On semicontinuity in impulsive dynamical systems

• Autorzy: Ciesielski K.
• Języki publikacji: EN

Abstrakty

EN

In the important paper on impulsive systems [Kl] several notions are introduced and several properties of these systems are shown. In particular, the function [phi] which describes "the time of reaching impulse points" is considered; this function has many important applications. In [Kl] the continuity of this function is investigated. However, contrary to the theorem stated there, the function [phi] need not be continuous under the assumptions given in the theorem. Suitable examples are shown in this paper. We characterize the function [phi] from the point of view of its semicontinuity. Also, we show the analogous properties for impulsive systems given by semidynamical systems. In the last section we investigate the continuity properties of the escape time function in impulsive systems.

Słowa kluczowe

EN

dynamical system; flow; impulse; continuity; semicontinuity; section

PL

układ dynamiczny; przepływ; impuls; ciągłość; półciągłość; podział

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2004
• Tom: Vol. 52, nr 1
• Strony: 71--80
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Ciesielski K.

• Mathematics Institute Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland,

Bibliografia

• [BH] N. P. Bhatia and O. Hajek, Local Semi-Dynamical Systems, Lecture Notes In Math. 90, Springer, 1970.
• [BS] N. P. Bhatia and G. P. Szegő, Stability Theory of Dynamical Systems, Springer, 2002.
• [C1] K. Ciesielski, Sections in semidynamical systems, Bull. Polish Acad. Sci. Math. 40 (1992), 297-307.
• [C2] -, On isomorphisms of impulsive dynamical systems, to be published.
• [K1] S. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl. 150 (1990), 120-128.
• [K2] -, On impulsive semidynamical systems II, Recursive properties, Nonlinear Anal. 16 (1991), 635-645.
• [LBS] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Sci., 1989.
• [NS] V. V. Nemytskiĭ and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton Univ. Press, 1960.
• [P1] A. Pelczar, General Dynamical Systems, Jagiellonian Univ. Lecture Notes 293, 1978 (in Polish).
• [P2] -, Introduction to the Theory of Differential Equations; Part II (in Polish), Bibl. Mat. 67, PWN-Polish Sci. Publ., 1989.
• [V] J. de Vries, Elements of Topological Dynamics, Math. Appl. 257, Kluwer, 1993.