Search Results (321)

This paper is devoted to the study of the following fractional Choquard equation with prescribed $L^2$ norm: $\left\{\begin{array}{cc}& {(-\Delta )}^{s}u+\mu u=\left(I_{\alpha }*F\left(u\right)\right)F^{\prime }\left(u\right)\mathrm{in}{\mathbb{R}}^N,\hfill \\ & {\parallel u\parallel }_{L^2\left({\mathbb{R}}^N\right)}=a,\hfill \end{array}\right.$ where $N\ge 2$, $s\in (0,1)$, $I_{\alpha }$ is the Riesz potential with $\alpha \in (0,N)$, and $F\in C^1(\mathbb{R},\mathbb{R})$ satisfies the general Berestycki–Lions-type assumptions. Here, the parameter $\mu \in \mathbb{R}$ will arise as a Lagrange multiplier. In the $L^2$-subcritical case, we establish the existence of normalized ground states in $H^s\left({\mathbb{R}}^N\right)$ by applying minimax arguments to the Lagrange formulation and using the concentration-compactness principle to restore compactness. Moreover, we show that the normalized ground states constructed here additionally satisfy the Pohozaev identity. This result is noteworthy, since it remains an open question as to whether general solutions of fractional Choquard equations satisfy the Pohozaev identity. Full article

With the rapid advancement of exoskeletons and rehabilitation robotics, modern healthcare increasingly demands high dynamic accuracy and reliability from medical devices. However, the dynamic response and durability of mechanical systems are greatly influenced by the inevitable existence of clearances in kinematic joints. Existing studies predominantly focus on simplified planar or spatial mechanisms, offering limited guidance for complex mechanical structures in medical applications. To address this issue, a unified modeling framework is proposed in this study to explore the nonlinear dynamic behavior and wear properties of bionic humanoid rigid mechanisms incorporating revolute joint clearances. A dynamic model that accounts for revolute joint clearances is established, employing the Lankarani–Nikravesh contact model alongside a refined Coulomb friction approach to characterize contact behavior. To characterize the wear progression between the shaft and the bushing, the Archard wear model is employed, while the system’s dynamic equations are formulated using the Lagrange multiplier approach. Systematic simulations are conducted to examine the effects of clearance size, location, and multi-clearance coupling on dynamic response and wear behavior. The results reveal that clearances at the hip joint have the most pronounced impact on system performance, tibiofemoral joint clearances exacerbate precision disturbances, and foot-end clearances considerably undermine system robustness. Increased clearance sizes and the coexistence of multiple clearances aggravate wear and induce more severe nonlinear dynamic phenomena. Phase portraits and Poincaré maps further illustrate that the system may exhibit complex or chaotic behavior under certain conditions. This study offers theoretical insights into performance degradation mechanisms in humanoid robots with joint clearances and introduces a modular “driving–mid–terminal” structure that enhances model generality, enabling its application to exoskeletons and rehabilitation devices for design optimization, service life prediction, and health monitoring. Full article

With the rapid evolution toward 6G networks, ensuring robust physical layer security (PLS) in highly dynamic and heterogeneous wireless environments has become a key challenge. Traditional security methods often struggle to adapt to time-varying channels, especially in the absence of perfect channel state information. Furthermore, the dynamic nature of node selection and power allocation in heterogeneous networks creates a complex hybrid action space operating across multiple timescales, significantly complicating the design of efficient and adaptive security strategies. To address this, this paper proposes a novel constrained hierarchical reinforcement learning (CHRL) framework for secure cooperative communications in next-generation wireless systems. The framework is designed to optimize secrecy performance within a hybrid action space comprising both discrete node selection and continuous power allocation, operating at different timescales. By hierarchically decoupling the joint optimization problem, the upper layer performs risk-aware node selection to maximize long-term secrecy capacity (SC) while guaranteeing a stable and secure link. At the lower layer, we develop a constrained MiniMax Multi-objective Deep Deterministic Policy Gradient (M3DDPG) algorithm that optimizes power allocation considering worst-case conditions. Lagrange multipliers are integrated to enforce a strictly positive SC constraint throughout transmission, effectively preventing security outages. Simulation results under time-varying Rayleigh fading channels demonstrate that the proposed CHRL framework outperforms existing HRL methods, achieving up to 17% improvement in SC while strictly maintaining security constraints. These results validate the effectiveness of the proposed approach for enhancing PLS in next-generation cooperative wireless networks. Full article

This paper proposes a safety-constrained Transformer–PPO framework for low-carbon economic dispatch with settable incentive-based demand response (DR) in wind–PV integrated electro–thermal–hydrogen industrial-park energy systems. Hydrogen is modeled as exogenous hydrogen-domain demand and is satisfied through electrolyzer production and hydrogen inventory dynamics. A causal Transformer captures long-horizon multi-energy coupling and intertemporal constraints and is trained with PPO under uncertainty. A dual-layer safety mechanism combines dual-variable (Lagrange multiplier) updates for statistical constraints with an execution-layer quadratic-programming action projection to enforce hard physical constraints, including operating limits, ramping, battery SOC, hydrogen inventory bounds, and energy balance. Baseline–verification–settlement rules and budget-ledger states are embedded to ensure verifiable response quantities and settlement outcomes that are traceable and independently recompilable. Case studies on a real industrial-park scenario in Inner Mongolia show reduced peak-hour maximum grid purchase demand and constraint violations, together with lower total cost, carbon cost, and curtailment penalties versus MILP, PPO-MLP, and Transformer–PPO without safety mechanisms. Full article

Column generation is a fundamental technique for solving large-scale combinatorial optimization problems such as unit commitment and vehicle routing, yet its performance is often limited by dual oscillation. This study explores the intrinsic cause of this phenomenon from the perspective of shadow price theory and demonstrates that dual oscillation arises from the lack of marginal interpretability of Lagrange multipliers when multiple dual solutions coexist. To address this issue, an improved column generation framework is proposed in which traditional multipliers are replaced with minimum-norm multipliers that possess clear economic meaning and act as directional shadow prices. A generalized pricing subproblem is formulated, and partial minimum-norm multipliers are obtained through convex quadratic optimization to guide column generation. Numerical experiments on a simplified single-period unit commitment case and large-scale cutting stock problems showed that the proposed approach eliminated invalid column generation and achieved speedy convergence to the optimal solution within only two iterations for the unit commitment case, and the classical column generation exhibited slow convergence with dual oscillation in large-scale scenarios while the improved algorithm achieved fast and stable convergence. The results indicate that the stabilization method enhances the consistency of dual variables and provides a more robust foundation for the theoretical and practical development of column generation algorithms. Full article

In this paper, we propose a Lagrange multiplier (LM)-type unit root test for functional time series. The key novelty lies not in introducing a new LM principle but in establishing the asymptotic validity of such a test under the functional random walk null hypothesis without relying on functional principal component analysis (FPCA) or finite-dimensional unit root subspace assumptions. We derive the limit distribution of our proposed test statistics under the null hypothesis of a random walk and its asymptotic behavior of alternative hypotheses of trend stationary, weakly dependent stationary, and autoregressive stationary models. Specifically, we establish the theoretical consistency of the test under all aforementioned alternative hypotheses. Simulation studies corroborate these theoretical findings and demonstrate the desirable finite-sample performance of the proposed functional unit root test. The proposed test is also applied to real data of intraday stock price curves, and the test results are plausible. Full article

This study proposes a real-time chiller capacity control strategy based on marginal Coefficient of Performance (COP) analysis to improve the energy efficiency of air-conditioning systems. The research focuses on the air-conditioning system (ACS) of an office building. Operational data, including chiller capacity and the corresponding COP, were collected to derive the chiller’s operating characteristic curve. The Optimal Capacity Control (OCC) strategy aims to maximize the total COP of all chillers, and the initial capacity allocation is determined using the Lagrange multiplier method. To further refine performance, a fine-tuning mechanism is introduced, calculating the ratio of COP variation to capacity variation (RC ratio) for each chiller to identify which unit should be loaded or unloaded. Based on the fine-tuning mechanism, a comprehensive OCC model is established to ensure that the chiller’s cooling output precisely matches the load demand, thereby maximizing system efficiency and reducing energy consumption. To validate the effectiveness of the proposed OCC strategy, a numerical analysis was implemented using real operational data from the existing ACS. Comparative simulations between the OCC and a Traditional Capacity Control (TCC) strategy were conducted. On a representative summer day, total power consumption decreased from 1534.0 kWh (TCC) to 1527.2 kWh (OCC), while total system COP increased from 113.9 to 114.8. Seasonal analysis further confirms consistent energy savings under varying load conditions. The results indicate that the OCC strategy significantly enhances system performance and reduces energy consumption under varying load conditions. Overall, the proposed method achieves a higher system COP, leading to notable electricity savings and improved operational efficiency of the air-conditioning system. Full article

Traditional mean–variance portfolio optimization proves inadequate for cryptocurrency markets, where extreme volatility, fat-tailed return distributions, and unstable correlation structures undermine the validity of variance as a comprehensive risk measure. To address these limitations, this paper proposes a unified entropy-based portfolio optimization framework grounded in the Maximum Entropy Principle (MaxEnt). Within this setting, Shannon entropy, Tsallis entropy, and Weighted Shannon Entropy (WSE) are formally derived as particular specifications of a common constrained optimization problem solved via the method of Lagrange multipliers, ensuring analytical coherence and mathematical transparency. Moreover, the proposed MaxEnt formulation provides an information-theoretic interpretation of portfolio diversification as an inference problem under uncertainty, where optimal allocations correspond to the least informative distributions consistent with prescribed moment constraints. In this perspective, entropy acts as a structural regularizer that governs the geometry of diversification rather than as a direct proxy for risk. This interpretation strengthens the conceptual link between entropy, uncertainty quantification, and decision-making in complex financial systems, offering a robust and distribution-free alternative to classical variance-based portfolio optimization. The proposed framework is empirically illustrated using a portfolio composed of major cryptocurrencies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)—based on weekly return data. The results reveal systematic differences in the diversification behavior induced by each entropy measure: Shannon entropy favors near-uniform allocations, Tsallis entropy imposes stronger penalties on concentration and enhances robustness to tail risk, while WSE enables the incorporation of asset-specific informational weights reflecting heterogeneous market characteristics. From a theoretical perspective, the paper contributes a coherent MaxEnt formulation that unifies several entropy measures within a single information-theoretic optimization framework, clarifying the role of entropy as a structural regularizer of diversification. From an applied standpoint, the results indicate that entropy-based criteria yield stable and interpretable allocations across turbulent market regimes, offering a flexible alternative to classical risk-based portfolio construction. The framework naturally extends to dynamic multi-period settings and alternative entropy formulations, providing a foundation for future research on robust portfolio optimization under uncertainty. Full article

Spacecraft safe proximity, as a critical component of on-orbit servicing missions, primarily encounters the following two challenges: the partial observability of the environment surrounding the service spacecraft and the necessity to evade uncertain obstacles. A safe reinforcement learning algorithm based on a graph neural network is proposed to address the constrained Markov decision problem in partially observable scenarios for spacecraft safe proximity missions. A graph neural network mechanism is introduced to solve the problem of dynamic variations in the quantity and location of obstacles in the observation area of the service spacecraft. The graph attention network is used to facilitate the extraction of feature information from the graph structure, which is then utilized as input for the subsequent reinforcement learning algorithm. The Soft Actor–Critic–Lagrangian algorithm is adopted to deal with the problems of tuning reward function parameters and balancing safety and optimality. By introducing Lagrange multipliers, the constrained optimization problem is transformed into an unconstrained optimization problem. In order to verify the effectiveness of the algorithm proposed in this paper, a spacecraft safe proximity environment model with dynamic obstacles is constructed, and the GAT-SACL algorithm proposed in this paper is validated by the Monte Carlo shooting method. The results show that the GAT-SACL algorithm possess excellent exploratory characteristics and delivers significant advantages in balancing optimality and safety. Full article

Unmanned Aerial Vehicle (UAV)-assisted edge computing supports latency-sensitive applications by offloading computational tasks to ground-based servers. However, determining optimal resource allocation under strict latency constraints and stochastic channel conditions remains challenging. This paper addresses the joint computation partitioning and power allocation problem for UAV-assisted edge computing systems. We formulate the problem as a Constrained Markov Decision Process (CMDP) that explicitly models latency constraints, rather than relying on implicit reward shaping. To solve this CMDP, we propose Constrained Soft Actor–Critic (C-SAC), a deep reinforcement learning algorithm that combines maximum-entropy policy optimization with Lagrangian dual methods. C-SAC employs a dedicated constraint critic network to estimate long-term constraint violations and an adaptive Lagrange multiplier that automatically balances energy efficiency against latency satisfaction without manual tuning. Extensive experiments demonstrate that C-SAC achieves an $18.9\%$ constraint violation rate. This represents a 60.6-percentage-point improvement compared to unconstrained Soft Actor–Critic, with $79.5\%$, and a 22.4-percentage-point improvement over deterministic TD3-Lagrangian, achieving $41.3\%$. The learned policies exhibit strong channel-adaptive behavior with a correlation coefficient of $-0.894$ between the local computation ratio and channel quality, despite the absence of explicit channel modeling in the reward function. Ablation studies confirm that both adaptive mechanisms are essential, while sensitivity analyses show that C-SAC maintains robust performance with violation rates varying by less than 2 percentage points even as channel variability triples. These results establish constrained reinforcement learning as an effective approach for reliable UAV edge computing under stringent quality-of-service requirements. Full article

An attosecond optical pulse can entangle coherently related states of different characters, such as electronic and vibrational, in a molecular system. Using a quantum information theoretic approach, we explicitly define and discuss the surprisal of such a system in the maximal entropy formalism and identify the constraints and their conjugate Lagrange multipliers. Surprisal analysis shows how these constraints become fewer and simpler in the sudden approximation of the dynamics, a limit often valid for an ultrafast excitation. The optically accessible lower electronic states of N₂ are used as a numerical example to show the compaction of the dynamics from $\mathcal{O}\left(n^2\right)$ down to $\mathcal{O}\left(n\right)$ constraints, where $n$ is the number of vibronic states. The von Neumann entropy is used to confirm the fidelity of the compaction. Full article

We investigate the existence of positive normalized (mass-constrained) solutions for the fractional nonlinear Schrödinger equation ${(-\Delta )}^{s}v+V\left(x\right)v=\lambda v+{\mu \left|v\right|}^{p-2}v+{\left|v\right|}^{2_s^{*}-2}v\mathrm{in}{\mathbb{R}}^N,{\parallel v\parallel }_2^2=b^2,$ where $N>2s$, $s\in (0,1)$, $\mu >0$, $p\in (2,2_s^{*})$, and $2_s^{*}={\displaystyle \frac{2N}{N-2s}}$. Here, $\lambda \in \mathbb{R}$ denotes the Lagrange multiplier associated with the prescribed mass $b>0$. The potential $V\in C^1\left({\mathbb{R}}^N\right)$ is allowed to be nonconstant and satisfies $V\left(x\right)\to V_{\infty }$ as $\left|x\right|\to \infty$; moreover, the perturbations induced by $V-V_{\infty }$ and $x·\nabla V$ are assumed to be small in the quadratic-form sense compared with the fractional Dirichlet form ${\parallel (-\Delta )}^{s/2}{v\parallel }_2^2$. Using the Caffarelli–Silvestre extension, we establish a Pohozaev identity adapted to the presence of $V\left(x\right)$ and introduce a Pohozaev manifold on the $L^2$-sphere. Combining Jeanjean’s augmented functional approach with a splitting analysis at the Sobolev-critical level, we construct compact Palais–Smale sequences below a suitable critical energy level. As a consequence, we prove the existence of positive normalized solutions for small masses $b\in (0,b_0)$ in the $L^2$-critical and $L^2$-supercritical regimes (with respect to the lower-order power p). Full article

This paper proposes a coordinated capacity configuration method for Virtual Power Plant (VPP) distributed resources that considers time-varying power coupling. The method addresses the inadequate economic efficiency and reliability of existing configuration schemes, which stems from insufficient attention to the time-varying power coupling characteristics of Distributed Energy Resources (DERs). Firstly, we define the concepts of direct and indirect power coupling among DERs, derive a Lagrange multiplier-based coupling coefficient model, and realize the quantification of time-varying coupling coefficients through sliding time window correlation analysis (STWCA). Next, a capacity correlation matrix integrating technical and economic synergies is constructed to map coupling characteristics to capacity configuration. Then, a coordinated configuration model with time-varying coupling constraints is established to minimize life-cycle cost and maximize power supply reliability, validated by case simulation. The results demonstrate that the proposed method effectively reduces VPP operation cost and improves resource utilization and reliability, providing theoretical support for the scientific configuration of DERs in VPPs. Full article

High penetration and output volatility of island wind and photovoltaics (PV) pose challenges to energy consumption and supply–demand balance, and cost-effective energy storage configuration. A coupled dispatch model for a wind–PV–storage system is proposed, which treats multiple canal units as virtual ‘loads’ that switch between generation and pumping under constraints of power balance and available water head model. Considering the variable reservoir–irrigation feature, a multi-objective model framework is developed to minimize both economic cost and storage capacity required. An augmented Lagrangian–Nash product enhanced NSGA-II (AL-NP-NSGA-II) algorithm enforces constraints of irrigation shortfall and overflow via an augmented Lagrangian term and allocates fair benefits across canal units through a Nash product reward. Moreover, updates of Lagrange multipliers and reward weights maintain power balance and accelerate convergence. Finally, a case simulation (3.7 MW wind, 7.1 MW PV, and 24 h rural load) is performed, where 440.98 kWh storage eliminates shortfall/overflow and yields 1.5172 × 10⁴ CNY. Monte Carlo uncertainty analysis (±10% perturbations in load, wind, and PV) shows that increasing storage to 680 kWh can stabilize reliability above 98% and raise economic benefit to 1.5195 × 10⁴ CNY. The dispatch framework delivers coordination of irrigation and power balance in island microgrids, providing a systematic configuration solution. Full article

Several iterative integro-differential formulations for two-point, second- and third-order, nonlinear, boundary-value problems of ordinary differential equations based on Green’s functions and the method of variation of parameters are presented. It is shown that the generalized or dual Lagrange multiplier method (GVIM) previously developed for the iterative solution of nonlinear, boundary-value problems of ordinary differential equations that makes use of modified functionals and two Lagrange multipliers, is nothing but an iterative Green’s function formulation that does not require Lagrange multipliers at all. It is also shown that the two Lagrange multipliers of GVIM are associated with the left and right Green’s functions. The convergence of iterative methods based on both the Green function and the method of variation of parameters is proven for nonlinear functions that depend on the dependent variable and is illustrated by means of two examples. Several new iterative integro-differential formulations based on Green’s functions that use a multiplicative function for convergence acceleration are also presented. Full article