Well-Posedness of Nematic Liquid Crystal Flow in {L^{3}_{\rm uloc}(\mathbb{R}^3)}

In this paper, we establish the local well-posedness for the Cauchy problem of a simplified version of hydrodynamic flow of nematic liquid crystals in ${\mathbb{R}^3}$ for any initial data (u ₀, d ₀) having small ${L^{3}_{\rm uloc}}$ -norm of ${(u_{0}, \nabla d_{0})}$ . Here ${L^{3}_{\rm uloc}(\mathbb{R}^3)}$ is the space of uniformly locally L ³-integrable functions. For any initial data (u ₀, d ₀) with small ${\|(u_0, \nabla d_0)\|_{L^{3}(\mathbb{R}^3)}}$ , we show that there exists a unique, global solution to the problem under consideration which is smooth for t > 0 and has monotone deceasing L ³-energy for ${t \geqq 0}$ .     

\mathcal {H}_{\infty } filtering for sample data systems with stochastic sampling and Markovian jumping parameters

This paper deals with the problem of \(\mathcal {H}_{\infty }\) filtering for sample data systems that possess random jumping parameters described by a finite-state Markov process with stochastic sampling. Multiple stochastic sampling periods are considered in which each sampling period is assumed to be time varying that switches between two different values in a random way with given probability. The aim of this paper is to design a filter such that the filtering error system is stochastically stable with a prescribed \(\mathcal {H}_{\infty }\) disturbance attenuation level. Sufficient conditions for the existence of \(\mathcal {H}_{\infty }\) filters are expressed in terms of linear matrix inequalities (LMIs), which can be solved by using Matlab LMI toolbox. Numerical examples are given to illustrate the effectiveness of the proposed result including a realistic Transmission Control Protocol network model.     

Delay-dependent stability analysis and \mathcal {H}_{\infty } control for LPV systems with parameter-varying state delays

This paper develops the stability analysis and delay-dependent \(\mathcal {H}_{\infty }\) control synthesis for linear parameter-varying (LPV) systems with time-varying state delays. On the basis of the Finsler’s lemma, sufficient conditions on \(\mathcal {H}_{\infty }\) performance analysis are formulated in terms of parameterized linear matrix inequalities. The interesting annihilator matrix is constituted by time-varying parameters of LPV systems to reduce the conservatism. A numerical example is presented to confirm the efficiency of the proposed method.     

Switching stabilization and { H}_{\varvec{\infty }} performance of a class of discrete switched LPV system with unstable subsystems

Stability and $H_\infty $ performance are analyzed in this paper for a class of discrete switched linear parameter-varying (LPV) systems in which all subsystems’ state-space matrices are parametrically affine, and any subsystem is not stable for parameters varying in a convex set. A switching law is designed to stabilize and satisfy the $H_\infty $ performance of the switched LPV system. By means of the multiple Lyapunov functions method, linear matrix inequality (LMI) conditions for the existence of parameter-dependent Lyapunov functions are proposed. An example shows the effectiveness of the proposed methods.     

Robust \mathcal{H}_{\infty} decentralized dynamic control for synchronization of a complex dynamical network with randomly occurring uncertainties

This paper considers synchronization problem of an uncertain complex dynamical network. The norm-bounded uncertainties enter into the complex dynamical network in randomly ways, and such randomly occurring uncertainties (ROUs) obey certain mutually uncorrelated Bernoulli distributed white noise sequences. Under the circumstances, a robust $\mathcal{H}_{\infty}$ decentralized dynamic feedback controller is designed to achieve asymptotic synchronization of the network. Based on Lyapunov stability theory and linear matrix inequality (LMI) framework, the existence condition for feasible controllers is derived in terms of LMIs. Finally, the proposed method is applied to a numerical example in order to show the effectiveness of our result.     

Observer-based quantized sliding mode $${\varvec{\mathcal {H}}}_{\varvec{\infty }}$$ control of Markov jump systems

An adjustable quantized approach is adopted to treat the \(\mathcal {H}_{\infty }\) sliding mode control of Markov jump systems with general transition probabilities. To solve this problem, an integral sliding mode surface is constructed by an observer with the quantized output measurement and a new bound is developed to bridge the relationship between system output and its quantization. Nonlinearities incurred by controller synthesis and general transition probabilities are handled by separation strategies. With the help of these measurements, linear matrix inequalities-based conditions are established to ensure the stochastic stability of the sliding motion and meet the required \(\mathcal {H}_{\infty }\) performance level. An example of single-link robot arm system is simulated at last to demonstrate the validity.     

Design of a steering control law for an autonomous underwater vehicle using nonlinear $$\mathscr {H}_{\infty }$$ state feedback technique

This paper proposes a new robust nonlinear \(\mathscr {H}_{\infty }\) state feedback (NHSF) controller for an autonomous underwater vehicle (AUV) in steering plane. A three-degree-of-freedom nonlinear model of an AUV has considered for developing a steering control law. In this, the energy dissipative theory is used which leads to form a Hamilton–Jacobi–Isaacs (HJI) inequality. The nonlinear \(\mathscr {H}_{\infty }\) control algorithm has been developed by solving HJI equation such that the AUV tracks the desired yaw angle accurately. Furthermore, a path following control has been implemented using the NHSF control algorithm for various paths in steering plane. Simulation studies have been carried out using MATLAB/Simulink environment to verify the efficacies of the proposed control algorithm for AUV. From the results obtained, it is concluded that the proposed robust control algorithm exhibits a good tracking performance ensuring internal stability and significant disturbance attenuation.     

Flow Visualization of Superbursts and of the Log-Layer in a DNS at $$ \operatorname{Re} _{\tau } = 950 $$

This paper uses direct numerical simulations (DNS) of turbulent flow in a channel at (Del álamo, Jiménez, Zandonade, Moser J Fluid Mech 500:135–144, 2004) to provide a picture of the turbulent structures making large contributions to the Reynolds shear stress. Considerable work of this type has been done for the viscous wall region at smaller , for which a log-layer does not exist. Recent PIV measurements of turbulent velocity fluctuations in a plane parallel to the direction of flow have emphasized the dominant contribution of large scale structures in the outer flow. This prompted Hanratty and Papavassiliou (The role of wall vortices in producing turbulence. In: Panton, R.L. (ed) Self-sustaining Mechanism of Wall Turbulence. Computational Mechanics Publications, Southampton, pp. 83–108, 1997) to use DNS at to examine these structures in a plane perpendicular to the direction of flow. They identified plumes which extend from the wall to the center of a channel. The data at are used to explore these results further, to examine the structure of the log-layer, and to test present notions about the viscous wall layer.     

Nonuniqueness for the Heat Flow¶of Harmonic Maps on the Disk

We prove that for suitable initial data the heat flow of harmonic maps, from the disk to the sphere, admits infinitely many solutions, characterised by “backward bubbling” at some arbitrarily large time, all having uniformly bounded energy.     

Output feedback \varvec{\mathcal {H}}_{{\varvec{\infty }}} control of stochastic nonlinear time-delay systems with state and disturbance-dependent noises

This paper is concerned with the output feedback \(\mathcal {H}_\infty \) control problem for a class of stochastic nonlinear systems with time-varying state delays; the system dynamics is governed by the stochastic time-delay It \(\hat{o}\) -type differential equation with state and disturbance contaminated by white noises. The design of the output feedback \(\mathcal {H}_\infty \) control is based on the stochastic dissipative theory. By establishing the stochastic dissipation of the closed-loop system, the delay-dependent and delay-independent approaches are proposed for designing the output feedback \(\mathcal {H}_\infty \) controller. It is shown that the output feedback \(\mathcal {H}_\infty \) control problem for the stochastic nonlinear time-delay systems can be solved by two delay-involved Hamilton–Jacobi inequalities. A numerical example is provided to illustrate the effectiveness of the proposed methods.     

Existence and Regularity of the Reflector Surfaces in {\mathbb{R}^{n+1}}

In this paper we study the problem of constructing reflector surfaces from the near field data. The light is transmitted as a collinear beam and the reflected rays illuminate a given domain on the fixed receiver surface. We consider two types of weak solutions and prove their equivalence under some convexity assumptions on the target domain. The regularity of weak solutions is a very delicate problem and the positive answer depends on a number of conditions characterizing the geometric positioning of the reflector and receiver. In fact, we show that there is a domain \({\mathcal{D}}\) in the ambient space such that the weak solution is smooth if and only if its graph lies in \({\mathcal{D}}\) .     

Necessary and Sufficient Conditions for the Invertibility of the Nonlinear Difference Operator \left( {\mathcal{D}x} \right)\left( t \right) = x\left( {t + 1}\right) - f\left( {x\left( t \right)} \right) in the Space of Functions Bounded and Continuous on the Axis

We find necessary and sufficient conditions for the nonlinear difference operator $\left( {\mathcal{D}x} \right)\left( t \right) = x\left( {t + 1} \right) - f\left( {x\left( t \right)} \right)$ $t \in \mathbb{R}$ , where $f:\mathbb{R} \to \mathbb{R}$ is a continuous function, to have the inverse in the space of functions bounded and continuous on $\mathbb{R}$ .     

Effect of Fuel Stratification on OH and $${\mathrm {CH}}_{2}\hbox {O}$$ CH 2 O PLIF Multiplication of Turbulent Hydrogen-Enriched Flames

Multiplication of hydroxyl and formaldehyde planar laser-induced fluorescence signals for turbulent hydrogen-enriched methane–air flames with compositionally inhomogeneous mixtures is investigated experimentally. Hydrogen-enriched methane–air flames with a global fuel–air equivalence ratio of 0.8 and hydrogen-enrichment percentage of 60% are examined. Two nozzles, each containing 4 fuel/air injection lobes are used in the experiments. The lobes of the first nozzle are straight, while those of the second nozzle are not, generating a swirling motion. The fuel is injected through several small diameter holes into the lobes. The amount of injected fuel flow rate varies between the lobes, generating stratified conditions. For each nozzle, two mean bulk flow velocities of 5 and 25 m/s are tested. Simultaneous hydroxyl and formaldehyde planar laser-induced fluorescence as well as separate stereoscopic particle image velocimetry are performed for the tested reacting conditions. For non-reacting flow tests, separate particle image velocimetry and acetone planar laser-induced fluorescence experiments are conducted to study the background turbulent flow characteristics and fuel/air mixing, respectively. The results show that stratification can lead to fragmentation of the flames and generation of islands with noticeable multiplication of hydroxyl and formaldehyde planar laser-induced fluorescence signals. Due to their significantly large number of occurrences, such flame structure can generate relatively large integral of the PLIF signals multiplication.

     

Convergence to Steady States of Solutions of Non-autonomous Heat Equations in $$\mathbb{R}^{N}$$

Under certain assumptions on f and g we prove that positive, global and bounded solutions u of the non-autonomous heat equation

Application of an Inverse Simulator of Single-Phase Flow Analysis for Leakage Pathway Estimation in a Multiphase System for Geologic $$\hbox {CO}_{2}$$ Storage

When \(\hbox {CO}_{2}\) is injected in a brine reservoir, brine or \(\hbox {CO}_{2}\) can be discharged into a permeable formation saturated with brine above the storage reservoir along a leakage pathway, if present. In most situations, the overlying formation can act as a single-phase aquifer with only brine leakage before \(\hbox {CO}_{2}\) leaks. This study examines the applicability of a developed inverse code for single-phase fluids to detect leakage pathway locations in view of the overlying formation using pressure anomalies induced by leaks. Before applying inverse analysis, forward modeling is performed using the TOUGH2 model to determine brine and \(\hbox {CO}_{2}\) leakage in a homogeneous conceptual model, and the simulated pressure profiles at monitoring wells are used as measurements in the inverse model. In the inverse code, an important consideration is that the vertical hydraulic conductivity and cross-sectional area of a leakage pathway that are inherent to a leakage term in the mass balance equation are integrated as a single parameter to estimate the leakage pathway locations. This method eliminates the impact of the uncertainty of the leakage pathway size on the accuracy of leakage pathway estimation. The inverse model examines the effect of the number of monitoring wells, monitoring periods and \(\hbox {CO}_{2}\) leakage into the overlying formation on the accuracy of leakage pathway estimation according to eleven application examples. The comparison between the results of the single-phase inverse code and iTOUGH2 code illustrates that the single-phase inverse model can be applicable to the leakage pathway estimation in a multiphase flow system.     

A Numerical Model of Tracer Transport in a Non-isothermal Two-Phase Flow System for {\text{ CO}}_2 Geological Storage Characterization

For the purpose of characterizing geologically stored $\text{ CO}_{2}Air sparging is an in situ soil/groundwater remediation technology, which involves the injection of pressurized air through air sparging well below the zone of contamination. To investigate the rate-dependent flow properties during multistep air sparging, a rule-based dynamic two-phase flow model was developed and applied to a 3D pore network which is employed to characterize the void structure of porous media. The simulated dynamic two-phase flow at the pore scale or microscale was translated into functional relationships at the continuum-scale of capillary pressure?Csaturation (P ^c?CS) and relative permeability??saturation (K _r?CS) relationships. A significant contribution from the air injection pressure step and duration time of each air injection pressure on both of the above relationships was observed during the multistep air sparging tests. It is observed from the simulation that at a given matric potential, larger amount of water is retained during transient flow than that during steady flow. Shorter the duration of each air injection pressure step, there is higher fraction of retained water. The relative air/water permeability values are also greatly affected by the pressure step. With large air injection pressure step, the air/water relative permeability is much higher than that with a smaller air injection pressure step at the same water saturation level. However, the impact of pressure step on relative permeability is not consistent for flows with different capillary numbers (N _ca). When compared with relative air permeability, relative water permeability has a higher scatter. It was further observed that the dynamic effects on the relative permeability curve are more apparent for networks with larger pore sizes than that with smaller pore sizes. In addition, the effect of pore size on relative water permeability is higher than that on relative air permeability.