zbMATH — the first resource for mathematics

Stability of discontinuous Cauchy problems in Banach space. (English) Zbl 1110.34036
The authors present stability results on a class of discontinuous dynamical systems in Banach spaces or Cauchy problems in abstract spaces. The results are applied in the theory of functional-differential equations, Volterra integro-differential equations and PDEs.

34G20 Nonlinear differential equations in abstract spaces
34K30 Functional-differential equations in abstract spaces
Full Text: DOI
[1] Zubov, V.I., Methods of A.M. Lyapunov and their applications, (1964), P. Noordhoff, Ltd. Groningen, The Netherlands · Zbl 0115.30204
[2] Hahn, W., Stability of motion, (1967), Springer-Verlag Berlin, Germany · Zbl 0189.38503
[3] Michel, A.N.; Wang, K.; Hu, B., Qualitative analysis of dynamical systems, (2001), Marcel Dekker New York
[4] Ye, H.; Michel, A.N.; Hou, L., Stability theory for hybrid dynamical systems, IEEE transactions on automatic control, 43, 4, 461-474, (1998) · Zbl 0905.93024
[5] Branicky, M.S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE transactions on automatic control, 43, 4, 475-482, (1998) · Zbl 0904.93036
[6] Michel, A.N., Recent trends in the stability analysis of hybrid dynamical systems, IEEE transactions on circuits and systems—I: fundamental theory and applications, 46, 1, 120-134, (1999) · Zbl 0981.93055
[7] Michel, A.N.; Hu, B., Towards a stability theory of general hybrid dynamical systems, Automatica, 35, 371-384, (1999) · Zbl 0936.93040
[8] Liberzon, D.; Morse, A.S., Basic problems in stability and design of switched systems, IEEE control systems magazine, 19, 5, 59-70, (1999) · Zbl 1384.93064
[9] DeCarlo, R.; Branicky, M.; Pettersson, S.; Lennartson, B., Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE, 88, 7, 1069-1082, (2000)
[10] Bainov, D.D.; Simeonov, P.S., Systems with impulsive effects: stability, theory and applications, (1989), Halsted Press New York · Zbl 0676.34035
[11] Krasovskii, N.N., Stability of motion, (1963), Stanford University Press Stanford, California · Zbl 0109.06001
[12] Yoshizawa, T., Stability theory by lyapunov’s second method, (1966), The Mathematical Society of Japan Tokyo, Japan · Zbl 0144.10802
[13] Hale, J.K., Functional differential equations, (1971), Springer-Verlag Berlin · Zbl 0213.36901
[14] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Inc. Cambridge, MA · Zbl 0777.34002
[15] E. Hille, R.S. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc. Colloquium Publ., vol. 33, Amer. Math. Soc., Providence, RI, 1957 · Zbl 0078.10004
[16] Massera, J.L., Contributions to stability theory, Annals of mathematics, 64, 182-206, (1956) · Zbl 0070.31003
[17] Melnikova, I.V.; Filinkov, A., Abstract Cauchy problems, (2000), Chapman & Hall/CRC New York · Zbl 1011.34048
[18] Haberman, R., Elementary applied partial differential equations: with Fourier series and boundary value problems, (1998), Prentice-Hall New Jersey · Zbl 0949.35001
[19] Neerven, J.V., The asymptotic behaviour of semigroups of linear operators, (1996), Birkhäsuser Verlag New York
[20] Michel, A.N.; Sun, Y., Stability analysis of discontinuous dynamical systems determined by semigroups, IEEE transactions on automatic control, 50, 9, 1277-1290, (2005) · Zbl 1365.34104
[21] Sun, Y.; Michel, A.N., Stability of discontinuous retarded functional differential equations with applications, IEEE transactions on automatic control, 50, 8, 1090-1105, (2006) · Zbl 1365.34127
[22] Godunov, A.N., Peano’s theorem in Banach spaces, Functional analysis and its applications, 9, 53-56, (1975) · Zbl 0314.34059
[23] Dieudonné, J., Foundations of modern analysis, (1960), Academic Press New York · Zbl 0100.04201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.