# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Stability of impulsive control systems with time delay. (English) Zbl 1081.93021
The following time delay impulsive control system is considered $x(t)=Ax(t)+Bx(t-r)$, $t\ne\tau_k$, $x(t^+)=x(t^-)+C_kx(t^-)$, $t= \tau_k$, where $A,B,C_k$, $k=1,2$, are some constant matrices, $r>0$ is a delay constant, $\{\tau_k,C_k(x(\tau_k^-))$, $k=1,2,\dots\}$ denotes the impulsive control law with $0<\tau_1<\tau_2<\cdots<\tau_k <\tau_{k+1}<\dots$, $\tau_k\to\infty$, as $k\to\infty$; $x(t^+)= \lim_{t\to\tau_k^+}x(t)$, $x(t^-)=\lim_{t\to\tau_k^-} x(t)$. Several criteria on asymptotic stability are established using the method of Lyapunov functions. It is shown that system can be stabilized even if it contains no stable matrix $A$.

##### MSC:
 93D20 Asymptotic stability of control systems 93D30 Scalar and vector Lyapunov functions 34K20 Stability theory of functional-differential equations 34K45 Functional-differential equations with impulses
##### Keywords:
asymptotic stability; Lyapunov functions
Full Text:
##### References:
 [1] Leela, S.; Mcrae, F. A.; Sivasundaram, S.: Controllability of impulsive differential equations. J. math. Anal. appl. 177, 24-30 (1993) · Zbl 0785.93016 [2] Liu, X.; Ballinger, G.: Uniform asymptotic stability of impulsive differential equations. Computers math. Applic. 41, No. 7/8, 903-915 (2001) · Zbl 0989.34061 [3] Liu, X.; Shen, J. H.: Razumikhin-type theorems on boundedness for impulsive functional differential equations. Dynamic systems & appl. 9, 389-404 (2000) · Zbl 0971.34059 [4] Liu, X.: Impulsive stabilization and applications to population growth models. Rocky mountain J. Math 25, 381-395 (1995) · Zbl 0832.34039 [5] Liu, X.; Rohlf, K.: Impulsive control of Lotka-Volterra models. IMA J. Math. control infor. 15, 269-284 (1998) · Zbl 0949.93069 [6] Liu, X.; Willms, A.: Impulsive stabilization. Comparison methods and stability theory, 269-276 (1994) · Zbl 0812.93054 [7] Liu, X.; Willms, A.: Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft. Math. problems in engineering 2, 277-299 (1996) · Zbl 0876.93014 [8] Liu, Y.; Teo, K. L.; Jennings, L. S.; Wang, S.: On a class of optimal control problems with state jumps. J. opti. Theo. appl. 98, 65-82 (1998) · Zbl 0908.49023 [9] Prussing, J. E.; Wellnitz, L. J.: Optimal impulsive time-fixed direct-ascent interaction. J. guidance control dynam. 12, 487-494 (1989) [10] Yang, T.: Impulsive control. IEEE tran. Automatic contr. 44, 1081-1083 (1999) · Zbl 0954.49022 [11] Ballinger, G.; Liu, X.: On boundedness of solutions of impulsive systems. Nonlinear studies 4, No. 1, 121-131 (1997) · Zbl 0879.34015 [12] Ballinger, G.; Liu, X.: Permanence of population growth models with impulsive effects. Mathl. comput. Modelling 26, No. 12, 59-72 (1997) · Zbl 1185.34014 [13] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002 [14] Lakshmikantham, V.; Liu, X.: Stability criteria for impulsive differential equations in terms of two measures. J. math. Anal. appl. 137, 591-604 (1989) · Zbl 0688.34031 [15] Ballinger, G.; Liu, X.: Existence and uniqueness results for impulsive delay differential equations. Dcdis 5, 579-591 (1999) · Zbl 0955.34068 [16] Ballinger, G.; Liu, X.: Existence, uniqueness and boundedness results for impulsive delay differential equations. Appl. anal. 74, 71-93 (2000) · Zbl 1031.34081 [17] Horn, R. A.; Johnson, C. R.: Matrix analysis. (1985) · Zbl 0576.15001 [18] Hale, J. K.; Lunel, S. M. V: Introduction to functional differential equations. (1993) · Zbl 0787.34002