Neural network ℋ<Subscript>∞</Subscript> chaos synchronization

This paper proposes a new neural network ?_∞ synchronization (NNHS) scheme for unknown chaotic systems. In the proposed framework, a dynamic neural network is constructed as an alternative to approximate the chaotic system. Based on this neural network and linear matrix inequality (LMI) formulation, the NNHS controller and the learning law are presented to reduce the effect of disturbance to an ?_∞ norm constraint. It is shown that finding the NNHS controller and the learning law can be transformed into the LMI problem and solved using the convex optimization method. A numerical example is presented to demonstrate the validity of the proposed NNHS scheme.     

Synchronization of unified chaotic system by sliding mode/mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control

This paper deals with the synchronization of uncertain unified chaotic system in the presence of two kinds of disturbances, white noise and bounded power signal. A sliding mode controller (SMC) is established to guarantee the sliding motion. Moreover, a proportional-integral (PI) switching surface is used to determine the performance of the system in the sliding motion. Also, by using a mixed H ₂/H _∞ approach, the effect of external disturbances on the sliding motion is reduced. The necessary parameters of constructing controller and switching surface are found via semidefinite programming (SDP) which can be solved effectively by a standard software. Finally, a numerical simulation is presented to show the effectiveness of the proposed method.     

Elastic properties of Al<Subscript>2</Subscript>O<Subscript>3</Subscript>–NiAl: a modified version of Hashin–Shtrikman bounds

The modified nonlinear relations for the estimation of elastic constants of Al₂O₃–NiAl composite material are developed. The concept of microstructure and interconnectivity of phases at the interface is used. Hashin–Shtrikman relations are described in their actual form and modified version of Hashin–Shtrikman relations for bulk and shear moduli are discussed. These relations for elastic and mechanical properties are applied mainly for Al₂O₃–NiAl composite material. Theoretical predictions using modified relations are compared with Hashin–Shtrikman bounds and experimental results of elastic properties for Al₂O₃–NiAl matrix-inclusion-based composite. It is found that the predicted values of elastic and mechanical properties using modified relations are quite close to the experimental results.     

Multivalued penalty method for evolution variational inequalities with <Emphasis Type="Italic">λ</Emphasis><Subscript>0</Subscript>-pseudomonotone multivalued maps

We consider evolution variational inequalities with λ ₀-pseudomonotone maps. The main properties of these maps are investigated. By using the finite-difference method, we prove the property of strong solvability for the class of evolution variational inequalities with λ ₀-pseudomonotone maps. Using the penalty method for multivalued maps, we show the existence of weak solutions of evolution variational inequalities on closed convex sets. The class of multivalued penalty operators is constructed. We also consider a model example to illustrate this theory.     

Stability and implementable ℋ<Subscript>∞</Subscript> filters for singular systems with nonlinear perturbations

In this paper, we investigate the problem of designing ℋ_∞ filter for a class of continuous-time uncertain singular systems with nonlinear perturbations, which can be realized in practice. The perturbation is a time-varying function of the system state and satisfies a Lipschitz constraint. The design objective is to guarantee that a prescribed upper bound on an ℋ_∞ performance of the robust filter is attained for all possible energy-bounded input disturbances and all admissible uncertainties and which can be implemented on-line to get a good replica of the state. We first establish sufficient condition for the existence and uniqueness of solution to the singular system connected with the normal filter. Using a linear matrix inequality (LMI) format, we then provide a sufficient condition for the asymptotic stability of the realizable ℋ_∞ filter. Then by means of a convex analysis procedure the filter gain matrices are derived and an important special case is readily deduced. Finally, a numerical example is presented to illustrate the theoretical developments.     

A Phase-Partitioning Model for CO<Subscript>2</Subscript>–Brine Mixtures at Elevated Temperatures and Pressures: Application to CO<Subscript>2</Subscript>-Enhanced Geothermal Systems

Correlations are presented to compute the mutual solubilities of CO₂ and chloride brines at temperatures 12–300°C, pressures 1–600 bar (0.1–60 MPa), and salinities 0–6 m NaCl. The formulation is computationally efficient and primarily intended for numerical simulations of CO₂-water flow in carbon sequestration and geothermal studies. The phase-partitioning model relies on experimental data from literature for phase partitioning between CO₂ and NaCl brines, and extends the previously published correlations to higher temperatures. The model relies on activity coefficients for the H₂O-rich (aqueous) phase and fugacity coefficients for the CO₂-rich phase. Activity coefficients are treated using a Margules expression for CO₂ in pure water, and a Pitzer expression for salting-out effects. Fugacity coefficients are computed using a modified Redlich–Kwong equation of state and mixing rules that incorporate asymmetric binary interaction parameters. Parameters for the calculation of activity and fugacity coefficients were fitted to published solubility data over the P–T range of interest. In doing so, mutual solubilities and gas-phase volumetric data are typically reproduced within the scatter of the available data. An example of multiphase flow simulation implementing the mutual solubility model is presented for the case of a hypothetical, enhanced geothermal system where CO₂ is used as the heat extraction fluid. In this simulation, dry supercritical CO₂ at 20°C is injected into a 200°C hot-water reservoir. Results show that the injected CO₂ displaces the formation water relatively quickly, but that the produced CO₂ contains significant water for long periods of time. The amount of water in the CO₂ could have implications for reactivity with reservoir rocks and engineered materials.     

Roughness effects in low-<Emphasis Type="Italic"> Re</Emphasis><Subscript><Emphasis Type="Italic"> θ</Emphasis></Subscript> open-channel turbulent boundary layers

This paper reports velocity measurements obtained on a smooth and two geometrically different types of rough surfaces in an open channel. The measurements were obtained using a laser-Doppler anemometer. The recent boundary layer theory proposed by George and Castillo (1997) and conventional scaling laws are used to analyze the data. The present flow shows a strong structural similarity to a canonical turbulent boundary layer in the inner layer. The results demonstrate that surface roughness increases the wake parameter. Surface roughness also enhances the levels of turbulence intensities, Reynolds shear stress and triple correlations over most of the boundary layer, but decreases the stress anisotropy.     

On the convergence of successive Galerkin approximation for nonlinear output feedback <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control

Although the formulation of the nonlinear theory of H _∞ control has been well developed, solving the Hamilton–Jacobi–Isaacs equation remains a challenge and is the major bottleneck for practical application of the theory. Several numerical methods have been proposed for its solution. In this paper, results on convergence and stability for a successive Galerkin approximation approach for nonlinear H _∞ control via output feedback are presented. An example is presented illustrating the application of the algorithm.     

Application of X-ray CT investigation of CO<Subscript>2</Subscript>–brine flow in porous media

A clear understanding of two-phase flows in porous media is important for investigating CO₂ geological storage. In this study, we conducted an experiment of CO₂/brine flow process in porous media under sequestration conditions using X-ray CT technique. The flow properties of relative permeability, porosity heterogeneity, and CO₂ saturation were observed in this experiment. The porous media was packed with glass beads having a diameter of 0.2 mm. The porosity distribution along the flow direction is heterogeneous owing to the diameter and shape of glass beads along the flow direction. There is a relationship between CO₂ saturation and porosity distribution, which changes with different flow rates and fractional flows. The heterogeneity of the porous media influences the distribution of CO₂; moreover, gravity, fractional flows, and flow rates influence CO₂ distribution and saturation. The relative permeability curve was constructed using the steady-state method. The results agreed well with the relative permeability curve simulated using pore-network model.     

Young’s modulus defect for the solid solution of hydrogen in palladium (PdH<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>)

We use the internal friction method in the free vibration mode to study the temperature dependence of the spectra of dissipative losses and shear modulus in the Pd polycrystalline system, the PdH _x penetration solid solution, and in the dehydrogenized Pd system. Studying the spectra and shear moduli, we discovered that the hydrogen desorption from the α- and β-phases of the PdH _x system occurs completely after the first cycle of heating of this system. The temperature dependences of the shear modulus of the original and dehydrogenized systems coincide. The existence of dissipative processes in the solid solution is related to the mobility of hydrogen atoms in the face-centered cubic lattice of the Pd structure and provokes the appearance of peaks of losses on the modulus defect spectrum in the temperature region. We consider the possibility of describing the modulus defect in the framework of phenomenological models of elasticity. Dissolution of hydrogen atoms in a metal structure results in changes in the physical-mechanical characteristics of the obtained solid solutions compared with the original dehydrogenized metals.     

Topological classification of affine mappings from ℝ<Superscript>2</Superscript> into ℝ<Superscript>2</Superscript>

We investigate affine mappings from ℝ² into ℝ² and establish necessary and sufficient conditions for the topological conjugacy of these mappings. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 472–480, October–December, 2008.     

MHD flow and heat transfer of a power-law non-Newtonian nanofluid (Cu–H<Subscript>2</Subscript>O) over a vertical stretching sheet

A steady-state mixed convection boundary layer flow of an electrically conducting nanofluid (Cu–H₂O) obeying a power-law model in the presence of an alternating magnetic field due to a stretching vertical heated sheet is investigated numerically through the use of Wolfram Mathematica. The surface stretching velocity and the surface temperature are assumed to vary as linear functions of the distance from the origin. A similarity solution is presented, which depends on the nanoparticle volume fraction, power-law parameter, magnetic field parameter, buoyancy convection parameter, and modified Prandtl number.     

Entropy Solutions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">∞</Emphasis></Superscript> for the Euler Equations in Nonlinear Elastodynamics and Related Equations

A compactness framework is established for approximate solutions to the Euler equations in one-dimensional nonlinear elastodynamics by identifying new properties of the Lax entropies, especially the higher order terms in the Lax entropy expansions, and by developing ways to employ these new properties in the method of compensated compactness. Then this framework is applied to establish the existence, compactness, and decay of entropy solutions in L for the Euler equations in nonlinear elastodynamics with a more general stress-strain relation than those for the previous existence results. This compactness framework is further applied to solving the Euler equations of conservation laws of mass, momentum, and energy for a class of thermoelastic media, and the equations of motion of viscoelastic media with memory.     

Two-Phase Problems for Linear Elliptic Operators with Variable Coefficients: Lipschitz Free Boundaries are <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">1,γ</Emphasis></Superscript>

We prove C ^1, regularity of Lipschitz free boundaries of two-phase problems for linear elliptic operators with Hölder continuous coefficients.     

The Navier-Stokes Equations in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with Linearly Growing Initial Data

Consider the equations of Navier-Stokes on ⁿ with initial data U₀ of the form U₀(x)=u₀(x)–Mx, where M is an n×n matrix with constant real entries and u₀ L^p(ⁿ). It is shown that under these assumptions the equations of Navier-Stokes admit a unique local solution in L^p(ⁿ). Moreover, if ||e^tM||1 for all t0, then this mild solution is even analytic in x. This is surprising since the underlying semigroup of Ornstein-Uhlenbeck type is not analytic, in contrast to the Stokes semigroup.Acknowledgement It is our pleasure to thank G. METAFUNE, E. PRIOLA and A. RHANDI for fruitful discussions on the Ornstein-Uhlenbeck     

A Global Foliation of Einstein–Euler Spacetimes with Gowdy-Symmetry on <Emphasis Type="Italic">T</Emphasis><Superscript>3</Superscript>

We investigate the initial value problem for the Einstein–Euler equations of general relativity under the assumption of Gowdy symmetry on T ³, and we construct matter spacetimes with low regularity. These spacetimes admit both impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularities propagating at light speed) and shock waves in the fluid (that is, discontinuities propagating at about the sound speed). Given an initial data set, we establish the existence of a future development, and we provide a global foliation in terms of a globally and geometrically defined time-function, closely related to the area of the orbits of the symmetry group. The main difficulty lies in the low regularity assumed on the initial data set which requires a distributional formulation of the Einstein–Euler equations.     

The Singular Set of <Emphasis Type="Italic">ω</Emphasis>-minima

We consider ω-minima of convex variational integrals in the vectorial case n,N≥2, and we provide estimates for the Hausdorff dimension of their singular sets.     

On Local Uniqueness of Weak Solutions to the Navier–Stokes System with <Emphasis Type="Italic">BMO</Emphasis><Superscript>−1</Superscript> Initial Datum

We address the problem of local uniqueness of weak solutions to the Navier–Stokes system, with the initial datum in a subspace of . The existence and uniqueness of local mild solutions has been proven by Koch and Tataru (Adv Math 157:22–35, 2001). We present a necessary and sufficient condition for two weak solutions to evolve from the same initial datum, and for weak solutions to be mild.      

Linear <Emphasis Type="Italic">Γ</Emphasis>-limits of multiwell energies in nonlinear elasticity theory

We derive linearized theories from nonlinear elasticity theory for multiwell energies. Under natural assumptions on the nonlinear stored energy densities, the properly rescaled nonlinear energy functionals are shown to Γ-converge to the relaxation of a corresponding linearized model. Minimizing sequences of problems with displacement boundary conditions and body forces are investigated and found to correspond to minimizing sequences of the linearized problems. As applications of our results we discuss the validity and failure of a formula that is widely used to model multiwell energies in the regime of linear elasticity. Applying our convergence results to the special case of single well densities, we also obtain a new strong convergence result for the sequence of minimizers of the nonlinear problem.      

Turbulent Flow Around Fluid–Porous Interfaces Computed with a Diffusion-Jump Model for <Emphasis Type="Italic">k</Emphasis> and <Emphasis Type="Italic">ε</Emphasis> Transport Equations

Flow over vegetation and bottom of rivers can be characterized by some sort of porous structure of irregular surface through which a fluid permeates. Also, in engineering systems, one can have components that make use of a working fluid flowing over irregular layers of porous material. This article presents numerical solutions for such hybrid medium, considering here a channel partially filled with a flat porous layer saturated by a fluid flowing in turbulent regime. One unique set of transport equations is applied to both the regions. A diffusion-jump model for both the turbulent kinetic energy and its dissipation rate, across the interface, is presented and discussed upon. The discretization steps taken for numerically accommodating such model in the system of algebraic equations are presented. Numerical results show the effects of Reynolds number, porosity, and permeability on mean and turbulence fields. Results indicate that when negative values for the stress jump coefficient are applied, the peak of the turbulent kinetic energy distribution occurs at the macroscopic interface.