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

Magnetic field induced transition in a wide parabolic well superimposed with superlattice

We study a $Al_{x}Ga_{x-1}As$ parabolic quantum wells (PQW) with $GaAs/Al_{x}Ga_{x-1}As$ square superlattice. The magnetotransport in PQW with intentionally disordered short-period superlattice reveals a surprising transition from electrons distribution over whole parabolic well to independent-layer states with unequal density. The transition occurs in the perpendicular magnetic field at Landau filling factor $\nu\approx3$ and is signaled by the appearance of the strong and developing fractional quantum Hall (FQH) states and by the enhanced slope of the Hall resistance. We attribute the transition to the possible electron localization in the x-y plane inside the lateral wells, and formation of the FQH states in the central well of the superlattice, driven by electron-electron interaction.Comment: 5 pages, 4 figure

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

Time-domain Brillouin Scattering as a Local Temperature Probe in Liquids

We present results of time-domain Brillouin scattering (TDBS) to determine the local temperature of liquids in contact to an optical transducer. TDBS is based on an ultrafast pump-probe technique to determine the light scattering frequency shift caused by the propagation of coherent acoustic waves in a sample. Since the temperature influences the Brillouin scattering frequency shift, the TDBS signal probes the local temperature of the liquid. Results for the extracted Brillouin scattering frequencies recorded at different liquid temperatures and at different laser powers - i.e. different steady state background temperatures- are shown to demonstrate the usefulness of TDBS as a temperature probe. This TDBS experimental scheme is a first step towards the investigation of ultrathin liquids measured by GHz ultrasonic probing.Comment: arXiv admin note: substantial text overlap with arXiv:1702.0107

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