Stability Conditions Research Articles

MODELING A REHAB–RECOVERY–RELAPSE CYCLE WITH COMMUNITY DEPENDENCE VIA ODES

In this paper, we introduce a rehab–recovery–relapse cycle model to study the interaction between healthy individuals, individuals that might develop substance use disorders (SUD), which for simplicity here we will call susceptible individuals, individuals with SUD, and individuals recovered in a rehab community. We measure the community’s health through a system defined by four non-linear ordinary differential equations in which we include a rehab community. After computing the SUD-free and coexistence equilibrium points, we find the feasibility and stability conditions of an equivalent and reduced system obtained by nondimensionalizing the equations of the original model. We have numerically investigated the importance of the parameter values on the model’s outcome via a one- and two-strain parameter analysis. From the numerical results, we observe that an efficient rehab community can be characterized by (i) a high recovery rate (the individuals stay in that community as little time as possible) and (ii) permanent rehabilitation being preferred at the expense of a low recovery rate.

Neural learning control for sampled‐data nonlinear systems based on Euler approximation and first‐order filter

AbstractThe primary focus of this research paper is to explore the realm of dynamic learning in sampled‐data strict‐feedback nonlinear systems (SFNSs) by leveraging the capabilities of radial basis function (RBF) neural networks (NNs) under the framework of adaptive control. First, the exact discrete‐time model of the continuous‐time system is expressed as an Euler strict‐feedback model with a sampling approximation error. We provide the consistency condition that establishes the relationship between the exact model and the Euler model with meticulous detail. Meanwhile, a novel lemma is derived to show the stability condition of a digital first‐order filter. To address the non‐causality issues of SFNSs with sampling approximation error and the input data dimension explosion of NNs, the auxiliary digital first‐order filter and backstepping technology are combined to propose an adaptive neural dynamic surface control (ANDSC) scheme. Such a scheme avoids the ‐step time delays associated with the existing NN updating laws derived by the common ‐step predictor technology. A rigorous recursion method is employed to provide a comprehensive verification of the stability, guaranteeing its overall performance and dependability. Following that, the NN weight error systems are systematically decomposed into a sequence of linear time‐varying subsystems, allowing for a more detailed analysis and understanding. In order to ensure the recurrent nature of the input variables, a recursive design is employed, thereby satisfying the partial persistent excitation condition specifically designed for the RBF NNs. Meanwhile, it can verify that the NN estimated weights converge to their ideal values. Compared with the common ‐step predictor technology, there is no need to redesign the learning rules due to the designed NN weight updating laws without time delays. Subsequently, after capturing and storing the convergence weights, a novel neural learning dynamic surface control (NLDSC) scheme is specifically formulated by leveraging the acquired knowledge. The introduced methodology reduces computational complexity and facilitates practical implementation. Finally, empirical evidence obtained from simulation experiments validates the efficacy and viability of the proposed methodology.

On the stability of a single-species model with a generic delay distribution kernel

The delayed logistic equation, also known as Hutchinson’s equation, is a simple and elegant model commonly used to capture critical features of complex phenomena in biology, medicine, and economics. This paper studies the stability of a single-species logistic model with a general delay distribution and a constant inflow of nutritional resources. We provide conditions for the linear stability of the positive equilibrium and the occurrence of Hopf bifurcation. The findings complement existing literature and are applied to specific delay distributions: Uniform, Dirac-delta, and gamma distributions. Without resource inflow, we find that the positive equilibrium is stable for short delays but loses stability through Hopf bifurcation as the mean delay increases. The model’s dynamics vary with resource inflow based on the delay distribution: in uniform and Dirac-delta distributions, the dynamics are similar to the no-inflow case, whereas for the gamma distribution, stability depends on the delay order p=1,2,3.

Observer-based quantized control for networked interconnected PDE systems with actuator failures

This paper proposes an observer-based quantized controller for parabolic partial differential equation systems interconnected by a nonlinear coupling protocol. First, a Markov jump model is introduced to describe various randomly occurring actuator failures, and an observer-based pointwise controller is designed under the averaged measurement scheme. Furthermore, taking into account the limitation of network communication resources, a quantization method is adopted to relieve bandwidth pressure. In addition, stability conditions of the closed-loop system with ℋ∞ disturbance attenuation performance are derived by utilizing appropriate Lyapunov functional and inequality techniques. Ultimately, the proposed method is applied to the Fisher equation to verify its feasibility and effectiveness.

Indirect to direct band gap engineering of cubic bromide perovskite AlMgBr3 under pressure: First-principles calculations for enhanced optoelectronic applications

Through first principles, the structural, electronic, optical, mechanical, and thermodynamic properties of AlMgBr3 at different pressures are investigated for the first time. At 0 GPa, cubic AlMgBr3 is an indirect band gap insulator (Eg = 3.958 eV). Interestingly, its band gap type changes from an indirect band gap to a direct band gap at 5 GPa, making it a more favorable material for optoelectronic device applications. In addition, as the pressure increases, the conduction band electrons move downward, resulting in a reduction of the band gap, and the band gap transition from the ultraviolet to visible region. The optical properties show that AlMgBr3 has a high dielectric constant, refractive index, and conductivity, which is favorable for its application in solar cells, waveguides and other optoelectronic devices. The phase transition point of AlMgBr3 was determined to be 39.6 GPa based on the mechanical stability conditions, and the mechanical properties revealed the ductility, anisotropy and machinability of AlMgBr3. By using the quasi-harmonic Debye model, the effect of temperature and pressure on the bulk modulus, coefficient of thermal expansion, isobaric heat capacity, Debye temperature, and entropy were also calculated. It is worth mentioning that the electronic, optical, mechanical and thermodynamic properties of AlMgBr3 have been significantly enhanced under pressure.

Predefined time quasi-sliding mode control with fast convergence based on a switchable exponent for nonlinear systems

This paper proposed a new predefined time nonsingular sliding mode control (SMC) method for nonlinear systems. Firstly, based on the definition of predefined time stability (PTS), a new sufficient condition is designed to ensure that the system states converge to the origin within a predefined time. The design of a simple variable exponent not only guarantees PTS, but also enables adaptive adjustment when the system states are far away from and near the equilibrium point. And compared with traditional methods, the proposed Lemma 2 enhances the control effect and achieves faster convergence whether the system states are far from or near to the equilibrium point. Secondly, based on the proposed stability condition, a new nonsingular SMC method is designed to ensure that the tracking error converges to an arbitrarily small region within a predefined time. Finally, the proposed method is verified to have good performance through simulation and physical experiments.

Algebraic conditions for stability in Runge-Kutta methods and their certification via semidefinite programming

In this work, we present approaches to rigorously certify A- and A(α)-stability in Runge-Kutta methods through the solution of convex feasibility problems defined by linear matrix inequalities. We adopt two approaches. The first is based on sum-of-squares programming applied to the Runge-Kutta E-polynomial and is applicable to both A- and A(α)-stability. In the second, we sharpen the algebraic conditions for A-stability of Cooper, Scherer, Türke, and Wendler to incorporate the Runge-Kutta order conditions. We demonstrate how the theoretical improvement enables the practical use of these conditions for certification of A-stability within a computational framework. We then use both approaches to obtain rigorous certificates of stability for several diagonally implicit schemes devised in the literature.

Stability and convergence rate analysis of continuous-time delay-difference systems with both point and distributed delays

This paper studies the exponential stability and convergence rate analysis of continuous-time delay-difference systems. Firstly, stability and convergence rate analysis of delay-difference systems with both point delays and distributed delays having exponential integral kernels are studied by using the weighted Lyapunov–Krasovskii functionals (LKFs) approach and the state transformation approach, respectively. Different sufficient stability conditions expressed by linear matrix inequalities (LMIs) are presented. Secondly, as a particular case, LMIs based conditions and spectral radius based conditions are given to ensure the exponential stability with a guaranteed convergence rate for delay-difference systems with both point delays and distributed delays having constant integral kernels, respectively. Finally, numerical examples illustrate the effectiveness of the obtained results.

On-demand mobility-as-a-Service platform assignment games with guaranteed stable outcomes

Mobility-as-a-Service (MaaS) systems are two-sided markets, with two mutually exclusive sets of agents, i.e., travelers/users and operators, forming a mobility ecosystem in which multiple operators compete or cooperate to serve customers under a governing platform provider. This study proposes a MaaS platform equilibrium model based on many-to-many assignment games incorporating both fixed-route transit services and mobility-on-demand (MOD) services. The matching problem is formulated as a convex multicommodity flow network design problem under congestion that captures the cost of accessing MOD services. The local stability conditions reflect a generalization of Wardrop's principles that include operators’ decisions. Due to the presence of congestion, the problem may result in non-stable designs, and a subsidy mechanism from the platform is proposed to guarantee local stability. A new exact solution algorithm to the matching problem is proposed based on a branch and bound framework with a Frank-Wolfe algorithm integrated with Lagrangian relaxation and subgradient optimization, which guarantees the optimality of the matching problem but not stability. A heuristic which integrates stability conditions and subsidy design is proposed, which reaches either an optimal MaaS platform equilibrium solution with global stability, or a feasible locally stable solution that may require subsidy. For the heuristic, a worst-case bound and condition for obtaining an exact solution are both identified. Two sets of reproducible numerical experiments are conducted. The first, on a toy network, verifies the model and algorithm, and illustrates the differences local and global stability. The second, on an expanded Sioux Falls network with 82 nodes and 748 links, derives generalizable insights about the model for coopetitive interdependencies between operators sharing the platform, handling congestion effects in MOD services, effects of local stability on investment impacts, and illustrating inequities that may arise under heterogeneous populations.

On the local quadratic stability of T–S fuzzy systems in the vicinity of the origin

The main goal of this paper is to introduce new local stability conditions for continuous-time Takagi-Sugeno (T-S) fuzzy systems. These stability conditions are based on linear matrix inequalities (LMIs) in combination with quadratic Lyapunov functions. Moreover, they integrate information on the membership functions at the origin and effectively leverage the linear structure of the underlying nonlinear system in the vicinity of the origin. As a result, the proposed conditions are proved to be less conservative compared to existing methods using fuzzy Lyapunov functions. Moreover, we establish that the proposed methods offer necessary and sufficient conditions for the local exponential stability of T-S fuzzy systems. Discussions on the inherent limitations associated with fuzzy Lyapunov approaches are also given. To illustrate the theoretical results, we provide comprehensive examples that demonstrate the core concepts and validate the efficacy of the proposed conditions.

Сравнительная картина и проблемы иностранного портфельного инвестирования в реальном секторе экономики / Comparative picture and problems of foreign portfolio investment

The article is devoted to the comparative picture and problems of foreign portfolio investment in the real sector of the economy. The article analyzes the main obstacles that arise in the process of attracting foreign portfolio investments. The factors influencing the increase in the flow of foreign portfolio investments, the possibility of their rapid attraction and effective distribution are considered. Sustainable development of the economy of any country today is impossible without active participation in world economic relations. Along with international trade, international flows of investment capital, carried out on the basis of effective cooperation between countries, are becoming increasingly important. In recent years, the issue of attracting external capital to the Armenian economy has become topical. Foreign investment is one of the most important conditions for stabilization and growth of the country's economy. This is primarily due to the fact that firms’ own financial resources are limited, which is due to a number of reasons in Armenian reality, such as high return on invested capital, high taxes, etc. The movement of global financial capital through the channels of the global financial system occurs in various forms, among which international portfolio investments, including investments in equity and debt securities, occupy a special place. Financial globalization is accelerating the trend towards expanding the market segment of international portfolio investments.

Optimism leads to optimality: Ambiguity in network formation

We analyze a model of endogenous two-sided network formation where players are affected by uncertainty about their opponents' decisions. We model this uncertainty using the notion of equilibrium under ambiguity as in Eichberger and Kelsey (2014). Unlike the set of Nash equilibria, the set of equilibria under ambiguity does not always include underconnected and thus inefficient networks such as the empty network. On the other hand, it may include networks with unreciprocated, one-way links, which comes with an efficiency loss as linking efforts are costly. We characterize equilibria under ambiguity and provide conditions under which increased player optimism comes with an increase in connectivity and realized benefits in equilibrium. Next, we analyze network realignment under a myopic updating process with optimistic shocks and derive a global stability condition of efficient networks in the sense of Kandori et al. (1993). Under this condition, a subset of the Pareto optimal equilibrium networks is reached, specifically, networks that maximize the players' total benefits of connections.

Almost sure exponential stability of stochastic nonlinear semi-Markov jump T-S fuzzy systems under intermittent EDF scheduling controller

This paper studies the almost sure exponential stability (ASES) with a random switching process of stochastic nonlinear semi-Markov jump T-S Fuzzy systems (SMJT-SFSs) based on an intermittent scheduling controller. The intermittent scheduling controller is established by the earliest deadline first (EDF) algorithm, where the controller is enabled to take a rest interval. By building a mode-dependent Lyapunov function and employing the Itô formula, the sufficient stability conditions of the SMJT-SFSs with intermittent scheduling controller are obtained about solvable forms of linear matrix inequalities (LMIs). Furthermore, we explore two special cases involving linear systems and deterministic systems. In addition, an application example of the nonlinear robot arm model of single-link is provided to illustrate the results.

Melting Curve of Black Phosphorus: Evidence for a Solid-Liquid-Liquid Triple Point.

Black phosphorus (bP) is a crystalline material that can be seen as an ordered stacking of two-dimensional layers, which results in outstanding anisotropic physical properties. The knowledge of its pressure (P)-temperature (T) phase diagram, and in particular, of its melting curve is fundamental for a better understanding of the synthesis and stability conditions of this element. Despite the numerous studies devoted to this subject, significant uncertainties remain regarding the determination of the position and slope of its melting curve. Here we measured the melting curve of bP in an extended P, T region from 0.10(3) to 5.05(40) GPa and from 914(25) to 1788(70) K, using in situ high-pressure and high-temperature synchrotron X-ray diffraction. We employed an original metrology based on the anisotropic thermoelastic properties of bP to accurately determine P and T. We observed a monotonic increase of the melting temperature with pressure and the existence of two distinct linear regimes below and above 1.35(15) GPa, with respective slopes of 348 ± 21 and of 105 ± 12 K·GPa-1. These correspond to the melting of bP toward the low-density liquid and the high-density liquid, respectively. The triple point at which solid bP and the two liquids meet is located at 1.35(15) GPa and 1350(25) K. In addition, we have characterized the solid phases after crystallization of the two liquids and found that, while the high-density liquid transforms back to solid bP, the low-density liquid crystallizes into a more complex, partly crystalline and partly amorphous solid. The X-ray diffraction pattern of the crystalline component could be indexed as a mixture of red and violet P.

Stability and stabilization of discrete-time linear compartmental switched systems via Markov chains

The stabilizing switching signal design of discrete-time linear compartmental switched systems (DT-LCSSs) has been heretofore unsolved. It has been proven that a DT-LCSS is stabilizable if and only if it is stabilizable by a periodic switching signal. However, it still needs to be determined whether the period of a stabilizing switching signal can be confined within a bound. Moreover, the existing design method for stabilizing periodic switching signals requires the diagonal entries of system matrices of all subsystems to be strictly positive. In this study, we propose a novel approach to solve this problem completely. We construct a discrete-time Markov chain for a given DT-LCSS, termed the associated Markov chain, and prove the equivalence of stability and stabilizability between the DT-LCSS and the associated Markov chain. Based on this, verifiable necessary and sufficient conditions for stability and stabilizability are derived. Especially, the period of a stabilizing switching signal for an n-dimensional DT-LCSS can always be chosen within the bound n2−n+1. We propose a state-independent stabilizing switching signal design method for general stabilizable DT-LCSSs. We also prove the equivalence between stabilizability by state-independent switching laws and stabilizability by state-dependent switching laws. A state-dependent global stabilizing switching signal design method is also proposed. Additionally, the proposed results are applied to the consensus analysis of discrete-time leader–follower multi-agent systems with switching communication digraphs. The effectiveness of the theoretical results is demonstrated by examples.

Mean-square exponential stabilization of mixed-autonomy traffic PDE system

Control of mixed-autonomy traffic where Human-driven Vehicles (HVs) and Autonomous Vehicles (AVs) coexist on the road has gained increasing attention over the recent decades. This paper addresses the boundary stabilization problem for mixed traffic on freeways. The traffic dynamics are described by uncertain coupled hyperbolic partial differential equations (PDEs) with Markov jumping parameters, which aim to address the distinctive driving strategies between AVs and HVs. Considering that the spacing policies of AVs vary in mixed traffic, the stochastic impact area of AVs is governed by a continuous Markov chain. The interactions between HVs and AVs such as overtaking or lane changing are mainly induced by impact areas. Using backstepping design, we develop a full-state feedback boundary control law to stabilize the deterministic system (nominal system). Applying Lyapunov analysis, we demonstrate that the nominal backstepping control law is able to stabilize the traffic system with Markov jumping parameters, provided the nominal parameters are sufficiently close to the stochastic ones on average. The mean-square exponential stability conditions are derived, and the results are validated by numerical simulations.

Research on Phase Diagram of an Improved Dual-Lane Heterogeneous Lattice Hydrodynamic Model Considering Curved Road

The actual road traffic is a heterogeneous traffic flow composed of various types of vehicles. Ensuring safe operation of different types of vehicles while improving traffic stabilization has become a very important issue. Inspired by this, taking into account the discrepancy between the safety headway and the maximal velocity of different types of vehicles, an improved dual-lane heterogeneous lattice hydrodynamic model (DLHLHM) considering curved road is put forward in this research. The improved model’s stabilization condition is inferred by reductive perturbation method. The mKdV equation is also inferred to describe the evolutionary processes of traffic density wave. Numerical examples probed into the impact of lane-changing rate, angle of curved road and different maximum speeds and safety distances on traffic stabilization. The hysteresis loop for new model also is explored to study traffic flow stabilization. Numerical simulation results accord with theoretical analysis, which indicates the effectiveness and feasibility of the DLHLHM. This study provides a new perspective on the dynamic evolution of heterogeneous traffic flow.

Dynamical behavior of obligatory mutualistic-cheater interactions under the influence of white noise

Intraspecific mutualism is vital for ecosystems. Some interactions involve one species benefiting without reciprocating. Recent experiments suggest stable co-existence between mutualists and cheaters. This paper focuses on interactions between obligate mutualistic species and their cheaters, using a modified Lotka-Volterra model to analyze the negative impact of cheaters. We establish conditions for stability and verify a diffusive system's stability under spatial effects. Additionally, we study population dynamics fluctuations in the presence of Gaussian noise.

Stability of few-cycle light bullets in nonlinear metamaterials beyond the slowly varying envelope approximation

This letter explores stable few-cycle light bullets in nonlinear metamaterials beyond the slowly varying envelope approximation. Using a (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation, the letter explores solitonic behaviors of light bullets in materials with a negative refractive index under the few-cycle regime. The analysis involves a Lagrangian variational approach designed for dissipative nonlinear systems and utilizes direct numerical simulations to demonstrate the possibility of stable light bullets forming in metamaterials and corroborate the stability analysis. Based on the stability conditions for the system parameters using the Routh-Hurwitz criterion, optimal values of the cubic and quintic self-steepening effects promote stable propagation of light bullets, while an imbalance may lead to collapse. These findings suggest that combining few-cycle light bullets with metamaterial-based optics may enable new possibilities for ultrafast and nonlinear optics.

Fair division with two-sided preferences

We study a fair division setting in which participants are to be fairly distributed among teams, where not only do the teams have preferences over the participants as in the canonical fair division setting, but the participants also have preferences over the teams. We focus on guaranteeing envy-freeness up to one participant (EF1) for the teams together with a stability condition for both sides. We show that an allocation satisfying EF1, swap stability, and individual stability always exists and can be computed in polynomial time, even when teams may have positive or negative values for participants. When teams have nonnegative values for participants, we prove that an EF1 and Pareto optimal allocation exists and, if the valuations are binary, can be found in polynomial time. We also show that an EF1 and justified envy-free allocation does not necessarily exist, and deciding whether such an allocation exists is computationally difficult.