Search Results (248)

We propose a life-expectancy pay-off function (LEP) for determining optimal cancer treatment within a control theory framework. The LEP averages life expectancy over all future outcomes, outcomes that are determined by key events during therapy such as tumour elimination (cure) and patient death (including treatment related mortality). We analyse this optimisation problem for tumours treated with chemotherapy using tumour growth models based on ordinary differential equations. To incorporate tumour elimination we draw on branching processes to compute the probability distribution of tumour population extinction. To demonstrate the approach, we apply the LEP framework to simplified one-compartment models of tumour growth that include three possible outcomes: cure, relapse, or death during treatment. Using Pontryagin’s maximum principle (PMP) we show that the best treatment strategies fall into three categories: (i) continuous treatment at the maximum tolerated dose (MTD), (ii) no treatment, or (iii) treat-and-stop therapy, where the drug is given at the MTD and then halted before the treatment (time) horizon. Optimal treatment strategies are independent of the time horizon unless the time horizon is too short to accommodate the most effective (treat-and-stop) therapy. For sufficiently long horizons, the optimal solution is either no treatment (when treatment yields no benefit) or treat-and-stop. Patients, thus, split into an untreatable class and a treatable class, with patient demographics, tumour size, tumour response, and drug toxicity determining whether a patient benefits from treatment. The LEP is in principle parametrisable from data, requiring estimation of the rates of each event and the associated life expectancy under that event. This makes the approach suitable for personalising cancer therapy based on tumour characteristics and patient-specific risk profiles. Full article

This work presents the results of a research study focused on the development and evaluation of an algorithmic optimal control framework for energy-efficient operation of screw compressors in smart power systems. The proposed approach is based on the Pontryagin maximum principle (PMP), which enables the synthesis of a mathematically grounded regulator that minimizes the total energy consumption of a nonlinear electromechanical system composed of a screw compressor and a variable-frequency induction motor. Unlike conventional PID controllers, the developed algorithm explicitly incorporates system constraints, nonlinear dynamics, and performance trade-offs into the control law, allowing for improved adaptability and energy-aware operation. Simulation results obtained using MATLAB/Simulink confirm that the PMP-based regulator outperforms classical PID solutions in both transient and steady-state regimes. Experimental tests conducted in accordance with standard energy consumption evaluation methods showed that the proposed PMP-based controller provides a reduction in specific energy consumption of up to 18% under dynamic load conditions compared to a well-tuned basic PID controller, while maintaining high control accuracy, faster settling, and complete suppression of overshoot under external disturbances. The control system demonstrates robustness to parametric uncertainty and load variability, maintaining a statistical pressure error below 0.2%. The regulator’s structure is compatible with real-time execution on industrial programmable logic controllers (PLCs), supporting integration into intelligent automation systems and smart grid infrastructures. The discrete-time PLC implementation of the regulator requires only 10³ arithmetic operations per cycle and less than 10² kB of RAM for state, buffers, and logging, making it suitable for mid-range industrial controllers under 2–10 ms task cycles. Fault-tolerance is ensured via range and rate-of-change checks, residual-based plausibility tests, and safe fallbacks (baseline PID or torque-limited speed hold) in case of sensor faults. Furthermore, the proposed approach lays the groundwork for hybrid extensions combining model-based control with AI-driven optimization and learning mechanisms, including reinforcement learning, surrogate modeling, and digital twins. These enhancements open pathways toward predictive, self-adaptive compressor control with embedded energy optimization. The research outcomes contribute to the broader field of algorithmic control in power electronics, offering a scalable and analytically justified alternative to heuristic and empirical tuning approaches commonly used in industry. The results highlight the potential of advanced control algorithms to enhance the efficiency, stability, and intelligence of energy-intensive components within the context of Industry 4.0 and sustainable energy systems. Full article

In this study, we develop a mathematical model to describe the transmission dynamics of the Respiratory Syncytial Virus (RSV), incorporating the coexistence of two distinct strains. The global stability of the disease-free and endemic equilibria is analyzed. Bifurcation analysis reveals the occurrence of a forward bifurcation. To control the spread of the infection, Pontryagin’s maximum principle is applied within the framework of optimal control theory, considering intervention strategies such as isolation, treatment, and vaccination. A detailed evaluation of the effectiveness of these control strategies is conducted for a specific population based on a nonlinear optimal control model. Moreover, a cost-effectiveness analysis is performed to identify the most economically viable intervention. The findings indicate that, among the studied interventions, isolation is the most cost-effective strategy for reducing RSV prevalence. The model is numerically solved using the fourth-order Runge–Kutta method, coupled with the forward–backward sweep algorithm, to assess the impact of various control combinations on the transmission dynamics of RSV. Full article

Fractional calculus of variations for a broad class of fractional operators with a general analytic kernel function is considered. Using techniques from variational analysis, we derive first- and second-order necessary optimality conditions, namely the Euler–Lagrange equation, the Weierstrass necessary condition, the Legendre condition, and finally the Legendre–Clebsch condition. Our results are new in the sense that the Euler–Lagrange equation is based on duality theory, and thus build up only with left fractional operators. The Weierstrass necessary condition is a variant of strong necessary optimality condition, and it is derived from maximum condition of Pontryagin for this general analytic kernels. The Legendre–Clebsch condition is obtained under normality assumptions on data because of equality constraints. Full article

This paper systematically reviews and analyzes various energy management strategies, as well as the characteristics, core challenges, and general processes of energy management for hybrid vehicles, aircraft, and ships. It also Analyzes the application scenarios, advantages, and limitations of rule-based energy management strategies. Based on the characteristics, design challenges, and general processes of optimized energy management strategies, a comparative analysis was conducted of mainstream strategies such as dynamic programming algorithms, Pontryagin’s minimum principle, equivalent energy consumption minimization, and multi-objective prediction. The focus was on analyzing intelligent control energy management strategies, including hybrid power system energy management strategies and their control effects based on neural network control, adaptive dynamic programming, reinforcement learning, and deep reinforcement learning. Finally, this paper addresses the challenges in applying energy management strategies, the limitations of modeling approaches, the validation of their effectiveness, and future research directions. Full article

In this study, we formulate and analyze a deterministic mathematical model describing the transmission dynamics of pneumonia. A comprehensive stability analysis is conducted for both the disease-free and endemic equilibrium points. The disease-free equilibrium is locally and globally asymptotically stable when the basic reproduction number R₀ < 1, while the endemic equilibrium is locally and globally asymptotically stable when R₀ > 1. To evaluate effective intervention strategies, an optimal control problem is formulated by introducing time-dependent control variables representing awareness campaigns, screening of carriers, and treatment of infected individuals. Applying Pontryagin’s Maximum Principle, the simulation results confirm the effectiveness of the proposed control strategies in reducing the number of infections and mitigating the overall disease burden. The findings offer valuable insights into the control of pneumonia and highlight the potential impact of strategic public health interventions. Full article

This article develops an innovative and intelligent method for analysing the criminal community’s influence on risk-forming parameters based on an analysis of regional economic processes. The research motivation was the need to create an intelligent method for quantitative assessment and risk control arising from the interaction between regional economic processes and criminal activity. The method includes a three-level mathematical model in which the economic activity dynamics are described by a modified logistic equation, taking into account the criminal activity’s negative impact and feedback through the integral risk. The criminal activity itself is modelled by a similar logistic equation, taking into account the economic base. The risk parameter accumulates the direct impact and delayed effects through the memory core. To numerically solve the spatio-temporal optimal control problem, a neural network based on the convolutional architecture was developed: two successive convolutional layers (N₁ with 3 × 3 filters and N₂ with 3 × 3 filters) extract local features, after which two 1 × 1 convolutional layers (FC₁ and FC₂) form a three-channel output corresponding to the control actions U_E, U_C, and U_I. The loss function combines the supervised component and the residual terms of the differential equations, which ensures the satisfaction of physical constraints. The computational experiment showed the high accuracy of the model: accuracy is 0.9907, precision is 0.9842, recall is 0.9983, and F1-score is 0.9912, with a minimum residual loss of 0.0093 and superiority over alternative architectures in key metrics (MSE is 0.0124, IoU is 0.74, and Dice is 0.83). Full article

This study presents a mathematical model to describe the transmission dynamics of asthma, explicitly accounting for the impact of dust waves and airborne particulate matter in the environment, recognized as key triggers of asthma exacerbations. The model incorporates a single endemic equilibrium point, which is shown to be locally asymptotically stable. To mitigate the burden of asthma, we employed the Pontryagin Maximum Principle within an optimal control framework, incorporating three time-dependent intervention strategies: vaccination, treatment, and avoidance of environmental triggers such as dust exposure. The model was numerically solved using the fourth-order Runge–Kutta method in conjunction with a forward–backward sweep algorithm to investigate the effects of various control combinations on the prevalence of asthma. Additionally, a comprehensive cost-effectiveness analysis was conducted to evaluate the economic viability of each strategy. The results indicate that the combined application of vaccination and treatment is the most cost-effective approach among the strategies analyzed, significantly reducing the number of asthma cases at minimal cost. All simulations and numerical experiments were performed to validate the theoretical findings and quantify the effectiveness of the proposed interventions under realistic environmental conditions driven by dust activity. The model highlights the importance of integrated medical and environmental control policies in mitigating asthma outbreaks, particularly in regions frequently exposed to dust storms. Full article

The application of contagion risk spread modeling within the banking sector is a relatively recent development, emerging as a response to the persistent threat of liquidity risk that has affected financial institutions globally. Liquidity risk is recognized as one of the most destructive financial threats to banks, capable of causing severe and irreparable damage if overlooked or underestimated. This study aims to identify the most effective control strategy for managing financial contagion using a Susceptible–Infected–Recovered (SIR) epidemic model, incorporating time delays in both state and control variables. The proposed strategy seeks to maximize the number of resilient (vulnerable) banks while minimizing the number of infected institutions at risk of bankruptcy. Our goal is to formulate intervention policies that can curtail the propagation of financial contagion and mitigate associated systemic risks. Our model remains a simplification of reality. It does not account for strategic interactions between banks (e.g., panic reactions, network coordination), nor for adaptive regulatory mechanisms. The integration of these aspects will be the subject of future work. We establish the existence of an optimal control strategy and apply Pontryagin’s Maximum Principle to characterize and analyze the control dynamics. To numerically solve the control system, we employ a discretization approach based on forward and backward finite difference approximations. Despite the model’s simplifications, it captures key dynamics relevant to major European banks. Simulations performed using Python 3.12 yield significant results across three distinct scenarios. Notably, in the most severe case (${\alpha }_3=1.0$), the optimal control strategy reduces bankruptcies from 25% to nearly 0% in Spain, and from 12.5% to 0% in France and Germany, demonstrating the effectiveness of timely intervention in containing financial contagion. Full article

This paper introduces a novel fractional-order model using the Caputo derivative operator to investigate the virus dynamics of adaptive immune responses. Two infection routes, namely cell-to-cell and virus-to-cell transmissions, are incorporated into the dynamics. Our research establishes the existence and uniqueness of positive and bounded solutions through the application of the generalized mean-value theorem and Banach fixed-point theory methods. The fractional-order model is shown to be Ulam–Hyers stable, ensuring the model’s resilience to small errors. By employing the normalized forward sensitivity method, we identify critical parameters that profoundly influence the transmission dynamics of the fractional-order virus model. Additionally, the framework of optimal control theory is used to explore the characterization of optimal adaptive immune responses, encompassing antibodies and cytotoxic T lymphocytes (CTL). To assess the influence of memory effects, we utilize the generalized forward–backward sweep technique to simulate the fractional-order virus dynamics. This study contributes to the existing body of knowledge by providing insights into how the interaction between virus-to-cell and cell-to-cell dynamics within the host is affected by memory effects in the presence of optimal control, reinforcing the invaluable synergy between fractional calculus and optimal control theory in modeling within-host virus dynamics, and paving the way for potential control strategies rooted in adaptive immunity and fractional-order modeling. Full article

This paper addresses the problem of optimizing obesity, which has been a challenging issue in the last decade based on recent data revealed in 2024 by the World Health Organization (WHO). The current work introduces a new mathematical model of the dynamics of weight over time with embedded control parameters to optimize the number of obese, overweight, and comorbidity populations. The mathematical formulation of the model is developed under certain sufficient conditions that guarantee the positivity and boundedness of solutions over time. The model structure exhibits inherent symmetry in population group transitions, particularly around the equilibrium state, which allows the application of analytical tools such as the Routh–Hurwitz and Metzler criteria. Then, the analysis of local and global stability of the obesity-free equilibrium state is discussed based on these criteria. Based on the Pontryagin maximum principle (PMP), the deviation from the obesity-free equilibrium state is controlled. The model’s effectiveness is demonstrated through simulation using the Forward–Backward Sweeping algorithm with parameters derived from recent research in human health. Incorporating symmetry considerations in the model enhances the understanding of system behavior and supports balanced intervention strategies. Results suggest that the model can effectively inform strategies to mitigate obesity prevalence and associated health risks. Full article

In this paper, we develop an HIV/AIDS epidemic model that incorporates a saturated incidence rate to reflect the limited transmission capacity and the impact of behavioral saturation in contact patterns. The model is formulated as a system of seven non-linear ordinary differential equations representing key population compartments. In addition to model formulation, we introduce an optimal control problem involving three control measures: educational campaigns, screening of unaware infected individuals, and antiretroviral treatment for aware infected individuals. We begin by establishing the positivity and boundedness of the model solutions under constant control inputs. The existence and local and global stability of both the disease-free and endemic equilibrium points are analyzed, depending on the effective reproduction number (${\mathcal{R}}_e$). Bifurcation analysis reveals that the model undergoes a forward bifurcation at ${\mathcal{R}}_e=1$. A local sensitivity analysis of ${\mathcal{R}}_e$ identifies the disease transmission rate as the most sensitive parameter. The optimal control problem is then formulated by incorporating the dynamics of infected subpopulations, control costs, and time-dependent controls. The existence of optimal control solutions is proven, and the necessary conditions for optimality are derived using Pontryagin’s Maximum Principle. Numerical simulations support the theoretical analysis and confirm the stability of the equilibrium points. The optimal control strategies, evaluated using the Incremental Cost-Effectiveness Ratio (ICER), indicate that implementing both screening and treatment (Strategy D) is the most cost-effective intervention. These results provide important insights for designing effective and economically sustainable HIV/AIDS intervention policies. Full article

The aim of this paper is to propose mathematical models to predict and optimize the cost of wastewater treatment using bacteria and oxygen under fluctuating resource and cultivation conditions. We have thus developed deterministic mathematical models based on dynamic systems and applied optimal control theory to reduce treatment costs. Two wastewater treatment models are proposed: one using only one type of aerobic bacteria, thermophilic bacteria; and the second using two types of aerobic bacteria, thermophilic and mesophilic bacteria. For each model, an optimal control problem is solved and numerical simulations illustrate the theoretical results. Full article

To address energy management challenges for intelligent connected automated range-extended electric vehicles under vehicle-road cooperative environments, a hierarchical energy management strategy (EMS) based on speed prediction is proposed from the perspective of multi-objective optimization (MOO), with comprehensive system performance being significantly enhanced. Focusing on connected car-following scenarios, acceleration sequence prediction is performed based on Kalman filtering and preceding vehicle acceleration. A dual-layer optimization strategy is subsequently developed: in the upper layer, optimal speed curves are planned based on road network topology and preceding vehicle trajectories, while in the lower layer, coordinated multi-power source allocation is achieved through EMS_MPC-P, a Bayesian-optimized model predictive EMS based on Pontryagin’ s minimum principle (PMP). A MOO model is ultimately formulated to enhance comprehensive system performance. Simulation and bench test results demonstrate that with SoC₀ = 0.4, 7.69% and 5.13% improvement in fuel economy is achieved by EMS_MPC-P compared to the charge depleting-charge sustaining (CD-CS) method and the charge depleting-blend (CD-Blend) method. Travel time reductions of 62.2% and 58.7% are observed versus CD-CS and CD-Blend. Battery lifespan degradation is mitigated by 16.18% and 5.89% relative to CD-CS and CD-Blend, demonstrating the method’s marked advantages in improving traffic efficiency, safety, battery life maintenance, and fuel economy. This study not only establishes a technical paradigm with theoretical depth and engineering applicability for EMS, but also quantitatively reveals intrinsic mechanisms underlying long-term prediction accuracy enhancement through data analysis, providing critical guidance for future vehicle–road–cloud collaborative system development. Full article

This paper constructs a fundamental mathematical model to depict the therapeutic effects of two drugs on breast cancer patients. The model is described by fractional order differential equations with two control variables. Two scenarios are considered: the constant control and the optimal control. For the constant control scenario, the existence and uniqueness of the solution of the system are proved by using the fixed point theorem and combining with the Caputo–Fabrizio fractional derivative; then, the sufficient conditions for the existence and stability of the system’s equilibriums are derived. For the optimal control scenario, the optimal control solution is obtained by using the Pontryagin’s maximum principle. To further validate the effectiveness of the theoretical results, numerical simulations were conducted. The results show that the parameters have significant sensitivity to the dynamic behavior of the system. Full article