Sufficient Conditions Research Articles

Secure impulsive tracking of multi-agent systems with directed hypergraph topologies against hybrid deception attacks

This research delves into the challenges of achieving secure consensus tracking within multi-agent systems characterized by directed hypergraph topologies, in the face of hybrid deception attacks. The hybrid discrete and continuous deception attacks are targeted at the controller communication channels and the hyperedges, respectively. To overcome these threats, an impulsive control mechanism based on hypergraph theory are introduced, and sufficient conditions are established, under which consensus can be maintained in a mean-square bounded sense, supported by rigorous mathematical proofs. Furthermore, the investigation quantifies the relationship between the mean-square bounded consensus of the multi-agent system and the intensity of the deception attacks, delineating a specific range for this error metric. The robustness and effectiveness of the proposed control method are verified through comprehensive simulation experiments, demonstrating its applicability in varied scenarios influenced by these sophisticated attacks. This study underscores the potential of hypergraph-based strategies in enhancing system resilience against complex hybrid attacks.

Virtually coupled train set control subject to space-time separation: A distributed economic MPC approach with emergency braking configuration

The emerging virtual coupling technology aims to operate multiple train units in a virtually coupled train set (VCTS) at a minimal but safe distance. To guarantee collision avoidance, the safety distance should be calculated using the state-of-the-art space-time separation principle that separates the emergency braking (EB) trajectories of two successive units during the whole EB process. In this case, the minimal safety distance is usually numerically calculated without an analytic formulation. Thus, the constrained VCTS control problem is hard to address with space-time separation, which is still a gap in the existing literature. To solve this problem, we propose a distributed economic model predictive control (DEMPC) approach with computation efficiency and theoretical guarantee. Specifically, to alleviate the computation burden, we transform implicit safety constraints into explicitly linear ones, such that the optimal control problem in DEMPC is a quadratic programming problem that can be solved efficiently. For theoretical analysis, sufficient conditions are derived to guarantee the recursive feasibility and stability of DEMPC, employing compatibility constraints, tube techniques and terminal ingredient tuning. Moreover, we extend our approach with globally optimal and distributed online EB configuration methods to shorten the minimal distance among VCTS. Finally, experimental results demonstrate the performance and advantages of the proposed approaches.

Event-triggered impulsive control for nonlinear stochastic delayed systems and complex networks

In this paper, we probe the pth moment exponential stability (p−ES) of stochastic delayed systems subject to event-triggered delayed impulsive control (ETDIC), where the impulsive intensities are assumed to be positive random variables. Based on event-triggered mechanism (ETM) in the sense of expectation, some new sufficient conditions are developed to ensure the stability of the addressed system with Zeno-free behavior. The Lyapunov-Razumikhin method is adopted to handle the time-varying delay in continuous dynamics, and the concept of average random impulsive estimation (ARIE) is introduced to reduce the design requirements of the controller. Especially, the proposed ETDIC strategy not only generates the impulse time sequence according to the predesigned ETM, but also removes the limitations on the size of time delays. Furthermore, the ETM serves as a solution for synchronization problems in complex neural networks. Finally, two examples are given to illustrate the effectiveness of our conclusions.

Characterization and dynamics of certain classes of polynomial vector fields on the torus

In this paper, we classify all polynomial vector fields in R3 of degree up to three such that their flow makes the torusT2={(x,y,z)∈R3:(x2+y2−a2)2+z2−1=0}witha∈(1,∞) invariant. We also classify cubic Kolmogorov vector fields on T2 and prove that they exhibit a rational first integral. We study ‘pseudo-type-n’ vector fields on T2 and show that any such vector field is completely integrable. We prove that the Lie bracket of any two quadratic vector fields on T2 is completely integrable. We explicitly find all cubic vector fields on T2 which achieve the sharp bounds for the number of invariant meridians and parallels. We present necessary and sufficient conditions when invariant meridians and parallels of cubic vector fields on T2 are periodic orbits or limit cycles. We discuss invariant meridians and parallels of pseudo-type-n vector fields as well. Moreover, we characterize the singular points of a class of polynomial vector fields on T2.

On the polynomiality conjecture of cluster realization of quantum groups

In this paper, we give a sufficient and necessary condition for a regular element of a quantum cluster algebra Oq(X) to be universally polynomial. This resolves several conjectures by the first author on the polynomiality of the cluster realization of quantum group generators in different families of positive representations.

Some remarks on relative modular categories

We study properties of relative modular categories and derive sufficient conditions for their existence. In particular, we derive sufficient conditions for relative pre-modular categories to be non-degenerate and sufficient conditions for the construction of generically semisimple categories.

Multistability and fixed-time multisynchronization of switched neural networks with state-dependent switching rules

This paper presents theoretical results on the multistability and fixed-time synchronization of switched neural networks with multiple almost-periodic solutions and state-dependent switching rules. It is shown herein that the number, location, and stability of the almost-periodic solutions of the switched neural networks can be characterized by making use of the state-space partition. Two sets of sufficient conditions are derived to ascertain the existence of 3n exponentially stable almost-periodic solutions. Subsequently, this paper introduces the novel concept of fixed-time multisynchronization in switched neural networks associated with a range of almost-periodic parameters within multiple stable equilibrium states for the first time. Based on the multistability results, it is demonstrated that there are 3n synchronization manifolds, wherein n is the number of neurons. Additionally, an estimation for the settling time required for drive–response switched neural networks to achieve synchronization is provided. It should be noted that this paper considers stable equilibrium points (static multisynchronization), stable almost-periodic orbits (dynamical multisynchronization), and hybrid stable equilibrium states (hybrid multisynchronization) as special cases of multistability (multisynchronization). Two numerical examples are elaborated to substantiate the theoretical results.

Robust [formula omitted] control for LFC of discrete T-S fuzzy MAPS with DFIG and time-varying delays

The H∞ LFC control problem for a class of nonlinear power systems with time-varying delays is under study. Considering uncertainties arising from nonlinear issues such as the generation rate constraint (GRC) and governor dead band (GDB), as well as the high variability of renewable energy sources like wind power, the model is transformed into a discrete-time Takagi–Sugeno (T-S) fuzzy model with parameter uncertainty. By constructing a Lyapunov-Krasovskii functional, and employing difference inequalities and generalized cross-convex matrix inequalities, sufficient conditions for the asymptotic stability of power systems are provided. Based on the obtained conditions, a controller is designed to ensure the asymptotic stability of Multi-Area Power Systems (MAPS), with the performance index being H∞. Finally, simulation results demonstrate the correctness and effectiveness of the theorem.

Effects of wariness on economic growth in overlapping generations models

We introduce the notion of wariness, defined as a concern for the lowest lifetime utility, in overlapping generations models and explore its effects on economic growth. In an exogenous growth model, under standard assumptions, we prove that the capital stock converges to a steady state. We then explore conditions under which this steady state is increasing (or decreasing) in the wariness level. We also provide a necessary and sufficient condition for the dynamic efficiency of the intertemporal equilibrium. In endogenous growth models (à la Romer (1986) or à la Barro (1990)), we show that the growth rate of capital stock per capita in the economy with wariness is lower (higher, respectively) than that in the economy without wariness if and only if the capital return is high (low, respectively).

Stability and Substitutability in Multi-period Matching Markets

We analyze a multi-period matching market where matching between agents is decided for each time period. To analyze this situation, we embed the situation into the framework of many-to-many matching with contracts where the contract includes the time period at which the matching occurs. While a general stability concept is already defined for the matching with contracts framework, in a multi-period matching model, a stable outcome may not exist when contracts exhibit complementarities across time periods. Thus, we define a weaker stability concept called temporal stability by taking into account the dynamic nature of the model. We provide sufficient conditions for the existence of a temporally stable outcome, including a corresponding substitutability condition, ordered substitutability, for the multi-period matching model.

Bipartite consensus of concatenated opinion dynamics for two antagonistic groups: A game theoretical perspective

Recent decades have witnessed a significant surge in scholarly attention toward formulating dynamic models for opinion formation on social networks. This paper considers the bipartite consensus problem over a series of issues for two antagonistic groups wherein two graphs are employed to model the interaction within groups and a noncooperative game to describe the interaction between groups. Firstly, we prove that the noncooperative game admits the unique Nash equilibrium. Secondly, we give sufficient conditions for achieving a bipartite consensus that both graphs are strongly connected. Additionally, we obtain that the game’s mechanism decides the convergence rate of bipartite consensus from one issue to the next. It is important to highlight that confrontation is key in elevating the most influential individuals to leadership positions within their groups. Thirdly, we construct the bipartite consensus state for a particular case where the networks of two groups are balanced. Finally, we present two examples to demonstrate the theoretical results.

Wolff potential estimates and Wiener criterion for nonlocal equations with Orlicz growth

We prove the Wolff potential estimates for nonlocal equations with Orlicz growth. As an application, we obtain the Wiener criterion in this framework, which provides a necessary and sufficient condition for boundary points to be regular. Our approach relies on the fine analysis of superharmonic functions in view of nonlocal nonlinear potential theory.

Sufficient condition for reliable logic operations in an over damped bistable system driven by Gaussian white noise.

In this paper, we investigate an overdamped bistable system subject to Gaussian white noise and logical inputs. By solving and estimating the steady distribution of the corresponding Fokker-Planck equation and considering the two essential features of the reliable logic operations (RLOs)-initial value independence and sign invariance-we establish the sufficient condition for RLO occurrence and identify parametric resonance phenomena in the system. Our numerical simulations confirm the reliability and accuracy of the theoretical results. This work offers insights into enhancing the accuracy of stochastic systems, particularly in the realm of logical stochastic resonance, thereby contributing to advancements in understanding and controlling stochastic dynamical systems.

Sampled-data control for Markovian switching neural networks with output quantization and packet dropouts

This paper explores sampled-data control for Markovian switching neural networks (MSNNs) with dynamic output quantization and packet dropouts. The primary goal is to construct a multi-mode, quantized sampled-data controller that ensures stochastic stability and H∞ disturbance-reduction performance of the closed-loop MSNN. A Bernoulli-distributed random variable with uncertain probability is introduced to characterize the incidence of packet dropouts. To describe potential mode inconsistencies that may occur between the MSNN and controller, an exponential hidden Markov model is employed. Furthermore, the quantizer’s dynamic scaling factor is intentionally built as a piecewise function to avoid the potential division-by-zero problem. A sufficient condition for stochastic stability and H∞ disturbance-reduction performance is proposed, utilizing a mode- and time-dependent Lyapunov-type functional and several stochastic analysis tools. Then, through decoupling nonlinearities, a numerically efficient approach for determining the desired controller gains and parameter range associated with the dynamic scaling factor is developed. In order to facilitate comparisons, the situation with no uncertainty in the probability of packet dropouts is studied, and both analysis and design approaches are offered. Finally, two simulation examples are provided to validate the effectiveness and applicability of the developed approaches.

Construction of pairwise orthogonal Parseval frames generated by filters on LCA groups

The generalized translation invariant (GTI) systems unify the discrete frame theory of generalized shift-invariant systems with its continuous version, such as wavelets, shearlets, Gabor transforms, and others. This article provides sufficient conditions to construct pairwise orthogonal Parseval GTI frames in L2(G) satisfying the local integrability condition (LIC) and having the Calderón sum one, where G is a second countable locally compact abelian group. The pairwise orthogonality plays a crucial role in multiple access communications, hiding data, synthesizing superframes and frames, etc. Further, we provide a result for constructing N numbers of GTI Parseval frames, which are pairwise orthogonal. Consequently, we obtain an explicit construction of pairwise orthogonal Parseval frames in L2(R) and L2(G), using B-splines as a generating function. In the end, the results are particularly discussed for wavelet systems.

Tilting theory for finite dimensional 1-Iwanaga-Gorenstein algebras

We study tilting objects of the stable category CM_ZA of graded Cohen-Macaulay modules over a finite dimensional graded Iwanaga-Gorenstein algebra A. We first show that if there exists a tilting object in CM_ZA, then its endomorphism algebra always has finite global dimension. Next, to study the existence of a tilting object, we introduce a numerical invariant g(A). In the case where A is 1-Iwanaga-Gorenstein, we give a sufficient condition on g(A) for the existence of a tilting object. As an application, for a truncated preprojective algebra Π(Q)w of a tree quiver Q, we prove that CM_ZΠ(Q)w always admits a tilting object.

A New Method for Constructing Self-Dual Codes over Finite Commutative Rings with Characteristic 2

In this work, we present a new method for constructing self-dual codes over finite commutative rings R with characteristic 2. Our method involves searching for k×2k matrices M over R satisfying the conditions that its rows are linearly independent over R and MM⊤=α⊤α for an R-linearly independent vector α∈Rk. Let C be a linear code generated by such a matrix M. We prove that the dual code C⊥ of C is also a free linear code with dimension k, as well as C/Hull(C) and C⊥/Hull(C) are one-dimensional free R-modules, where Hull(C) represents the hull of C. Based on these facts, an isometry from Rx+Ry onto R2 is established, assuming that x+Hull(C) and y+Hull(C) are bases for C/Hull(C) and C⊥/Hull(C) over R, respectively. By utilizing this isometry, we introduce a new method for constructing self-dual codes from self-dual codes of length 2 over finite commutative rings with characteristic 2. To determine whether the matrix MM⊤ takes the form of α⊤α with α being a linearly independent vector in Rk, a necessary and sufficient condition is provided. Our method differs from the conventional approach, which requires the matrix M to satisfy MM⊤=0. The main advantage of our method is the ability to construct nonfree self-dual codes over finite commutative rings, a task that is typically unachievable using the conventional approach. Therefore, by combining our method with the conventional approach and selecting an appropriate matrix construction, it is possible to produce more self-dual codes, in contrast to using solely the conventional approach.

A complete analysis of a slow–fast modified Leslie–Gower predator–prey system

In this paper, we transform a modified Leslie–Gower predator–prey system into the corresponding fast–slow version by assuming that prey reproduces much faster than predator, and then perform a complete analysis about its dynamics. More specifically, we find the necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its (or their) location, and then we further fully determine its (or their) dynamics under explicit conditions. Besides, by converting the slow–fast system into its slow–fast normal form, we are able to characterize its rich dynamics completely, including relaxation oscillation, singular Hopf bifurcation, canard explosion, homoclinic orbits, heteroclinic orbits and global attraction of equilibrium. Moreover, the sufficient conditions to guarantee these various rich dynamics are explicitly given, including the explicit conditions to determine whether singular Hopf bifurcation is supercritical or subcritical, which generally cannot be explicitly derived in the existing literatures. Additionally, the cyclicity of diverse canard cycles is found under explicit conditions. Of particular interest is that we show the existence and uniqueness of one canard cycle without head whose cyclicity is at most two under explicit parameters conditions. Our results complement and enrich the previous work (relaxation oscillation, and global attraction of a boundary equilibrium) about this fast–slow system. We also employ numerical simulations to illustrate our theoretical analysis.

Stochastic comparisons of second largest order statistics with dependent heterogeneous random variables

In the context of actuarial science, the second largest claim amount is crucial to insurance analysis since they provide useful information for determining annual premium. In this article, we provide sufficient conditions of the second largest claim amounts arising from two sets of dependent and heterogeneous individual risk models according to various stochastic orders. It is first shown that the reversed hazard order between the occurrence probabilities vectors implies the reversed hazard order of the second largest claim amounts under certain conditions. Second, sufficient conditions are established for the usual stochastic ordering of the second largest claim amounts arising from heterogeneous dependent individual risk models under copula dependence. Finally, several examples illustrating the theoretical results are presented as well.

A fast smoothing newton method for bilevel hyperparameter optimization for SVC with logistic loss

Support vector classification (SVC) with logistic loss has excellent theoretical properties in classification problems where the label values are not continuous. In this paper, we reformulate the hyperparameter selection for SVC with logistic loss as a bilevel optimization problem in which the upper-level problem and the lower-level problem are both based on logistic loss. The resulting bilevel optimization model is converted to a single-level nonlinear programming (NLP) based on the KKT conditions of the lower-level problem. Such NLP contains a set of nonlinear equality constraints and a simple lower-bound constraint. The second-order sufficient condition is characterized, which guarantees that the strict local optimizers are obtained. To solve such NLP, we apply the smoothing Newton method proposed in [Liang L, Sun D., Toh KC. A squared smoothing Newton method for semidefinite programming, 2023] to solve the KKT conditions, which contain one pair of complementarity constraints. We show that the smoothing Newton method has a superlinear convergence rate. Extensive numerical results verify the efficiency of the proposed approach and strict local minimizers can be achieved both numerically and theoretically. In particular, compared with other methods, our algorithm can achieve competitive results while consuming less time than other methods.