This paper pioneers the application of algebraic matrix partial order, Löwner partial order, to the multi-player nonzero-sum games of nonlinear systems (NMPNZS) with unmatched uncertainty, with extended investigations incorporating event-triggered mechanism (ETM) control method integrated with actor-critic neural networks (NNs). To handle unmatched uncertainties and construct appropriate auxiliary system are critically important issues. However, the construction of the auxiliary system by exploiting the seemingly unrelated Löwner partial order constitutes a particularly innovative and technically challenging methodology proposed in this study, distinguishing it from existing approaches. Moreover, through rigorous theoretical analysis, it is proved that under the framework of Löwner partial order-constructed auxiliary system with redefined value function, robust control issue can be transformed into optimal control issue. Then based on the designed ETM condition, in contrast to traditional time-triggered mechanism (TTM) control it reduces both computational burden and communication bandwidth. Furthermore, a kind of composite actor-critic NNs architecture is synergistically embedded within the control-theoretic framework to operationalize Nash equilibrium attainment, with optimization of control policy. By constructing suitable Lyapunov functions, it is demonstrated that both the weight approximation errors and the considered NMPNZS system are uniformly ultimately bounded (UUB). Meanwhile, the Zeno phenomenon is excluded by proving the inter-event time is bounded. Finally, a nonlinear numerical example and the canonical reinforcement learning testing environment ‘Pendulum-v1’ are employed to validate the operational feasibility and effectiveness of the developed control strategy.

In this article, the synchronization problem of hybrid stochastic coupled systems with semi-Markov jump and time-varying delays via adaptive aperiodically intermittent control is studied, which satisfies the nonlinear coupling condition. Compared with the existing works on hybrid coupling, the hybrid coupling in this paper is not limited to linear coupling conditions. Additionally, the model incorporates time-varying delays, stochastic perturbations, and semi-Markov jump topology to enhance its generality. It is worth noting that addressing the combined effects of these factors to achieve synchronization poses significant challenges. Through the Lyapunov method and graph theory, some sufficient conditions for adaptive aperiodically intermittent controller are obtained. Furthermore, several synchronization criteria related to the system's topology and the maximum uncontrolled ratio are established. Finally, an illustrative example is given to show the effectiveness and applicability of the theoretical results.

Using the ∂¯-steepest descent method associated with the Riemann-Hilbert problem, we study the long-time asymptotics of solutions to the defocusing fifth-order modified Korteweg-de Vries (mKdV) equation with a non-vanishing background, which admits discrete spectra. Taking the self-similar variable ξ=x/(5t), we derive distinct long-time asymptotic behaviors in two solitonless regions with bounded ξ: ξ > 6 and ξ < 1.2, respectively. In the region ξ > 6, the phase function possesses four real and six purely imaginary phase points. Accordingly, the asymptotic expansion of the solution consists of the leading term from the non-vanishing background, an order term from the continuous spectrum and a residual error from the ∂¯-problem. In the region ξ < 1.2, ten stationary points of the phase function lie off the jump contour, and the corresponding asymptotic expansion contains the nonvanishing background plus a residual error of order O(t−1).

Gait generation for multi-legged robots is a core research topic in the field of robotic locomotion control. The central pattern generator (CPG) control system has widespread applications in gait generation for multi-legged robots. Generally, oscillators such as van der Pol (VDP) and Hopf are used as coupling units in the CPG control system because they can generate periodic rhythmic signals that simulate gait patterns. However, the CPG systems constructed with these oscillators usually have high system dimensions and multiple control parameters, which make the control process more complex. In fact, the one-dimensional Hopfield-like unit can also achieve periodic rhythms under the regulation of time delay. Taking the eight-legged robot as an example, this paper proposes an effective gait control method based on the delayed Hopfield-like CPG control system. First, a delayed Hopfield-like CPG control system is constructed through a unidirectional ring network structure. Moreover, the control mechanism of time-delay on the spatiotemporal patterns of rhythmic oscillations is revealed using the theory of symmetric Hopf bifurcation. Based on this mechanism, the eight-legged robot successfully replicates many of the classic locomotion gaits of eight-legged animals. Furthermore, the joint mapping function and time-scale transformation method are designed to achieve the coordination of legged joints and the stability of the gait patterns of the eight-legged robot. Finally, the experimental results on the Webots simulation platform show that the delayed Hopfield-like CPG control system can effectively generate multiple gaits for the eight-legged robot.

This paper uses an adaptive noise control strategy to investigate the almost sure exponential stability of a class of stochastic coupled networks influenced by jump diffusion. Due to the strong dependence of the diffusion coefficient on adaptive signals, traditional Lyapunov methods face significant limitations in handling stochastic noise. Different from most existing literature, a novel martingale-based approach that leverages the positive effects of control-dependent stochastic noise is presented to maintain system stability. Besides, the adaptive noise control mechanism adjusts dynamically based on the system state, enhancing the stability of the system against unpredictable jumps. Finally, an example of single-link arms is provided to argue the feasibility and efficacy of the proposed control strategy.

In this paper, we propose the use of matrix partial order: L & ouml;wner and GL partial orders from algebra to investigate some classes of generalized nonlinear systems with uncertainties. Due to the presence of unmatched and structural uncertainties in control and disturbance channels, the considered class of nonlinear systems often faces challenges such as modeling difficulties, increased complexity in robust control design, and significant obstacles in stability analysis. Then, to address the aforementioned challenges, from the novel theoretical perspective of matrix partial orders, corresponding L & ouml;wner and GL partial orders techniques are employed to construct the auxiliary systems and introduce the corresponding performance indices, thereby reformulating the original robust control problems as optimal control problems. Next, the trajectory consistency between the auxiliary systems and the original systems is rigorously established through the theorems concerning global asymptotic stability (GAS). Furthermore, based on the constructed auxiliary systems and the new designed controllers, the long-time passivity (LTP) behavior of the original systems is analyzed. Leveraging GAS and LTP, the analyses of the uniform ultimate boundedness (UUB) are conducted. Finally, several practical control problems are presented to demonstrate the feasibility and effectiveness of the proposed theoretical results.

In this article, the average tracking consensus problem of multi-agent systems (MASs) with denial-ofservice(DoS) attacks and time-varying delays via a dynamic quantized control is investigated. First, an auxiliary detector incorporating internal variables is designed for each agent to track the average signals relying on topological structure of MASs. The dynamic quantization mechanism is designed for the quantizer, which can avoid saturation and eliminate the quantization error asymptotically. Then, we rigorously prove that the proposed dynamic quantization mechanism can handle general linear MASs with constrained bandwidth of digital communication network. Furthermore, we extend the quantization framework to guarantee average tracking consensus even under simultaneous DoS attacks and time-varying delays. By constructing appropriate Lyapunov functions and employing mathematical recursion, we derive sufficient conditions concerning constrained bandwidth, DoS attacks and time-varying delays. Finally, a practical example of spacecraft formation tracking in low Earth orbit is presented to validate the theoretical conclusions of this study.

In the framework of Q-learning, the research in this paper investigates a data-based discrete-time two-player zero-sum delayed game system. Crucial issue of the delayed game system is to solve the discrete-time delayed game algebraic Riccati equation (DTDGARE). However, to solve this equation, it is typically necessary to have complete system dynamics information and to account for the impact of delay terms, which introduces certain challenges during the solution process. To address the aforementioned challenges, a sophisticated delay operator technique is employed to address the challenges posed by the delay terms. Subsequently, the Qfunction incorporates delayed terms, which, unlike the classical Q-function from reinforcement learning, is specifically designed to circumvent the need for directly solving the DTDGARE. Consequently, we propose a novel data-based delayed policy iteration Q-learning (DPIQL) algorithm, designed to learn the optimal Q-function directly from real data, thereby overcoming the limitations associated with requiring complete knowledge of system dynamics. Furthermore, by constructing a Banach space, a rigorous and innovative mathematical proof leveraging Kantorovich's theorem ensures the convergence of the newly proposed DPIQL algorithm. Finally, the effectiveness and feasibility of the developed algorithm are validated through numerical simulations based on a real drone flight dataset.

This paper investigates almost sure synchronization of T-S fuzzy semi-Markov asynchronous switched systems under deception attacks. In the considered system, the mode transitions follow a semi-Markov process with generally distributed sojourn times, while deception attacks occur at stochastic discrete time instants, independent of the system’s switching process. First, the deception attack is modeled as a renewal-driven stochastic impulsive process, where the attack occurrence times follow a renewal process and the attack intensities vary randomly. Then, by constructing a mode-dependent Lyapunov function and utilizing the strong law of large numbers, sufficient conditions for almost sure synchronization are derived. Compared with existing studies, our work models the asynchrony between semi-Markov switching and deception attacks, leading to a more general and realistic framework for analyzing complex networked systems under adversarial conditions. Finally, numerical simulations on an oscillator model are conducted to validate the effectiveness of the proposed method.

In this article, we address the almost sure exponential synchronization issue of stochastic strict-feedback systems (SSFSs) with semi-Markov jump. It is the first time to consider semi-Markov jump into SSFSs. Different from previous works, the sojourn time of semi-Markov jump is influenced by the present and next states. Meanwhile, with the help of multiple mode-dependent Lyapunov-like functions, the relevant conditions for linear comparability become more relaxed. We propose a new controller design based on the idea of backstepping due to the introduction of a semi-Markov jump in the system. Then, we calculate differential inequalities for Lyapunov functions. Moreover, based on Lyapunov functions, differential inequality techniques, and the design controllers, some sufficient criteria are required to achieve almost sure exponential synchronization. In addition, the theoretical results are extended to SSFSs with Markov jump and some sufficient conditions are attained. Finally, a specific numerical example and an application about single-link robot manipulator are employed to demonstrate the effectiveness and feasibility of our results.

This paper concentrates on the almost sure synchronization for a class of stochastic multi-links coupled semi-Markov jump systems through aperiodically intermittent control. For these stochastic switching systems, almost sure synchronization is investigated by employing mode-dependent multiple Lyapunov-like function method and stochastic analysis. Notably, mode-dependent multiple Lyapunov-like function method is designed to provide more flexibility. In addition, when considering the sojourn time distribution of the semi-Markov jump, dependence on both the current state and the next state can effectively reduce unnecessary restrictions compared with previous assumption of the sojourn time distribution function. Ultimately, the Chua’s circuit along with numerical simulations are provided to validate the effectiveness of the theoretical results.

For the Fornberg–Whitham equation, the local well-posedness in the critical Besov space B_p, 1^1+1/p(ℝ) with 1≤ p <∞ has been studied in Guo (Nonlinear Anal RWA 70:103791, 2023). However, for the endpoint case p=∞ , whether it is locally well-posed or ill-posed in B_∞ , 1^1(ℝ) is still open. In this paper, we prove that the Fornberg–Whitham equation is well-posed in the critical Besov space B_∞ , 1^1(ℝ) with solutions depending continuously on initial data, which is different from that of the Camassa–Holm equation (Guo et al. in J Differ Equ 327:127, 2022). In addition, we show that this dependence is sharp by showing that the solution map is not uniformly continuous on the initial data.

The (2+1) -dimensional nonlocal derivative nonlinear Schrödinger equation, correlated with Lax pair involving partial reverse space–time symmetric potential, is investigated in this paper. The special eigenfunctions of spectral problem are required to ensure the validity of the nonlocal reduction conditions. In view of Darboux transformation method and different seed solutions, some explicit solutions are derived for the reduced system. As its applications, periodic waves, breathers, dark solitons, anti-dark solitons, shock wave, interactions and parallel propagations of the above waves are plotted to distinguish different parameter values. By taking limit technology and Taylor expansion, the N -order generalized Darboux transformation, multi-line waves and multi-parabola waves solutions are obtained. Compared with local ones, the nonlocal equations possess more plentiful pattern dynamics for the waves as it should take nonlocal term and local terms into account rather than just local terms. These interesting results may be useful for future experimental study.

This paper investigates the pth moment exponential synchronization problem of multi-links stochastic delayed complex networks with semi-Markov jump via aperiodically intermittent control. Combining random disturbances, time-varying delay and semi-Markov jump with multi-links systems, our work is more relevant than previous work. Based on Lyapunov method and graph theory, a novel inequality for disposing of the problem of pth exponential synchronization is established under the aperiodically intermittent control and some sufficient criteria are derived. The theoretical results supply a new perspective showing the synchronization criterion and the topological structure of multi-links systems related closely. Furthermore, the value of our results is exhibited by applying them to the single-link robot arms in engineering. Eventually, a numerical simulation is provided to demonstrate the validity of our results.

In this paper, we investigate the exponential synchronization problem for multi-link and multi-delayed inertial neural networks with Markov jump (MMDINNMJ) using an aperiodically intermittent adaptive control strategy. Different from most research on inertial neural networks, we take multi-link, multi-delay and Markov jump into account. The obstacle caused by the coexistence of Markov jump and multi-delay is avoided by using the delayed integral method while considering the exponential synchronization of MMDINNMJ. Additionally, under graph theory, Lyapunov stability theory and the developed control scheme, some novel sufficient conditions for synchronization at exponential rate in pth (p>0) moment of underlying networks are determined, which are strongly related to multi-link topological structure, time delay, and Markov jump. Finally, two examples are given to demonstrate the viability of the theoretical conclusions.

This paper investigates the stability of semi-Markovian jump stochastic coupled nonlinear systems with mode-dependent sojourn time distributions. It is challenging to consider coupling in the semi-Markovian jump systems due to the different networked topologies. First, more realistic complex networks with coupling and semi-Markovian jump are established. Based on Lyapunov method and graph theory, a novel stochastic analysis method and linear comparable Lyapunov like functions are employed to ensure almost surely exponential stability (ASES). Then, without additional restrictions on the sojourn time, the sufficient criterion for ASES is developed. Compared with the most of existing works, which are restrictions on the independence and the distribution function of the sojourn time and the coupling are not considered, our results are meaningful and they can be directly applied to the coupled oscillator model. Finally, some numerical simulations are provided to demonstrate the validity of our results.

In this paper, we investigate the issue of almost sure stability (ASS) for a class of stochastic strict-feedback semi-Markov jump systems (SSSJSs) under periodic intermittent control (PIC). This control method and dynamic properties are studied for the first time in strict-feedback systems. First of all, based on the structural properties of the SSSJSs, virtual controllers are designed step by step and eventually deduced into the actual controller. Furthermore, by using stochastic analysis theory and multiple Lyapunov function method, we obtain sufficient conditions for ASS via PIC, which have a close relationship with control width and control period. By virtue of multiple Lyapunov functions that depend on the system states, the conservatism caused by mode-independent cases can be reduced effectively. Finally, the effectiveness of the presented results is illustrated by simulation examples.

In this paper, we will focus on the reverse space–time Fokas–Lenells equation. According to the Lax pair, the semi-degenerate Darboux transformation will be constructed by Taylor expansions. Based on the determinant form, some rogue wave solutions will be derived on the three types of backgrounds, including M-shaped periodic waves, mixed M-shaped periodic waves and breather, mixed single-periodic and double-periodic waves. And the rogue waves in various backgrounds can be transformed into the corresponding backgrounds waves with parameters selection. The dynamic features of those solutions will be discussed via graphical illustration. It is worth mentioning that these results are novel.

This paper focuses on exponential synchronization for multi-link and multi-delayed large-scale systems with semi-Markov jump (MLMDLSSMJ) via adaptive aperiodically intermittent control. It is challenging to overcome the impacts of multiple time delays and semi-Markov jump because of the intermittent property. The multi-delayed differential inequalities with semi-Markov jump are established to address this issue, which extend the existing Halanay-type differential inequalities. Several exponential synchronization criteria for intermittently controlled MLMDLSSMJ are built by using the inequality technique, Lyapunov method and graph theory. Notably, the exponential convergence rate is more accurate than the previous literature, which may aptly show the synchronization capability of MLMDLSSMJ. The exponential synchronization of Chua’s circuits network is investigated as a practical implementation of the theoretical results. Finally, to demonstrate the efficiency of the theoretical results, a few numerical simulations are provided.

In this paper, the issue of exponential synchronization in Markov switching inertial neural networks with mixed delays is investigated via aperiodically on–off adaptive control. The inertial term is considered, which extends the existing network modes with first-order differential term. Combined with the Lyapunov method, graph theory, and the differential inequalities technique, two types of synchronization criteria are presented which take into account all of the time delay information and reduce the conservativeness. Finally, some numerical simulations are provided in order to show the validity of the theoretical results.