This study focuses on achieving hierarchical optimal synchronization in a fractional-order nonlinear multi-agent system (FONMAS) composed of a single leader and multiple followers, analyzed within the framework of Stackelberg–Nash game theory. The leader makes decisions by anticipating the optimal responses of all followers, whereas each follower concurrently reacts optimally to the leader's strategy by engaging in a Nash game. To obtain the optimal control policy of the FONMAS, a regularized fuzzy reinforcement learning approach is proposed. First, a fractional-order (FO) Hamilton–Jacobi–Bellman (HJB) equation in coupled form is formulated, which serves as the basis for deriving the optimal control policies of both the leader and the followers. It is further proven that these strategies constitute a Stackelberg–Nash equilibrium (SNE). Next, due to the asymmetry among agents, solving the FO coupled HJB equations becomes challenging. To address this, a hierarchical learning framework grounded in FO value iteration is proposed, which depends solely on partial knowledge of the system dynamics. We demonstrate that, given mild coupling assumptions, this method converges asymptotically to equilibrium policies. Furthermore, a regularized actor–critic (A-C) framework with fuzzy logic is employed to estimate the cost function and optimal control policy, and the FO weight update rules are developed by formulating a Lyapunov function for the optimal fuzzy weight deviation, ensuring convergence to optimal weight values. Ultimately, both theoretical analysis and simulation results support the efficiency of the proposed method.

In this study, we employed the batch gradient method to investigate the monotonicity and convergence of the Elman neural network (ENN) based on the entropy error function (EEF) and regularization methods. This enhances network stability and sparsity while also boosting its ability to generalize. Traditional mean square error (mse) functions in complex networks often result in slower convergence during training, prone-to-local minima, and even incorrect saturation issues. To address this drawback, we propose a novel EEF for training ENN, effectively avoiding the problem of learning speed degradation. Furthermore, by leveraging smoothing group $L_{1/2}$ regularization $(\text {SGL}_{1/2})$ methods in studying ENN based on EEF, we effectively overcome the drawbacks of traditional group $L_{1/2}$ regularization $(\text {GL}_{1/2})$ leading to error function oscillations. In addition, we optimize the network architecture effectively in two key ways: reducing redundant nodes to near 0 and driving redundant weights toward 0 for remaining nodes, further boosting network sparsity. This article rigorously proves the monotonicity of the error function, alongside presenting strong and weak convergence outcomes for the novel method. The effectiveness and correctness of our approach are clearly illustrated through experimental results. The simulation results align with the theoretical findings.

In this article, we investigate the boundedness and convergence of the online gradient method with the smoothing group regularization for the sigma-pi-sigma neural network (SPSNN). This enhances the sparseness of the network and improves its generalization ability. For the original group regularization, the error function is nonconvex and nonsmooth, which can cause oscillation of the error function. To ameliorate this drawback, we propose a simple and effective smoothing technique, which can effectively eliminate the deficiency of the original group regularization. The group regularization effectively optimizes the network structure from two aspects redundant hidden nodes tending to zero and redundant weights of surviving hidden nodes in the network tending to zero. This article shows the strong and weak convergence results for the proposed method and proves the boundedness of weights. Experiment results clearly demonstrate the capability of the proposed method and the effectiveness of redundancy control. The simulation results are observed to support the theoretical results.

On September 6, 2008, in Taiyuan Satellite Launch Center, HJ-1A/B satellites (HJ-1A/B), China's first two satellites of Environment & Disasters Monitoring and Predicting Microsatellite Constellations, were successfully launched with the technique of “one rocket, two satellites”. The HyperSpectral Imager(HSI) on-board HJ-1A satellite is a Fourier Transform HyperSpectral Imager(FTHSI) built by Xian Institute of Optics and Precision Mechanics(XIOPM) of Chinese Academy of Sciences(CAS). This paper briefly introduces the spectral and radiometric properties of HSI, basic imaging theories of the spatially modulated Fourier transform imaging spectrometer, and then discusses the algorithms of spectrum reconstruction. In the end, the result of the operational spectrum reconstruction for the raw data of the HJ-1A satellite Fourier transform HSI is presented.