Search Results (224)

Here, we consider a nonlinear hydrodynamic model with mixed dispersion–temporal evolution as the scalar version of the generalized shallow-water wave equation, which specifically provides a comprehensive and versatile framework for studying energy propagation in nonlinear fluids of constrained depth. This equation is acknowledged as an integrable model in the analysis of tidal wave dynamics and in simulations of weather variations, tsunami prediction, and irrigation flows. We also investigate a few of its singular and periodic solitary wave solutions by employing various Riccati-based ansatzes. These results highlight the necessity of studying various nonlinear wave phenomena, which may have potential applications in various domains of physics and applied mathematics. These results extend the variety of its solutions and also enrich the existing knowledge about its solutions with various profiles. To improve visual clarity and to facilitate structural understanding, the solution profiles are represented graphically using Maple software (version 2023.2) in 3D, 2D, and contour plots.We also discuss its invariance under infinitesimal transformations, which yields a one-dimensional Hamilton–Jacobi-like equation. Full article

The rapid growth of the sharing economy has improved resource utilization in car-sharing, yet it has also sharpened market competition and diversified user demand. A persistent obstacle is the low coordination efficiency between asset-heavy operating companies and traffic-driven platforms, whose misaligned objectives waste social resources. This paper uses differential game theory to analyze their dynamic coordination strategies and benefit allocation mechanisms. The Nerlove–Arrow model captures the evolution of brand goodwill, while the company’s decisions on station layout, vehicle dispatch, and pricing, together with the platform’s advertising investment, form the core decision variables in a two-party game framework linking the asset side and the traffic side. Compared with the non-cooperative Nash equilibrium, the cooperative mode removes the double marginalization effect, strengthens the investment incentives of both parties, and raises the system’s steady-state goodwill and total profit, achieving a Pareto improvement. To ground the cooperative framework in rigorous theory, we supply a verification theorem confirming that the linear candidate value functions satisfy the Hamilton–Jacobi–Bellman equations over the entire admissible state space. A formal proof of instantaneous rationality ensures that neither party falls into a cooperation trap on the horizon $[0,T]$, and the asymptotic stability of the steady-state goodwill trajectory is established. We further endogenize the revenue-sharing coefficient through a generalized Nash bargaining model that admits asymmetric bargaining structures, and introduce a Stackelberg leadership benchmark as a third comparative regime. Sensitivity analyses with respect to the discount rate and user heterogeneity confirm the robustness of the findings. A dedicated discussion section bridges the gap between idealized parameterization and data-driven calibration, describing practical pathways via A/B testing, user churn metrics, and econometric estimation of demand parameters. The results offer a scientific decision-making reference for strategic cooperation in the car-sharing industry. Full article

The adaptive regulation of student productivity remains a challenging problem in technology-enhanced learning environments due to the continuous and uncertain nature of cognitive effort, attention, and behavioral fluctuations. While existing educational intervention models are predominantly based on discrete-time decision frameworks, they often provide limited support for the representation of stochastic productivity dynamics and continuous effort adaptation. This paper proposes a continuous-time stochastic optimal control framework for adaptive effort allocation in student productivity regulation. The learner productivity level is modeled as a bounded stochastic diffusion process evolving on the interval ([0, 1]), where the drift and diffusion coefficients depend on both effort allocation and learner-specific psychological characteristics. The control objective is formulated as the maximization of an expected cumulative productivity reward penalized by excessive cognitive effort over a finite study horizon. Using the Hamilton–Jacobi–Bellman (HJB) framework, we derive an optimal state-dependent feedback policy that dynamically adjusts effort allocation according to the current productivity level, the remaining study horizon, and the learner profile. We establish the well-posedness of the controlled stochastic dynamics and show that the productivity state remains invariant within the admissible interval. The resulting HJB equation is solved numerically using a semi-implicit finite-difference approximation combined with iterative feedback updates. Simulation experiments conducted on synthetic learner profiles illustrate the qualitative behavior of the proposed controller under heterogeneous psychological configurations. Compared with constant-effort and threshold-based heuristic strategies, the adaptive feedback policy produces smoother productivity trajectories and more stable effort allocation patterns under stochastic perturbations. The proposed framework provides a mathematically grounded approach for studying adaptive productivity regulation under uncertainty and establishes a foundation for future data-driven calibration and personalized intervention systems. Full article

Traditional credit scoring models treat lending decisions as static classification, ignoring the dynamic evolution of borrower risk and long-term profit optimisation. This paper reinterprets credit risk management as a discrete-time stochastic optimal control problem and integrates the Hamilton–Jacobi–Bellman (HJB) framework with deep reinforcement learning. Theoretically, we establish the equivalence between a discrete Markov decision process and the HJB equation, prove the existence and uniqueness of the optimal value function, derive the closed-form Riccati solution under linear-quadratic assumptions, and provide a convergence analysis of neural network value iteration. Empirically, using LendingClub loan data (2016–2018), we implement a PPO-based dynamic credit policy. The proposed model achieves an average reward of 1.6726 and a total reward of 867,613, significantly outperforming static baselines as well as a DQN baseline. Ablation experiments show that replacing the policy network with a linear mapping reduces the average reward by 40.8%, confirming the necessity of nonlinear function approximation. Sensitivity analysis and statistical tests (p < 0.001) confirm the robustness and significance of the gains. This work provides a rigorous mathematical foundation and empirical evidence for shifting credit scoring from static classification to dynamic optimisation. Full article

Traditional credit scoring treats lending as static classification and lacks the ability to adjust risk preferences dynamically. This paper develops a dynamic credit decision framework based on the entropy-regularized Hamilton–Jacobi–Bellman (ER-HJB) equation. Theoretically, we prove the existence and uniqueness of a solution to the ER-HJB equation, show that under exact tabular assumptions the soft policy iteration underlying Soft Actor-Critic (SAC) converges to this solution, and derive a closed-form analytical solution under linear-quadratic conditions. Empirically, using LendingClub loan panel data (2016–2018), we show that a single entropy coefficient continuously modulates the risk–return trade-off. As this coefficient increases from 0.01 to 1.00, tail risk (CVaR 95%) steadily improves, while the Sortino ratio peaks near 0.20. The dynamic SAC model outperforms static baselines (logistic regression, XGBoost, LightGBM) in average reward and, by tuning the entropy coefficient, achieves significant downside risk reduction without retraining. This framework transforms credit scoring into dynamic optimal control with continuously adjustable and interpretable risk preferences, offering a theoretically grounded tool for refined risk management. Full article

This paper proposes an optimal tracking control framework for a permanent magnet synchronous motor (PMSM) drive integrated with a quasi-Z-source (QZS) inverter for all-electric aircraft applications. Two tracking control strategies are developed: (i) an online adaptive optimal control (OAC) method for tracking motor speed and (ii) a linear quadratic integral (LQI) controller for regulating the peak DC-link voltage (PDV) of the QZS. Due to the nonlinear characteristics, parameter uncertainties, and external disturbances inherent in PMSM systems, achieving accurate speed tracking and stable DC-link voltage (DCV) regulation using a PDV control strategy under varying power flow conditions remains a significant challenge. In this study, the PMSM model is represented as a nonlinear system with strict feedback. Augmented feedforward control signals are incorporated to restructure the conventional cascade control architecture into a novel optimal control framework. Based on this formulation, a saturated adaptive optimal control law is proposed, relying on a near-optimal solution to the Hamilton–Jacobi–Isaacs (HJI) equation. This solution is approximated using an online approximator combined with an integral reinforcement learning technique. Meanwhile, an LQI controller is employed to regulate the PDV and suppress voltage fluctuations in the QZS. Simulation results demonstrate that the proposed approach significantly improves speed tracking accuracy, DCV stability, and disturbance rejection capability while improving the overall performance and reliability of PMSM drive systems. The simulation results demonstrate that the proposed control strategies have strong potential for effective application in all-electric aircraft systems, meeting the requirements of high performance and energy efficiency. Full article

We study a continuous-time cooperative differential game of pollution control in which the pollution stock accumulates emissions and affects long-run welfare. The key feature is a one-time random increase in the public damage weight, interpreted as a regime shift in environmental policy, social damage assessment, or regulatory pressure. Using dynamic programming, we characterize the grand-coalition feedback solution from the Hamilton–Jacobi–Bellman equations and derive closed-form expressions for cooperative emissions, pollution dynamics, regime-specific steady states, and transition paths. Under emission caps, we construct the coalition characteristic function using a conservative worst-case benchmark for outsider behavior rather than an unlimited-pollution assumption. For payoff allocation, we derive a dynamic payment schedule that implements the Shapley allocation along the stochastic pollution path and keeps the remaining payoff consistent with the corresponding continuation game. Finally, we extend the framework to a threshold-triggered shifted-exponential switching mechanism. This extension gives a computable objective for the optimal threshold-hitting time and clarifies how the pollution threshold and switching hazard can be interpreted as policy-relevant indicators of regulatory or ecological regime change. Full article

This study formulates a corporate abatement decision problem under carbon price uncertainty as a continuous-time stochastic control model. To this end, the carbon price is modeled as a geometric Brownian motion, while abatement capacity is accumulated through costly effort and depreciates over time. Specifically, the firm chooses its abatement effort to maximize expected discounted profits while accounting for allowance purchasing costs, compliance-related penalties, abatement costs, and potential revenues from surplus allowances. The paper contributes by integrating stochastic carbon prices, endogenous abatement-capacity accumulation, allowance-shortage/allowance-surplus asymmetry, and surplus allowance monetization into a unified corporate abatement framework. Applying the dynamic programming principle, the associated Hamilton–Jacobi–Bellman equation is derived, and the bounded optimal abatement effort is characterized in feedback form. Since the resulting nonlinear HJB equation generally does not admit a closed-form solution, a finite-difference scheme with damped policy iteration is used for numerical analysis. The results show that optimal abatement effort is strongly state-dependent. Higher carbon prices strengthen abatement incentives in the allowance-shortage region, whereas effort declines sharply after reaching allowance neutrality if surplus allowances cannot be monetized. Moreover, partial monetization of surplus allowances significantly increases abatement effort in the surplus region and can shift firms’ behavior from passive compliance to active low-carbon investment. Overall, these findings suggest that surplus allowance monetization plays an important role in sustaining firms’ abatement incentives under carbon price uncertainty. Full article

This study develops a production-based asset pricing model that incorporates time-varying disaster risk together with model uncertainty. Within an extended relative-entropy framework, agents’ distorted beliefs and ambiguity aversion are characterized, and the corresponding Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation is derived under a stochastic robust-control setting. The framework implies that the equity premium can be decomposed into three components: diffusion and jump risk premiums associated with conventional risk aversion and an additional rare-event premium generated by ambiguity aversion. Numerical experiments show that ambiguity aversion reduces the equilibrium risk-free rate, whereas aversion to rare disasters significantly raises compensation for bearing risk, helping reconcile both the equity premium puzzle and the risk-free rate puzzle. In addition, equity return volatility increases with the probability of disaster events, but at a diminishing rate. Overall, the results underscore the importance of model uncertainty and time-varying disaster risk in the determination of asset prices and risk premia. Full article

We study an infinite-horizon consumption–investment problem in which an investor endogenously manages a consumption comfort zone above a fixed subsistence benchmark. Consumption can move freely within the prevailing admissible interval, while upward expansions of the upper endpoint are irreversible and costly. This captures downward rigidity not through a single ratcheting reference level but through the endogenous management of a sustainable expenditure range. Using the dual martingale method together with singular stochastic control, we reduce the problem to a one-sided singular control problem for the comfort-zone width process. The associated dual Hamilton–Jacobi–Bellman equation becomes a gradient-constrained free-boundary problem, which admits a one-dimensional reduction under CRRA utility. We characterize the optimal comfort-zone expansion rule, consumption policy, risky portfolio rule, and value function. Economically, the model implies infrequent upward revisions of the sustainable consumption ceiling, smoother consumption than in the frictionless Merton benchmark, and state-dependent portfolio behavior. A key implication of the additive specification is that proportional consumption flexibility shrinks as the upper endpoint rises, so higher consumption states become endogenously tighter and require a larger wealth buffer to sustain. The infinite-horizon formulation is interpreted as a stationary benchmark that isolates the economics of costly lifestyle upgrading. Full article

This paper presents an adaptive optimal consensus tracking control scheme for canonical nonlinear multi-agent systems (MASs) with unknown dynamics, employing an actor–critic reinforcement learning (RL) framework. The scheme integrates a sliding mode mechanism to suppress tracking errors and ensure consensus tracking between the followers and the leader. Additionally, optimal control is designed to find a Nash equilibrium in a graphical game. To address the intractability of obtaining an analytical solution for the coupled Hamilton–Jacobi–Bellman (HJB) equation, a policy iteration algorithm is utilized. Within this algorithm, a critic neural network (NN) approximates the gradient of the optimal value function, while an actor NN approximates the optimal control policy. Together, these networks form a compact actor–critic (AC) architecture that achieves optimal consensus tracking. Furthermore, the proposed method guarantees the boundedness of all closed-loop signals while ensuring consensus tracking. Finally, two simulations are conducted to verify the effectiveness and advantages of the proposed method. Full article

The proper-time method for constructing perturbative dynamical gravitational fields is presented. Using the proper-time method, a perturbative analytical model of gravitational waves against the backdrop of an exact wave solution of Einstein’s equations in a Bianchi IV universe is constructed. To construct the perturbative analytical wave model a privileged wave coordinate system and a synchronous time function associated with the proper time of an observer freely moving in a gravitational wave are used. Reduction of the field equations, taking into account compatibility conditions, reduces the mathematical model of gravitational waves to a system of coupled ordinary differential equations for functions of the wave variable. Analytical solutions for the components of the gravitational wave metric have been found. The stability of the resulting perturbative solutions for the continuum domain of parameters is proven. The linear stability of the exact solution for a gravitational wave in the anisotropic Bianchi IV universe for the continuum domain of parameters is demonstrated. Full article

DC motors are widely used in mechatronic systems; however, their performance degrades significantly in the presence of nonlinear mechanical loads, parameter variations and sensing uncertainties. This paper proposes three control strategies (i.e., PID, optimal, and hybrid controllers) for discrete-time DC motor systems to overcome the disturbances caused by nonlinear mechanical loads and parameter variations. Optimal control of nonlinear discrete-time systems is formally characterized by the Hamilton–Jacobi–Bellman (HJB) equation, whose analytical solution is generally intractable. To address this challenge, a learning-based optimal control strategy based on the Heuristic Dynamic Programming (HDP) framework is developed to approximate the HJB equation, supported by a formal convergence proof. For that purpose, Neural Networks (NNs) are employed to approximate both the cost function and the optimal control policy, enabling near-optimal performance with manageable computational complexity. Although the resulting optimal control achieves fast convergence, it may introduce overshoot and steady-state offset under nonlinear disturbances. To address this limitation, a hybrid control framework is proposed, where nonlinear optimal corrections are integrated with the robustness and adaptability of Proportional–Integral–Derivative (PID) control through error-dependent gating and gain-scheduling mechanisms. A structured evaluation framework is conducted, including nominal analysis, motor-parameter stress testing across nine nonlinear scenarios, controller-design sensitivity analysis, and stochastic measurement-noise assessment under filtered sensing conditions. Results demonstrate that the hybrid controller preserves transient speeds within 5–10% of the optimal controller while effectively eliminating overshoot and steady-state offset under nominal conditions. The hybrid design reduces the accumulated tracking error by more than 95% compared to the optimal controller, while incurring only negligible additional control effort. Under aggressive supply-sag disturbances, the hybrid controller significantly limits peak deviation and reduces accumulated tracking error by over 90%, while maintaining comparable control cost. Overall, the hybrid framework provides a convergence-proven and practically deployable control solution that combines near-optimal convergence speed with robust, overshoot-free performance for intelligent motion-control and robotics applications. Full article

This paper addresses the stochastic optimal control of a closed-loop hybrid manufacturing–remanufacturing system (HMRS) operating under random machine failures and greenhouse gas (GHG) emission constraints in the context of sustainable industrial operations. The system consists of two dedicated machines for manufacturing and remanufacturing that jointly produce a single product in a dynamic production environment. The objective is to minimize the long-run expected total cost, including inventory holding and shortage costs, manufacturing and remanufacturing costs, and penalties associated with emissions exceeding a prescribed limit. The structure of the optimal production control policy is determined using a stochastic optimal control framework based on Hamilton–Jacobi–Bellman equations, whose optimality conditions are solved numerically. A sensitivity analysis is then conducted to examine the behavior of the resulting control policy under variations in key system parameters. The results show how coordinated manufacturing and remanufacturing decisions can be regulated through emission- and inventory-dependent thresholds in failure-prone hybrid production systems. This work contributes to the literature on sustainable manufacturing by providing a rigorous modeling and control framework for environmentally regulated hybrid manufacturing–remanufacturing systems. Full article

This paper investigates the optimal investment and reinsurance problem for insurers in a financial market with contagion risk. The prices of risky assets are assumed to follow a jump–diffusion model, where the jump component is driven by a multidimensional dynamic contagion process with diffusion (DCPD). This process simultaneously captures jumps triggered by endogenous and exogenous excitations, effectively characterizing the dynamic contagion effects arising from the joint influence of multiple factors in financial markets. The insurer aims to maximize a mean-variance (MV) utility function by purchasing per-loss reinsurance and investing the surplus in the contagion financial market. By solving the extended Hamilton–Jacobi–Bellman (HJB) equations, we derive the time-consistent equilibrium investment and reinsurance strategies, as well as explicit expressions for the equilibrium value function. These results are characterized by two nonlocal partial differential equations (PDEs), whose probabilistic solutions are obtained through the Feynman–Kac formula. Finally, numerical experiments illustrate how equilibrium strategies respond to changes in contagion intensity and confirm the effectiveness of the proposed model. Full article