United classification of cosmic gamma-ray bursts and their counterparts

United classification of gamma-ray bursts and their counterparts is established on the basis of measured characteristics: photon energy E and emission duration T. The founded interrelation between the mentioned characteristics of events consists in that, as the energy increases, the duration decreases (and vice versa). The given interrelation reflects the nature of the phenomenon and forms the E-T diagram, which represents a natural classification of all observed events in the energy range from 10E9 to 10E-6 eV and in the corresponding interval of durations from about 10E-2 up to 10E8 s. The proposed classification results in the consequences, which are principal for the theory and practical study of the phenomenon.Comment: Keywords Gamma rays: burst

Robustness in the graph topology of a common adaptive controller

For any m-input, m-output, finite-dimensional, linear, minimum-phase plant P with first Markov parameter having spectrum in the open right-half complex plane, it is well known that the adaptive output feedback control C, given by u = -ky, k = ||y||2, yields a closed-loop system [P,C] for which the state converges to zero, the signal k converges to a finite limit, and all other signals are of class L2. It is first shown that these properties continue to hold in the presence of L2-input and L2-output disturbances. By establishing gain function stability of an appropriate closed-loop operator, it is proved that these properties also persist when the plant P is replaced by a stabilizable and detectable linear plant P1 within a sufficiently small neighbourhood of P in the graph topology, provided that the plant initial data and the L2 magnitude of the disturbances are sufficiently small. Example 9 of Georgiou & Smith (IEEE Trans. Autom. Control 42(9) 1200-1221, 1997) is revisited to which the above L2-robustness result applies. Unstable behaviour for large initial conditions and/or large L2 disturbances is shown, demonstrating that the bounds obtained from the L2 theory are qualitatively tight: this contrasts with the L∞-robustness analysis of Georgiou & Smith which is insufficiently tight to predict the stable behaviour for small initial conditions and zero disturbances