Maximum Principle Research Articles

Sandflies spread the neglected vector-borne disease anthroponotic cutaneous leishmaniasis (ACL), which only affects humans. Despite decades of control, asymptomatic carriers, vector pesticide resistance, and low public awareness prevent eradication. This study proposes a fractional-order optimal control model that integrates biological and behavioral aspects of ACL transmission to better understand its complex dynamics and intervention responses. We model asymptomatic human illnesses, insecticide-resistant sandflies, and a dynamic awareness function under public health campaigns and collective behavioral memory. Four time-dependent control variables—symptomatic treatment, pesticide spraying, bed net use, and awareness promotion—are introduced under a shared budget constraint to reflect public health resource constraints. In addition, Caputo fractional derivatives incorporate memory-dependent processes and hereditary effects, allowing for epidemic and behavioral states to depend on prior infections and interventions; on the other hand, standard integer-order frameworks miss temporal smoothness, delayed responses, and persistence effects from this memory feature, which affect optimal control trajectories. Next, we determine the optimality conditions for fractional-order systems using a generalized Pontryagin’s maximum principle, then solve the state–adjoint equations numerically with an efficient forward–backward sweep approach. Simulations show that fractional (memory-based) dynamics capture behavioral inertia and cumulative public response, improving awareness and treatment efforts. Furthermore, sensitivity tests indicate that integer-order models do not predict the optimal allocation of limited resources, highlighting memory effects in epidemiological decision-making. Consequently, the proposed method provides a realistic and flexible mathematical basis for cost-effective and sustainable ACL control plans in endemic settings, revealing how memory-dependent dynamics may affect disease development and intervention efficiency.

This article presents an analysis of President Trump's statements on the Panama Canal from the perspective of Mearsheimer's Offensive Realism, examining their impact on Panamanian national identity and perception of sovereignty complemented by Dependency Theory to account for Panama's position as a peripheral and dependent state. The research is based on statistical analysis of two public opinion surveys ( n = 906, n = 732) conducted in February and April 2025, applying chi-square tests, correspondence analysis, and Spearman's correlation. Results reveal that Panamanians interpret these statements as a geopolitical strategy aimed at containing Chinese influence in the region, confirming Offensive Realism principles of power maximization and control of spheres of influence. Findings indicate that the Panama Canal is a symbol of Panamanian national identity and territorial sovereignty. The temporal analysis shows an evolution in public perception, where the narrative about the “Chinese threat” progressively lost credibility ( p < 0.024), while identification with the Canal as a national symbol remained strong (Spearman's correlation = 0.591, p < 0.001). Provinces differed significantly in how they saw presidential comments and how the media handled information. Popular response based on historical memory and collective identity demonstrates national symbols used as defense mechanisms against external hegemonic pressures in the reconfiguration of the international order.

Stochastic Maximum Principle for Equations with Delay: Going to Infinite Dimensions to Solve the Nonconvex Case

In this article, we proposed a new mathematical model to investigate the dynamics of the infectious disease, control, and general disease transmission. The model exhibits two distinct non-trivial equilibrium states. As a fundamental prerequisite for stability analysis, we first derive the epidemiological threshold parameter R0 through next-generation matrix methodology.According to our investigation, R0 plays an essential role in describing the model’s dynamics. We demonstrate that in the case when R0 takes values less or greater than unity, the endemic (disease-free) condition is asymptotically stable both locally and globally. To try to stop the general disease from spreading throughout a community, we add control parameters, create a control model, and suggest control techniques. The maximum principle of Pontryagin is used to derive the optimality system. Finally, the numerical simulations are performed using the fourth-order Runge-Kutta technique to validate and confirm our analytical conclusions. Phase portrait analysis further illustrates the convergence of system trajectories toward disease-free or endemic equilibria under different control scenarios, reinforcing the stability criteria derived for R0.

This paper presents a statistically grounded algorithm for surface imaging with linear frequency-modulated continuous wave synthetic aperture radar. The approach is based on the maximum likelihood principle, where solving the optimization problem naturally leads to the introduction of a spectral decorrelation filter. The proposed method increases the effective number of statistically independent samples, reduces speckle, and improves the accuracy of radar cross section estimation. Simulation experiments demonstrate consistent advantages over classical SAR processing: the proposed method achieves up to a 21% improvement in feature similarity metrics and an average 4% improvement across standard quantitative image quality measures.

Global maximum principle for optimal control of stochastic Volterra equations with singular kernels: An infinite dimensional approach

Sufficient stochastic maximum principle for mean-field control problems with regime switching in an infinite horizon

This paper develops and analyzes a deterministic SEIHR (Susceptible-Exposed-Infectious-Hospitalized-Recovered) model to investigate the transmission dynamics of cerebrospinal meningitis (CSM) and evaluate optimal control strategies. The framework incorporates three time-dependent control variables: mass vaccination of susceptible individuals, enhanced treatment for hospitalized patients, and public awareness campaigns. Using Pontryagin's Maximum Principle, we formulate an optimal control problem to minimize the number of infected individuals and the costs associated with the interventions. The basic reproduction number ($R_0$) is derived, and its sensitivity to key parameters is analyzed. Numerical simulations, using data relevant to the Yobe State context, demonstrate that a combined strategy of early, intensive vaccination, sustained treatment efforts, and effective public awareness is the most effective approach to mitigate the burden of a CSM outbreak. These findings provide quantitative support for evidence-based public health policies aimed at controlling meningitis in high-risk regions.

The aim of this study is to examine the dynamics of deposits, loans, and equity on the balance sheet of the bank under capital adequacy constraints. The intention is to provide a tractable framework for assessing solvency and regulatory policies. Therefore, a nonlinear continuous-time model was developed with logistic growth, credit risk, and capital adequacy conditions. The model was calibrated using monthly data for Indonesian commercial banks from 2022 to 2024, and the parameters were estimated using particle swarm optimization. The results showed that the model replicates observed trajectories with mean absolute percentage errors below 2.1% to confirm its empirical validity. The simulations showed that stricter capital requirements slowed equity growth while moderate requirements supported long-run capitalization. A time-varying capital adequacy policy was formulated as an optimal control problem, and the Pontryagin maximum principle was applied to derive an optimal regulatory path. The results showed that adaptive regulation stabilized capitalization while limiting policy costs. The trend reflected the value of the continuous-time control theory in financial regulation.

This paper presents a novel approach to the controllability of nonlinear dynamic systems using recurrent neural networks (RNNs). We develop a comprehensive theoretical framework that integrates controllability analysis, stability verification via Lyapunov functions, and the derivation of optimal control laws based on Pontryagin’s Maximum Principle. Our methodology not only ensures theoretical soundness but also offers practical effectiveness. To demonstrate its applicability, we conduct simulations using real-world data from the AAPL stock database. The proposed RNN-based control framework significantly reduces the deviation between predicted system outputs and actual observations. We further enhance performance through two complementary strategies, a direct control method and a parameter optimization approach, both of which contribute to the accuracy and adaptability of the control system. These results confirm the potential of neural network-based control in managing complex nonlinear dynamics

This paper introduces a thermodynamic unified field theory (UFT) that derives all physical phenomena from fundamental principles of entropy maximization, positing the photrino—a composite of photons, neutrinos, and anti-neutrinos—as the elementary entity. Grounded in three axioms—entropy maximization for equilibrium, Gibbs free energy linked to photon frequency with helicity encoding, and symmetries governing rotational and translational fluxes—the theory synthesizes quantum mechanics, general relativity, electromagnetism, and cosmology into a cohesive framework. By treating reality as a multi-phase flux sea, it eliminates arbitrary parameters in the Standard Model, deriving particle masses, interactions, and cosmic structures from thermodynamic scales. Key derivations include the reactive Hessian partial differential equations for flux dynamics, emergent time from angular-to-linear momentum "snaps," and the structural parameter β=5 incorporating gluon degrees of freedom for dimensional stability. The UFT resolves longstanding puzzles, such as the proton spin crisis (via rotational flux contributions), neutrino handedness (helicity signs), cosmological lithium anomaly (flux dilution), Yang-Mills mass gap (finite Hessian minima), and phase transition discontinuities (Gibbs continuity). It reinterprets quantum effects like double-slit interference and entanglement as phase equilibria, and classical laws (e.g., Maxwell's equations, Navier-Stokes) as dilute limits. Empirical alignments with observations in atomic spectra, QGP experiments, and cosmology are demonstrated, with testable predictions including photrino signatures in neutrino detectors, blueprints for room-temperature superconductors, optimized catalysis, and fusion plasma stability. This paradigm shifts physics toward a thermodynamic foundation, offering practical applications in materials science, biology, and energy while implying a cyclic, multiverse cosmology.

In the presents investigation a fractional-order SIR mathematical model formulated with the Caputo-Fabrizio derivative to investigate the transmission dynamics of Lumpy Skin Disease (LSD) in domestic cattle. Model parameter characteristic included as positivity, boundedness and the basic reproduction number are analytically established along with existence and uniqueness proven by the fixed-point theory. Optimal control strategies derived using Pontryagin's maximum principle target infection reduction through prevention, pre-treatment and enhanced treatment measures. Numerical simulations have been carried out the fractional-order dynamics which capture the real-world complexities with lower fractional order accelerating disease spread. Comparative analysis with integer-order models confirms the superior applicability of fractional calculus. The obtained results demonstrate that the proposed control measures can substantially reduce infection levels, offering practical guidance for LSD mitigation in cattle populations.

ABSTRACT In this paper, we analyze a deterministic model of malaria and Corona Virus Disease 2019 co-infection within a homogeneous population. We first studied the single infection model of each disease and then the co-infection dynamics. We calculate the basic reproduction number of each model and study the existence and stability of the steady states. Then, we show that under some suitable conditions, both the malaria single infection model and co-infection model exhibit backward bifurcation. Furthermore, we analyze the conditions of extinction, competitive exclusion and co-existence of these two diseases. In addition, a local sensitivity analysis of the basic reproduction numbers is performed to explore the impact of the different parameters’ variability on the dynamics of each disease. Moreover, we apply Pontryagin’s maximum principle to determine optimal strategies to control the two diseases in case of co-infection. Finally, some numerical simulation results are presented to support the theoretical findings.

This paper investigates the optimal control problem associated with the dynamic frictionless contact process of a Gao beam, which incorporates double Signorini conditions. By employing the Dubovitskiĭ-Milyutin functional analytical approach, the necessary optimality conditions in the form of the Pontryagin maximum principle are derived for the case of fixed final time horizon. The analysis takes into account the nonsmooth characteristics of the contact constraints and provides a rigorous mathematical framework for describing optimal controls. The paper concludes with an algorithm highlighting the potential applications of the derived results.

In this paper, we develop a deterministic mathematical model to study the transmission dynamics of diphtheria, incorporating optimal control strategies and cost-effectiveness analysis. The model consists of six compartments: susceptible, exposed, infectious, asymptomatic, hospitalized, and recovered individuals. We analyze the model’s qualitative behavior, including the existence of an invariant region, the positivity of solutions, and the identification and stability (both local and global) of two equilibrium points: the disease-free equilibrium and the endemic equilibrium. The effective reproduction number is derived to assess the potential spread of the disease. To determine optimal control strategies, we apply Pontryagin’s Maximum Principle to obtain the Hamiltonian, adjoint equations, control characterizations, and the resulting optimality system. Various combinations of control strategies are evaluated to assess their impact on diphtheria transmission. We use the incremental cost-effectiveness ratio (ICER) to identify the most effective and efficient intervention. Sensitivity analysis and numerical simulations are conducted to support the findings. The simulation results indicate that a combined strategy of prevention and vaccination is the most cost-effective approach. The implementation of control strategies is shown to play a critical role in reducing the burden of diphtheria in the community.

Marine insecurity, particularly the rising incidence of piracy and illegal maritime activities, poses a major threat to global trade, regional economies, and human safety. This study develops a nonlinear dynamical model that captures the interactions between vulnerable vessels, pirate groups, and naval patrol forces. The model incorporates resource allocation as an optimal control variable, representing patrol intensity, which simultaneously reduces attack success and suppresses pirate activity. Using Pontryagin’s Maximum Principle, an optimal control formulation is derived to minimize the number of successful attacks while balancing the operational costs of patrol deployment. Numerical simulations demonstrate the impact of varying patrol strategies on the dynamics of marine insecurity, highlighting threshold conditions that determine whether piracy persists or declines. The results provide insight into effective resource allocation strategies for maritime security agencies, emphasizing the balance between economic costs and long-term deterrence. This work offers a quantitative framework for decision-making in combating marine insecurity.

The paper is devoted to the study of the two-phase free boundary problem for nonlinear partial differential equations describing the evolution of a foam drainage in the one dimensional case which was proposed by Goldfarb et al. in 1988 in order to investigate the flow of a liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. In a series of papers, the authors have already solved the same problems without free boundary and with free boundary situated at the lower and the upper parts in the foam column, respectively. In this paper it is shown that the free boundary problem for the foam drainage equations with a sharp interface between dry and wet foams admits a unique global-in-time classical solution; this is done by a standard classical mathematical method, the maximum principle, and the comparison theorem. Moreover, the existence of the steady solution and its stability are shown.

Mass-balanced compartmental systems defy classical deterministic entropy measures since both metric and topological entropy vanish in dissipative dynamics. By interpreting open compartmental systems as absorbing continuous-time Markov chains that describe the random journey of a single representative particle, we allow established information-theoretic principles to be applied to this particular type of deterministic dynamical system. In particular, path entropy quantifies the uncertainty of complete trajectories, while entropy rates measure the average uncertainty of instantaneous transitions. Using Shannon’s information entropy, we derive closed-form expressions for these quantities in equilibrium and extend the maximum entropy principle (MaxEnt) to the problem of model selection in compartmental dynamics. This information-theoretic framework not only provides a systematic way to address equifinality but also reveals hidden structural properties of complex systems such as the global carbon cycle.

We investigate the set-theoretic strength of several maximality principles that play an important role in the study of modal and intuitionistic logics. We focus on well-known Fine’s and Esakia’s Maximality Principles, present two formulations of each, and show that the stronger formulations are equivalent to the Axiom of Choice (AC), while the weaker ones to the Boolean Prime Ideal Theorem (BPI).

Bifurcation and optimal control analysis for a fractional-order model of drug-resistant HBV infection.