THE LIMITS OF APPLICABILITY OF THE LINEARIZATION METHOD IN CALCULATING SMALL–TIME REACHABLE SETS

The reachable sets of nonlinear systems are usually quite complicated. They, as a rule, are non-convex and arranged to have rather complex behavior. In this paper, the asymptotic behavior of reachable sets of nonlinear control-affine systems on small time intervals is studied. We assume that the initial state of the system is fixed, and the control is bounded in the $\mathbb{L}_2$-norm. The subject of the study is the applicability of the linearization method for a sufficiently small length of the time interval. We provide sufficient conditions under which the reachable set of a nonlinear system is convex and asymptotically equal to the reachable set of a linearized system. The concept of asymptotic equality is defined in terms of the Banach-Mazur metric in the space of sets.  The conditions depend on the behavior of the controllability Gramian of the linearized system – the smallest eigenvalue of the Gramian should not tend to zero too quickly when the length of the time interval tends to zero.  The indicated asymptotic behavior occurs for a reasonably wide class of second-order nonlinear control systems but can be violated for systems of higher dimension.  The results of numerical simulation illustrate the theoretical conclusions of the paper

On the possibilities of large-scale radio and fiber optics detectors in cosmic rays

Different variants of radio and fiber optics detectors for registration of super high energy cascades in the atmosphere and in dense media are discussed. Particularly the possibilities for investigation of quasi horizontal cosmic ray showers (CRS) and simulated muons from these CRS with the help of radio detectors and fiber optics detectors located on the ice surface are considered

Calculation of resonances in the Coulomb three-body system with two disintegration channels in the adiabatic hyperspherical approach

The method of calculation of the resonance characteristics is developed for the metastable states of the Coulomb three-body (CTB) system with two disintegration channels. The energy dependence of K-matrix in the resonance region is calculated with the use of the stabilization method. Resonance position and partial widths are obtained by fitting the numerically calculated K(E)-matrix with the help of the generalized Breit-Wigner formula.Comment: Latex, 11 pages with 5 figures and 2 table

High order fractional microwave induced resistance oscillations in 2D systems

We report on the observation of microwave-induced resistance oscillations associated with the fractional ratio n/m of the microwave irradiation frequency to the cyclotron frequency for m up to 8 in a two-dimensional electron system with high electron density. The features are quenched at high microwave frequencies independent of the fractional order m. We analyze temperature, power, and frequency dependencies of the magnetoresistance oscillations and discuss them in connection with existing theories.Comment: 5 pages, 5 figure

Non-linear transport phenomena in a two-subband system

We study non-linear transport phenomena in a high-mobility bilayer system with two closely spaced populated electronic subbands in a perpendicular magnetic field. For a moderate direct current excitation, we observe zero-differential-resistance states with a characteristic 1/B periodicity. We investigate, both experimentally and theoretically, the Hall field-induced resistance oscillations which modulate the high-frequency magneto-intersubband oscillations in our system if we increase the current. We also observe and describe the influence of direct current on the magnetoresistance in the presence of microwave irradiation.Comment: 8 pages, 6 figure

Microwave-induced Hall resistance in bilayer electron systems

The influence of microwave irradiation on dissipative and Hall resistance in high-quality bilayer electron systems is investigated experimentally. We observe a deviation from odd symmetry under magnetic field reversal in the microwave-induced Hall resistance $\Delta R_{xy}$ whereas the dissipative resistance $\Delta R_{xx}$ obeys even symmetry. Studies of $\Delta R_{xy}$ as a function of the microwave electric field and polarization exhibit a strong and non-trivial power and polarization dependence. The obtained results are discussed in connection to existing theoretical models of microwave-induced photoconductivity.Comment: 8 pages, 5 figure