On Semicontinuity in Impulsive Dynamical Systems

Krzysztof Ciesielski¹

¹ Mathematics Institute Jagiellonian University Reymonta 4 30-059 Kraków, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) no. 1, pp. 71-80.

Résumé

In the important paper on impulsive systems [K1] 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 [K1] 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.

Détail

DOI : 10.4064/ba52-1-8

Keywords: important paper impulsive systems several notions introduced several properties these systems shown particular function phi which describes time reaching impulse points considered function has many important applications continuity function investigated however contrary theorem stated there function phi continuous under assumptions given theorem suitable examples shown paper characterize function phi point view its semicontinuity analogous properties impulsive systems given semidynamical systems section investigate continuity properties escape time function impulsive systems