined by continuous function $t\to M(t)$, where for every $t\in\mathbb R$ the set $M(t)$ is nonempty and compact. We obtain conditions for the positive invariance of this set, the uniform Lyapunov stability and the uniform asymptotic stability. We make a comparison with the researches of other authors who have considered the zero solution stability for similar systems.
Keywords:
control systems with impulsive actions, Lyapunov function, differential inclusions.
@article{VUU_2015_25_1_a5,
author = {Ya. Yu. Larina},
title = {Lyapunov functions and comparison theorems for control systems with impulsive actions},
journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
pages = {51--59},
publisher = {mathdoc},
volume = {25},
number = {1},
year = {2015},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VUU_2015_25_1_a5/}
}
TY - JOUR AU - Ya. Yu. Larina TI - Lyapunov functions and comparison theorems for control systems with impulsive actions JO - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki PY - 2015 SP - 51 EP - 59 VL - 25 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/VUU_2015_25_1_a5/ LA - ru ID - VUU_2015_25_1_a5 ER -
%0 Journal Article %A Ya. Yu. Larina %T Lyapunov functions and comparison theorems for control systems with impulsive actions %J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki %D 2015 %P 51-59 %V 25 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/item/VUU_2015_25_1_a5/ %G ru %F VUU_2015_25_1_a5
Ya. Yu. Larina. Lyapunov functions and comparison theorems for control systems with impulsive actions. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 1, pp. 51-59. http://geodesic.mathdoc.fr/item/VUU_2015_25_1_a5/