Lyapunov functions for stable cascades and applications to global stabilization

This paper generalizes recent Lyapunov constructions for a cascade of two nonlinear systems, one of which is stable rather than asymptotically stable. A new cross-term construction in the Lyapunov function allows us to replace earlier growth conditions by a necessary boundedness condition. This method is instrumental in the global stabilization of feedforward systems, and new stabilization results are derived from the generalized construction.

A differential lyapunov framework for contraction analysis

Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves. © 2013 IEEE.

A derivation of the source-channel error exponent using nonidentical product distributions

This paper studies the random-coding exponent of joint source-channel coding for a scheme where source messages are assigned to disjoint subsets (referred to as classes), and codewords are independently generated according to a distribution that depends on the class index of the source message. For discrete memoryless systems, two optimally chosen classes and product distributions are found to be sufficient to attain the sphere-packing exponent in those cases where it is tight. © 2014 IEEE.

Continuous spectrum growth and modal instability in swirling duct flow

We present results on the stability of compressible inviscid swirling flows in an annular duct. Such flows are present in aeroengines, for example in the by-pass duct, and there are also similar flows in many aeroacoustic or aeronautical applications. The linearised Euler equations have a ('critical layer') singularity associated with pure convection of the unsteady disturbance by the mean flow, and we focus our attention on this region of the spectrum. By considering the critical layer singularity, we identify the continuous spectrum of the problem and describe how it contributes to the unsteady field. We find a very generic family of instability modes near to the continuous spectrum, whose eigenvalue wavenumbers form an infinite set and accumulate to a point in the complex plane. We study this accumulation process asymptotically, and find conditions on the flow to support such instabilities. It is also found that the continuous spectrum can cause a new type of instability, leading to algebraic growth with an exponent determined by the mean flow, given in the analysis. The exponent of algebraic growth can be arbitrarily large. Numerical demonstrations of the continuous spectrum instability, and also the modal instabilities are presented.

Influence of particle properties on the erosive wear of sintered boron carbide

Sintered boron carbide is very hard, and can be an attractive material for wear-resistant components in critical applications. Previous studies of the erosion of less hard ceramics have shown that their wear resistance depends on the nature of the abrasive particles. Erosion tests were performed on a sintered boron carbide ceramic with silica, alumina and silicon carbide erodents. The different erodents caused different mechanisms of erosion, either by lateral cracking or small-scale chipping; the relative values of the hardness of the erodent and the target governed the operative mechanism. The small-scale chipping mechanism led to erosion rates typically an order of magnitude lower than the lateral fracture mechanism. The velocity exponents for erosion in the systems tested were similar to those seen in other work, except that measured with the 125 to 150 μm silica erodent. With this erodent the exponent was initially high, then decreased sharply with increasing velocity and became negative. It was proposed that this was due to deformation and fragmentation of the erodent particles. In the erosion testing of ceramics, the operative erosion mechanism is important. Care must be taken to ensure that the same mechanism is observed in laboratory testing as that which would be seen under service conditions, where the most common erodent is silica.

Strength mobilization in clays and silts

A large database of 115 triaxial, direct simple shear, and cyclic tests on 19 clays and silts is presented and analysed to develop an empirical framework for the prediction of the mobilization of the undrained shear strength, cu, of natural clays tested from an initially isotropic state of stress. The strain at half the peak undrained strength (γM=2) is used to normalize the shear strain data between mobilized strengths of 0.2cu and 0.8cu. A power law with an exponent of 0.6 is found to describe all the normalized data within a strain factor of 1.75 when a representative sample provides a value for γM=2. Multi-linear regression analysis shows that γM=2 is a function of cu, plasticity index Ip, and initial mean effective stress p′0. Of the 97 stress-strain curves for which cu, Ip, and p′0 were available, the observed values of γM=2 fell within a factor of three of the regression; this additional uncertainty should be acknowledged if a designer wished to limit immediate foundation settlements on the basis of an undrained strength profile and the plasticity index of the clay. The influence of stress history is also discussed. The application of these stress-strain relations to serviceability design calculations is portrayed through a worked example. The implications for geotechnical decision-making and codes of practice are considered.

Turbulent flow structure in experimental laboratory wind-generated gravity waves

This paper is the third part of a report on systematic measurements and analyses of wind-generated water waves in a laboratory environment. The results of the measurements of the turbulent flow on the water side are presented here, the details of which include the turbulence structure, the correlation functions, and the length and velocity scales. It shows that the mean turbulent velocity profiles are logarithmic, and the flows are hydraulically rough. The friction velocity in the water boundary layer is an order of magnitude smaller than that in the wind boundary layer. The level of turbulence is enhanced immediately beneath the water surface due to micro-breaking, which reflects that the Reynolds shear stress is of the order u *w 2. The vertical velocities of the turbulence are related to the relevant velocity scale at the still-water level. The autocorrelation function in the vertical direction shows features of typical anisotropic turbulence comprising a large range of wavelengths. The ratio between the microscale and macroscale can be expressed as λ/Λ=a Re Λ n, with the exponent n slightly different from -1/2, which is the value when turbulence production and dissipation are in balance. On the basis of the wavelength and turbulent velocity, the free-surface flows in the present experiments fall into the wavy free-surface flow regime. The integral turbulent scale on the water side alone underestimates the degree of disturbance at the free surface. © 2012 Elsevier B.V.

Electrical transport and percolation in magnetoresistive manganite / insulating oxide composites: case of La0.7Ca0.3MnO3 / Mn3O4

We report the results of electrical resistivity measurements carried out on well-sintered La0.7Ca0.3MnO3 / Mn3O4 composite samples with almost constant composition of the magnetoresistive manganite phase (La0.7Ca0.3MnO3). A percolation threshold (fc) occurs when the La0.7Ca0.3MnO3 volume fraction is ~ 0.19. The dependence of the electrical resistivity as a function of La0.7Ca0.3MnO3 volume fraction (fLCMO) can be described by percolation-like phenomenological equations. Fitting the conducting regime (fLCMO > fc) by the percolation power law returns a critical exponent t value of 2.0 +/- 0.2 at room temperature and 2.6 +/-0.2 at 5 K. The increase of t is ascribed to the influence of the grain boundaries on the electrical conduction process at low temperature.

Shielding efficiency and E(J) characteristics measured on large melt cast Bi-2212 hollow cylinders in axial magnetic fields

We show that tubes of melt cast Bi-2212 used as current leads for LTS magnets can also act as efficient magnetic shields. The magnetic screening properties under an axial DC magnetic field are characterized at several temperatures below the liquid nitrogen temperature (77 K). Two main shielding properties are studied and compared with those of Bi-2223, a material that has been considered in the past for bulk magnetic shields. The first property is related to the maximum magnetic flux density that can be screened, Blim; it is defined as the applied magnetic flux density below which the field attenuation measured at the centre of the shield exceeds 1000. For a cylinder of Bi-2212 with a wall thickness of 5 mm and a large ratio of length over radius, Blim is evaluated to 1 T at T = 10 K. This value largely exceeds the Blim value measured at the same temperature on similar tubes of Bi-2223. The second shielding property that is characterized is the dependence of Blim with respect to variations of the sweep rate of the applied field, dBapp/dt. This dependence is interpreted in terms of the power law E = Ec(J/Jc)^n and allows us to determine the exponent n of this E(J) characteristics for Bi-2212. The characterization of the magnetic field relaxation involves very small values of the electric field. This gives us the opportunity to experimentally determine the E(J) law in an unexplored region of small electric fields. Combining these results with transport and AC shielding measurements, we construct a piecewise E(J) law that spans over 8 orders of magnitude of the electric field.

Magnetic shielding properties of high- Tc superconducting hollow cylinders: Model combining experimental data for axial and transverse magnetic field configurations

Magnetic shielding efficiency was measured on high- Tc superconducting hollow cylinders subjected to either an axial or a transverse magnetic field in a large range of field sweep rates, dBapp/dt. The behaviour of the superconductor was modelled in order to reproduce the main features of the field penetration curves by using a minimum number of free parameters suitable for both magnetic field orientations. The field penetration measurements were carried out on Pb-doped Bi-2223 tubes at 77K by applying linearly increasing magnetic fields with a constant sweep rate ranging between 10νTs-1 and 10mTs-1 for both directions of the applied magnetic field. The experimental curves of the internal field versus the applied field, Bin(Bapp), show that, at a given sweep rate, the magnetic field for which the penetration occurs, Blim, is lower for the transverse configuration than for the axial configuration. A power law dependence with large exponent, n′, is found between Blim and dBapp/dt. The values of n′ are nearly the same for both configurations. We show that the main features of the curves B in(Bapp) can be reproduced using a simple 2D model, based on the method of Brandt, involving a E(J) power law with an n-exponent and a field-dependent critical current density, Jc(B), (following the Kim model: Jc = Jc0(1+B/B1)-1). In particular, a linear relationship between the measured n′-exponents and the n-exponent of the E(J) power law is suggested by taking into account the field dependence of the critical current density. Differences between the axial and the transverse shielding properties can be simply attributed to demagnetizing fields. © 2009 IOP Publishing Ltd.

Creep and oxygen diffusion in magnetite

The creep rate of polycrystalline Fe3O4 has been measured as a fonction of stress and oxygen partial pressure in the temperature range 480-1100°C. A regime of power law creep is found at high stress, with a stress exponent of ≈- 3.1 and an activation energy of 264 kJ/mol. A regime of diffusional flow is found at lower stresses and is interpreted as Nabarro-Herring creep. The data for the two regimes are combined to deduce an oxygen diffusion coefficient of ≈-10-5 exp(-264 kJ/mol/RT) m2s-1, with oxygen vacancies suggested as the mobile species. © 1990.

Consensus in non-commutative spaces

Convergence analysis of consensus algorithms is revisited in the light of the Hilbert distance. The Lyapunov function used in the early analysis by Tsitsiklis is shown to be the Hilbert distance to consensus in log coordinates. Birkhoff theorem, which proves contraction of the Hilbert metric for any positive homogeneous monotone map, provides an early yet general convergence result for consensus algorithms. Because Birkhoff theorem holds in arbitrary cones, we extend consensus algorithms to the cone of positive definite matrices. The proposed generalization finds applications in the convergence analysis of quantum stochastic maps, which are a generalization of stochastic maps to non-commutative probability spaces. ©2010 IEEE.

Spatial models of bistability in biological collectives

We explore collective behavior in biological systems using a cooperative control framework. In particular, we study a hysteresis phenomenon in which a collective switches from circular to parallel motion under slow variation of the neighborhood size in which individuals tend to align with one another. In the case that the neighborhood radius is less than the circular motion radius, both circular and parallel motion can occur. We provide Lyapunov-based analysis of bistability of circular and parallel motion in a closed-loop system of self-propelled particles with coupled-oscillator dynamics. ©2007 IEEE.