Some Aspects of Stability for Semigroup Actions and Control Systems

The present paper introduces both the notions of Lagrange and Poisson stabilities for semigroup actions. Let \(S\) be a semigroup acting on a topological space \(X\) with mapping \(\sigma :S\times X\rightarrow X\) , and let \(\mathcal {F}\) be a family of subsets of \(S\) . For \(x\in X\) the motion \(\sigma _{x}:S\rightarrow X\) is said to be forward Lagrange stable if the orbit \(Sx\) has compact closure in \(X\) . The point \(x\) is forward \(\mathcal {F}\) -Poisson stable if and only if it belongs to the limit set \(\omega \left( x,\mathcal {F}\right) \) . The concept of prolongational limit set is also introduced and used to describe nonwandering points. It is shown that a point \(x\) is \( \mathcal {F}\) -nonwandering if and only if \(x\) lies in its forward \(\mathcal {F} \) -prolongational limit set \(J\left( x,\mathcal {F}\right) \) . The paper contains applications to control systems.     

A fast algorithm to calculate the critical coupling strength for synchronization in a chain of Kuramoto oscillators

Synchronization in a one-dimensional chain of Kuramoto oscillators with periodic boundary conditions is studied. An algorithm to rapidly calculate the critical coupling strength \(K_c\) for complete frequency synchronization is presented according to the mathematical constraint conditions and the periodic boundary conditions. By this new algorithm, we have checked the relation between \(\langle K_c\rangle \) and \(N\) , which is \(\langle K_c\rangle \sim \sqrt{N}\) , not only for small \(N\) , but also for large \(N\) . We also investigate the heavy-tailed distribution of \(K_c\) for random intrinsic frequencies, which is obtained by showing that the synchronization problem is equivalent to a discretization of Brownian motion. This theoretical result was checked by generating a large sample of \(K_c\) for large \(N\) from our algorithm to get the empirical density of \(K_c\) . Finally, we derive the permutation for the maximum coupling strength and its exact expression, which grows linearly with \(N\) and would provide the theoretical support for engineering applications.     

Stable disarrangement phases of elastic aggregates: a setting for the emergence of no-tension materials with non-linear response in compression

A recent field theory of elastic bodies undergoing non-smooth submacroscopic geometrical changes (disarrangements) provides a setting in which, for a given homogeneous macroscopic deformation \(F\) of the body, there are typically a number of different states \(G\) of smooth, submacroscopic deformation (disarrangement phases) available to the body. A tensorial consistency relation and the inequality \(\det G\le \det F\) that guarantees that \(F\) accommodates \(G\) determine the totality of disarrangement phases \(G\) corresponding to \(F\) , and it is natural to seek for a given \(F\) those disarrangement phases that minimize the Helmholtz free energy (stable disarrangement phases). We introduce these concepts in the particular context of continuous bodies comprised of many small elastic bodies (elastic aggregates) and in the context where disarrangements do not contribute to the Helmholtz free energy (purely dissipative disarrangements). In this setting, the Helmholtz free energy response \(G\longmapsto \varPsi (G)\) of the pieces of the aggregate determines the totality of disarrangement phases corresponding to \(F\) , which necessarily includes the phase \(G=F\) (compact phase) in which every piece of the aggregate undergoes the given macroscopic deformation \(F\) . When the response function \(\varPsi \) is isotropic and smooth, and when \(\varPsi \) possesses standard semiconvexity and growth properties, the body also admits phases of the form \(G=\zeta _{\min }R\) (loose phases) with \(R\) an arbitrary rotation, provided that \(\zeta _{\min }R \) satisfies the accommodation inequality \(\zeta _{\min }^{3}\le \det F\) . Loose phases, when available, achieve the global minimum \(\varPsi (\zeta _{\min }R)\) of the free energy and consequently are stable and stress-free. When \( \varPsi (G)\) has the specific form \(\varPsi _{\alpha \beta }(G)=(\alpha /2)(\det G)^{-2}+(\beta /2)tr(GG^{T})\) , with \(\alpha \) , \(\beta \) given elastic constants, we determine all of the disarrangement phases corresponding to \(F\) . These include not only the compact and loose phases, but also disarrangement phases \(G\) in which the stress \(D\varPsi (G)\) is uniaxial or planar. Our main result (“stability implies no-tension”) is the assertion that every stable disarrangement phase for \(\varPsi _{\alpha \beta }\) cannot support tensile tractions, and our treatment of elastic aggregates thus provides a natural setting for the emergence of no-tension materials whose response in compression is non-linear. Existing treatments of no-tension materials assume at the outset that the body cannot support tension and that the response in compression is linear.     

Expansivity and Cone-fields in Metric Spaces

Due to the results of Lewowicz and Tolosa expansivity can be characterized with the aid of Lyapunov function. In this paper we study a similar problem for uniform expansivity and show that it can be described using generalized cone-fields on metric spaces. We say that a function \(f:X\rightarrow X\) is uniformly expansive on a set \(\varLambda \subset X\) if there exist \(\varepsilon >0\) and \(\alpha \in (0,1)\) such that for any two orbits \(\hbox {x}:\{-N,\ldots ,N\} \rightarrow \varLambda \) , \(\hbox {v}:\{-N,\ldots ,N\} \rightarrow X\) of \(f\) we have $$\begin{aligned} \sup _{-N\le n\le N}d(\hbox {x}_n,\hbox {v}_n) \le \varepsilon \implies d(\hbox {x}_0,\hbox {v}_0) \le \alpha \sup _{-N\le n\le N}d(\hbox {x}_n,\hbox {v}_n). \end{aligned}$$ It occurs that a function is uniformly expansive iff there exists a generalized cone-field on \(X\) such that \(f\) is cone-hyperbolic.     

Robust mixed H_2 /H_\infty active control for offshore steel jacket platform

This paper presents a robust mixed \(H_2 /H_\infty \) control method for wave-excited offshore jacket platforms. Its objective was to design a controller that minimizes the upper bound of the \(H_2 \) performance measure on platform dynamics satisfying some \(H_\infty \) norm bound constraint simultaneously. Based on mixed \(H_2 /H_\infty \) control theory and linear matrix inequality techniques, a novel approach to stabilize offshore platform vibration with constrained \(H_2 /H_\infty \) performances is proposed. Uncertainties of the wave excitation are considered in dynamic performance analysis of offshore platforms. A reduced mode offshore platform structure under wave excitation is analyzed, and simulations are used to verify the effectiveness of the proposed approach. Compared with existing \(H_\infty \) control methods, the proposed approach makes a significant improvement for dynamic performances of offshore platforms under random wave excitation.     

Gas Sorption and the Consequent Volumetric and Permeability Change of Coal II: Numerical Modeling

The permeability of coalbed methane reservoirs may evolve during the recovery of methane and injection of gas, due to the change of effective stress and gas adsorption and desorption. Experimental and numerical studies were conducted to investigate the sorption-induced permeability change of coal. This paper presents the numerical modeling part of the work. It was found that adsorption of pure gases on coal was well represented by parametric adsorption isotherm models in the literature. Based on the experimental data of this study, adsorption of pure \(\hbox {N}_2\) was modeled using the Langmuir equation, and adsorption of pure \(\hbox {CO}_2\) was well represented by the N-Layer BET equation. For the modeling of CO \(_2\) & N \(_2\) binary mixture adsorption, the ideal adsorbed solution (IAS) model and the real adsorbed solution (RAS) model were used. The IAS model estimated the total amount of mixture adsorption and the composition of the adsorbed phase based on the pure adsorption isotherms. The estimated total adsorption and adsorbed-phase composition were very different from the experimental results, indicating nonideality of the CO \(_2\) –N \(_2\) –Coal-adsorption system. The measured sorption-induced strain was linearly proportional to the total amount of adsorption despite the species of the adsorbed gas. Permeability reduction followed a linear correlation with the volumetric strain with the adsorption of pure \(\hbox {N}_2\) and the tested CO \(_2\) & N \(_2\) binary mixtures, and an exponential correlation with the adsorption of pure \(\hbox {CO}_2\) .     

Integrability with symbolic computation on the Bogoyavlensky–Konoplechenko model: Bell-polynomial manipulation,bilinear representation,and Wronskian solution

With symbolic computation, this paper investigates some integrable properties of a two-dimensional generalization of the Korteweg-de Vries equation, i.e., the Bogoyavlensky–Konoplechenko model, which can govern the interaction of a Riemann wave propagating along the \(y\) -axis and a long wave propagating along the \(x\) -axis. Within the framework of Bell-polynomial manipulations, Bell-polynomial expressions are firstly given, which then are cast into bilinear forms. The \(N\) -soliton solutions in the form of an \(N\) th-order polynomial in the \(N\) exponentials and in terms of the Wronskian determinant are, respectively, constructed with the Hirota bilinear method and Wronskian technique. Bilinear Bäcklund transformation is also derived with the achievement of a family of explicit solutions.     

Standard decomposition of expansive ergodically supported dynamics

In this work, we introduce the notion of weak quasigroups, which are quasigroup operations defined almost everywhere on some set. Then, we prove that the topological entropy and the ergodic period of an invertible expansive ergodically supported dynamical system \((X,T)\) with the shadowing property establish a sufficient criterion for the existence of quasigroup operations defined almost everywhere outside of universally null sets and for which \(T\) is an automorphism. Furthermore, we find a decomposition of the dynamics of \(T\) in terms of \(T\) -invariant weak topological subquasigroups.     

Anisotropic Estimates for the Two-Dimensional Kuramoto–Sivashinsky Equation

We address the global solvability of the Kuramoto–Sivashinsky equation in a rectangular domain \([0,L_1]\times [0,L_2]\) . We give sufficient conditions on the width \(L_2\) of the domain, depending on the length \(L_1\) , so that the obtained solutions are global. Our proofs are based on anisotropic estimates.     

On the neutralization of bacterial spores in post-detonation flows

In multiple operational scenarios, explosive charges are used to neutralize confined or unconfined stores of bacterial spores. The spore destruction is achieved by post-detonation combustion and mixing of hot detonation product gases with the ambient flow and spore clouds. In this work, blast wave interaction with bacterial spore clouds and the effect of post-detonation combustion on spore neutralization are investigated using numerical simulations. Spherical explosive charges (radius, \(R_\mathrm{C}\) = 5.9 cm) comprising of nitromethane are modeled in the vicinity of a spore cloud, and the spore kill in the post-detonation flow is quantified. The effect of the mass of the spores and the initial distance, \(d^0\) , of the spore cloud from the explosive charge on the percentage of spores neutralized is investigated. When the spores are initially placed within a distance of 3.0 \(R_\mathrm{C}\) , within 0.1 ms after detonation of the charge, all the spores are neutralized by the blast wave and the hot detonation product gases. In contrast, almost all the spores survived the explosion when \(d^0\) is greater than 8.0 \(R_\mathrm{C}\) . The percentage of intact spores varied from 0 to 100 for 3.0 \(R_\mathrm{C}\) \( 8.0 \(R_\mathrm{C}\) with spore neutralization dependent on time spent by the spores in the post-detonation mixing/combustion zone.     

Fractional models of anomalous relaxation based on the Kilbas and Saigo function

We revisit the Kilbas and Saigo functions of the Mittag-Leffler type of a real variable \(t\) , with two independent real order-parameters. These functions, subjected to the requirement to be completely monotone for \(t>0\) , can provide suitable models for the responses and for the corresponding spectral distributions in anomalous (non–Debye) relaxation processes, found e.g. in dielectrics. Our analysis includes as particular cases the classical models referred to as Cole–Cole (the one-parameter Mittag-Leffler function) and to as Kohlrausch (the stretched exponential function). After some remarks on the Kilbas and Saigo functions, we discuss a class of fractional differential equations of order \(\alpha \in (0,1]\) with a characteristic coefficient varying in time according to a power law of exponent \(\beta \) , whose solutions will be presented in terms of these functions. We show 2D plots of the solutions and, for a few of them, the corresponding spectral distributions, keeping fixed one of the two order-parameters. The numerical results confirm the complete monotonicity of the solutions via the non-negativity of the spectral distributions, provided that the parameters satisfy the additional condition \(0<\alpha +\beta \le 1\) , assumed by us.     

Robust {\varvec{L}}_{\varvec{2}}-gain control for a class of cascade switched nonlinear systems

In this paper, we study the robust finite \(L_2 \) -gain control for a class of cascade switched nonlinear systems with parameter uncertainty. Each subsystem of the switched system under consideration is composed of a zero-input asymptotically stable nonlinear part which is a lower dimension switched system, and of a linearizable part. The uncertainty appears in the control channel of each subsystem. We give sufficient conditions under which the nonlinear feedback controllers are derived to guarantee that the \(L_2 \) -gain of the closed-loop switched system is less than a prespecified value for all admissible uncertainty under arbitrary switching. Moreover, we also develop the \(L_2\) -gain controllers for the switched systems with nonminimum phase case.     

Global stabilization of inherently non-linear systems using continuously differentiable controllers

This paper concerns the problem of constructing \(C^1\) (continuously differentiable) controllers to stabilize a class of uncertain non-linear systems whose linearization around the origin may contain uncontrollable modes. Based on a new definition of homogeneity with monotone degrees, a polynomial Lyapunov function and a \(C^1\) global stabilizer are constructed recursively. Moreover, several special cases are investigated to show the advantages of the proposed approaches using the generalized homogeneity compared to the existing approaches using the traditional homogeneity.     

Estimability Analysis for Optimization of Hysteretic Soil Hydraulic Parameters Using Data of a Field Irrigation Experiment

To improve the quality of parameter optimization, estimability analysis has been proposed as the first step before inverse modeling. When using field data of irrigation experiments for the determination of soil hydraulic parameters, wetting and drying processes may complicate optimization. The objectives of this study were to compare estimability analysis and inverse optimization of the soil hydraulic parameters in the models with and without considering hysteresis of the soil water retention function. Soil water pressure head data of a field irrigation experiment were used. The one-dimensional vertical water movement in variably saturated soil was described with the Richards equation using the HYDRUS-1D code. Estimability of the unimodal van Genuchten–Mualem hydraulic model parameters as well as of the hysteretic parameter model of Parker and Lenhard was classified according to a sensitivity coefficient matrix. The matrix was obtained by sequentially calculating effects of initial parameter variations on changes in the simulated pressure head values. Optimization was carried out by means of the Levenberg-Marquardt method implemented in the HYDRUS-1D code. The parameters \(\alpha , K_{s}, \theta _{s}\) , and \(n\) in the nonhysteretic model were found sensitive and parameter \(\theta _{s}\) strongly correlated with parameter \(n\) . When assuming hysteresis, the estimability was decreased with soil depth for \(K_{s}\) and \(\alpha ^{d}\) , and increased for \(\theta _{s}\) and n. Among the shape parameters, \(\alpha ^{w}\) was the most estimable. The hysteretic model could approximate the pressure heads in the soil by considering parameters from wetting and drying periods separately as initial estimates. The inverse optimization could be carried out more efficiently with most estimable parameters. Despite the remaining weaknesses of the local optimization algorithm and the inflexibility of the unimodal van Genuchten model, the results suggested that estimability analysis could be considered as a guidance to better define the optimization scenarios and then improved the determination of soil hydraulic parameters.     

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.     

Effects of nonlinear damping suspension on nonperiodic motions of a flexible rotor in journal bearings

Nonlinear damping suspension is a promising method to be used in a rotor-bearing system for vibration isolation between the bearing and environment. However, the nonlinearity of the suspension may influence the stability of the rotor-bearing system. In this paper, the motions of a flexible rotor in short journal bearings with nonlinear damping suspension are studied. A computational method is used to solve the equations of motion, and the bifurcation diagrams, orbits, Poincaré maps, and amplitude spectra are used to display the motions. The results show that the effect of the nonlinear damping suspension on the motions of the rotor-bearing system depends on the speed of rotor: (a) For low speeds, the rotor- bearing system presents the same motion pattern under the nonlinear damping ( \(p=0.5, 2, 3\) ) suspension as for the linear damping ( \(p=1\) ) suspension; (b) For high speeds, the effect of nonlinear damping depends on a combination of the damping exponent and damping coefficient. The square root damping model ( \(p=0.5\) ) shows a wider stable speed range than the linear damping for large damping coefficients. The quadratic damping ( \(p=2\) ) shows similar results to linear damping with some special damping coefficients. The cubic damping ( \(p=3\) ) shows more stable response than the linear damping in general.     

Microsize and initial pressure effects on shock wave propagation in a tube

In this paper, the shock wave propagation in a channel with a micrometric hydraulic diameter is numerically simulated for an initial Mach number \(M_{s}=2.61\) . The obtained values of the Mach number along the tube are compared to experimental and numerical data given in the literature. The microscale effects on the flow behavior, such as shock wave attenuation and pressure increase behind the shock wave, are amplified by further reducing the scaling ratio (or Reynolds number) of the flow. This reduction is obtained by either decreasing the hydraulic diameter \(D_\mathrm{H}\) or the initial driven gas pressure \(P_1\) . Under these conditions, the flow behavior changes drastically.     

On simulation of a bistable system with fractional damping in the presence of stochastic coherence resonance

A bistable dynamical system with the Duffing potential, fractional damping, and random excitation has been modelled. To excite the system, we used a stochastic force defined by Wiener random process of Gaussian distribution. As expected, stochastic resonance appeared for sufficiently high noise intensity. We estimated the critical value of the noise level as a function of derivative order \(q\) . For smaller order \(q\) , damping enhancement was reported.     

On solutions of two coupled fractional time derivative Hirota equations

We consider the well-known nonlinear Hirota equation (NLH) with fractional time derivative and derive its periodic wave solution and approximate analytic solitary wave solution using the homotopy analysis method (HAM). We also apply HAM to two coupled time fractional NLHs and construct their periodic wave solution and approximate solitary wave solution. We observe that the obtained periodic wave solution in both cases can be written in terms of the Mittag–Leffler function when the convergence control parameter \({c}_0\) equals \(-1\) . Convergence of the obtained solution is discussed. The derived approximate analytic solution and the effect of time-fractional order \(\alpha \) are shown graphically.     

Intermittent control on switched networks via {\varvec{\omega }}-matrix measure method

This paper is concerned with global exponential synchronization problem for a class of switched delay networks with interval parameters uncertainty, different from the most existing results, without constructing complex Lyapunov–Krasovskii functions; \(\omega \) -matrix measure method is firstly introduced to switched interval networks, combining Halanay inequality technique, designing proper intermittent and non-intermittent control strategy; some easy-to-verify synchronization criteria are given to ensure the global exponential synchronization of switched interval networks under arbitrary switching rule and for admissible interval uncertainties. Moreover, as an application, the proposed scheme can be applied to chaotic neural networks. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results and show the obtained results via employing \(\omega \) -measure are superior to previous results by using \(1\) -measure.