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.     

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.     

{L}_{2}-gain analysis and output feedback control for continuous-time switched systems with actuator saturation

This paper is concerned with the problem of \(L_2\) -gain analysis for a class of continuous-time switched systems with actuator saturation. Different from the existing results using the state-dependent switching approach, a linear parameter-varying output feedback controller with the time-dependent switching mechanism is designed such that the overall system is exponentially stable and with disturbance tolerance capacity and \(L_2\) gain. Further, the proposed controller is insured by introducing the slack matrices to have less conservatism, which maximizes the ellipsoid where the state trajectories starting from the origin will remain. Sufficient conditions for the existence of the switching and output feedback control law are derived in terms of linear matrix inequalities. Finally, a practical example is given to show the effectiveness of the proposed method.     

Synchronization of inferior olive neurons via {\mathcal {L}}_1 adaptive feedback

This paper considers the synchronization of inferior olive neurons based on the \({\mathcal {L}}_1\) adaptive control theory. The ION model treated here is the cascade connection of two nonlinear subsystems, termed ZW and UV subsystems. It is assumed that the structure of the nonlinear functions and certain parameters of the IONs are not known, and disturbance inputs are present in the system. First, an \({\mathcal {L}}_1\) adaptive control system is designed to achieve global synchrony of the ZW subsystems using a single control input. This controller can accomplish local synchrony of the UV subsystems if the linearized UV subsystem is exponentially stable. For global synchrony of the UV subsystems, an \({\mathcal {L}}_1\) adaptive control law is designed. Each of these controllers includes a state predictor, an update law, and a control law. In the closed-loop system, global synchrony of the complete models of the IONs (the interconnected ZW and UV subsystems) is accomplished using these two adaptive controllers. Simulations results show that in the closed-loop system, the IONs are synchronized, despite unmodeled nonlinearities, disturbance inputs, and parameter uncertainties in the system.     

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.     

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.     

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.     

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.     

Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics

This paper is on the so called inverse problem of ordinary differential equations, i.e. the problem of determining the differential system satisfying a set of given properties. More precisely we characterize under very general assumptions the ordinary differential equations in \(\mathbb {R}^N\) which have a given set of either \(M\) partial integrals, or \(M first integral, or \(M partial and first integrals. Moreover, for such systems we determine the necessary and sufficient conditions for the existence of \(N-1\) independent first integrals. We give two relevant applications of the solutions of these inverse problem to constrained Lagrangian and Hamiltonian systems respectively. Additionally we provide the general solution of the inverse problem in dynamics.     

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.     

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.     

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.     

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.     

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.     

Statistical behavior of post-shock overpressure past grid turbulence

When a shock wave ejected from the exit of a 5.4-mm inner diameter, stainless steel tube propagated through grid turbulence across a distance of 215 mm, which is 5–15 times larger than its integral length scale \(L_{u}\) , and was normally incident onto a flat surface; the peak value of post-shock overpressure, \(\Delta P_{\mathrm{peak}}\) , at a shock Mach number of 1.0009 on the flat surface experienced a standard deviation of up to about 9 % of its ensemble average. This value was more than 40 times larger than the dynamic pressure fluctuation corresponding to the maximum value of the root-mean-square velocity fluctuation, \(u^{\prime }= 1.2~\hbox {m}/\hbox {s}\) . By varying \(u^{\prime }\) and \(L_{u}\) , the statistical behavior of \(\Delta P_{\mathrm{peak}}\) was obtained after at least 500 runs were performed for each condition. The standard deviation of \(\Delta P_{\mathrm{peak}}\) due to the turbulence was almost proportional to \(u^{{\prime }}\) . Although the overpressure modulations at two points 200 mm apart were independent of each other, we observed a weak positive correlation between the peak overpressure difference and the relative arrival time difference.     

Properties of Monodromic Points on Center Manifolds in {\mathbb {R}}^3 via Lie Symmetries

In this work we use Lie symmetries to investigate monodromic points on center manifolds of a singularity of an analytic vector field ${\mathcal {X}}$ in ${\mathbb {R}}^3$ . We investigate how whether the singularity is a focus or a center, is analytically normalizable or not, and is linearizable or not is reflected in the centralizer and normalizer of ${\mathcal {X}}$ .     

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.