Nonlinear Analysis: Modelling and Control
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Trajectory controllability of semilinear dynamic systems on time scales
Full text linkThis paper explores the trajectory controllability of semilinear dynamic systems defined over time scales, which is an important aspect in understanding and manipulating the behavior of such systems across discrete and continuous domains. We address the controllability of these systems under the assumption that the nonlinearities satisfy a Lipschitz-type condition. Our approach involves a detailed analysis of how these conditions impact the ability to steer the system’s state along a desired trajectory within a finite-time horizon. We establish sufficient conditions for trajectory controllability (T-controllability), providing a theoretical framework that extends classical results from differential and difference equations to the broader context of time-scale calculus. To illustrate the practical implications of our theoretical findings, we include several numerical examples that demonstrate the application of our results to specific semilinear dynamic systems, highlighting the versatility and effectiveness of our approach
New strong convergence algorithms for general equilibrium and variational inequality problems and resolvent operators in Banach spaces
Full text linkIn this paper, we introduce two new algorithms for solving variational inequalities in Banach spaces. Our aim is finding a common element of the solution set of variational inequalities (for two inverse-strongly monotone operators) and an equilibrium problem and the set of fixed points of two relatively nonexpansive mappings and a family of resolvent operators. Then the strong convergence of the sequences generated by these algorithms to this element will be proved under suitable conditions. Finally, we provide a numerical example to illustrate our main results
Nonparametric changed segment detection in functional data
Full text linkWe address the epidemic change point detection problem without parametric assumptions. We propose statistics based on Cramér–von Mises-type statistic and reproducing kernel Hilbert space that iterate through all interval subsets, rescaling them to remain sensitive to both short and long epidemics. We prove limit theorems and provide quantiles for both statistics under the different parametrizations. The simulations show consistent power across a wide range of scenarios, and an application to electricity balancing prices consistently detects a market disturbance
Quantization-based event-triggered control for synchronization of 2-D discrete-time switched master-slave systems in Roesser model
Full text linkThis paper investigates synchronization control for 2-D discrete-time switched master-slave systems modeled by the Roesser framework, which is classic for spatiotemporal dynamics in 2-D systems. A novel quantization-based event-triggered control strategy is proposed to handle complexities from switching dynamics, discrete-time features, and spatial coupling, while considering limited communication resources. By designing a mode-dependent event-triggered strategy and constructing mode-dependent Lyapunov functions for horizontal and vertical dynamics, new sufficient conditions are derived to ensure global exponential synchronization (GES) of the system. The approach relaxes strict stability requirements for individual modes, allowing global stability even with unstable modes. Additionally, the integration of quantization techniques and event-triggered mechanisms significantly reduces data transmission, thereby optimizing network bandwidth usage. Numerical simulations verify the method’s effectiveness
Existence and stability results for triple systems of fractional Sturm–Liouville–Langevin equations with cyclic boundary conditions
Full text linkWe investigate a triple system of fractional Sturm–Liouville–Langevin equations with cyclic antiperiodic boundary conditions. The fixed point theorem serves as a tool to establish the existence and uniqueness criteria for solutions. By applying the Banach contraction principle, we also obtain the Ulam–Hyers stability of the proposed system. Finally, examples are provided to illustrate main results
Energetic formulation of the subgroup commutativity degree
Full text linkFinite groups in which every pair of subgroups (H, K) satisfies H K = K H have been classified by Iwasawa, but only in the last decade it was introduced the notion of subgroup commutativity degree sd(G) of groups G. From restrictions of numerical nature on sd(G) one usually derives structural conditions on G; in fact, among groups G with sd(G) = 1, one finds those originally studied by Iwasawa. Here we offer a new perspective of study for sd(G); we use a recently introduced graph, which is called nonpermutability graph of subgroups ΓL(G) of G, in order to restrict sd(G) via the notion of energy of ΓL(G) and by means of methods of spectral graph theory. In particular, we find new criteria of nilpotence for G along with new bounds for sd(G)
An advanced visualization of self-organizing maps by determining data clusters
Full text linkThis paper proposes a novel approach to improve the visualization capabilities of self-organizing maps and facilitate the identification of the resulting clusters. Unlike other clustering algorithms, self-organizing maps lack the feature to select a predefined number of clusters, and the boundaries of the clusters are not explicitly represented on the self-organizing maps. The main advantage of our proposed approach is that the option for selecting the desired number of clusters has been implemented. The experimental investigation was performed using four datasets with different characteristics. The improved visualization leverages various similarity distances to assess their impact on performance. The effectiveness of the novel approach to clustering results has been compared with those of the well-known k-means and hierarchical clustering methods, which allow for the selection of the desired number of clusters. Additionally, the visualization results, obtained by the proposed approach, were compared with those produced using the Orange Data Mining tool, where the u-matrix is applied to visualize a self-organizing map. The advantage of our approach compared to the u-matrix visualization has been highlighted in this paper. The performance of clustering algorithms has been measured by calculating the ratio of data items correctly assigned to clusters in the case when the clusters are predefined in the analyzed dataset. The results obtained showed that the most effective similarity distances are the cosine and correlation distances, which help to detect the correctly predefined clusters in the visualization of self-organizing maps
Absolute exponential stability of switching time-delay Lurie systems with the application to switching Hopfield neural networks
Full text linkThis paper investigates the problem of absolute exponential stability analysis for switching time-delay Lurie system (STDLS) with all modes unstable. By proposing a novel switching time-varying Lyapunov–Razumikhin function, a computable sufficient condition is formulated to guarantee absolute exponential stability of STDLS under mode-dependent range dwell-time (MDRDT) switching. Especially, theoretical results are applied to switching delay Hopfield neural network. Simulations are served to illustrate the developed theory
A new class of integral-multipoint boundary value problems for nonlinear Hadamard fractional differential equations on a semiinfinite domain
Full text linkIn this paper, we introduce and investigate a new class of Hadamard fractional differential equation with integral-multipoint boundary conditions on a positive semiinfinite domain. We use the contraction mapping principle and the fixed point index theorem, respectively, to prove the uniqueness and the existence of at least two positive solutions to the given problem. Our results are new and enrich the literature on Hadamard-type fractional differential equations on unbounded domains. Some examples illustrating the main results are presented
A novel fractional operator-based model for Parkinson’s disease: Analyzing abnormal beta-oscillation and the influence of synaptic parameters
Full text linkWith the aggravation of the global aging trend, Parkinson’s disease has become a hot spot of scientific research all over the world. Abnormal β-oscillation in the basal ganglia region is considered to be a major inducement of Parkinson’s disease. In this paper, a new and more complete Parkinson’s model based on fractional operators is proposed to study the oscillation behavior of the basal ganglia region. The correctness of this new fractional model is validated by the simulation of Nambu and Tachibana’s experiment [A. Nambu, Y. Tachibana, Mechanism of parkinsonian neuronal oscillations in the primate basal ganglia: some considerations based on our recent work, Front. Syst. Neurosci., 8:74, 2014]. Then we carry out the Hopf bifurcation analysis of the fractional model and derive the critical conditions for periodic oscillation. The influence of important parameters on the oscillation behavior of the system is analyzed by numerical simulations. It is found that proper control of synaptic transmission delay and synaptic connection strength can improve the abnormal β-oscillation behavior in the basal ganglia region effectively. In addition, the fractional Parkinson’s model in this paper provides more flexibility for model fitting and parameter estimation. The choice of the fractional order α plays a crucial role in the analysis of system oscillation