Maximum Principle Research Articles

Dynamics of the optimality control of transmission of infectious disease: a sensitivity analysis

Over the course of history global population has witnessed deterioration of unprecedented scale caused by infectious transmission. The necessity to mitigate the infectious flow requires the launch of a well-directed and inclusive set of efforts. Motivated by the urge for continuous improvement in existing schemes, this article aims at the encapsulation of the dynamics of the spread of infectious diseases. The objectives are served by the launch of the infectious disease model. Moreover, an optimal control strategy is introduced to ensure the incorporation of the most feasible health interventions to reduce the number of infected individuals. The outcomes of the research are facilitated by stratifying the population into five compartments that are susceptible class, acute infected class, chronic infected class, recovered class, and vaccinated class. The optimal control strategy is formulated by incorporating specific control variables namely, awareness about medication, isolation, ventilation, vaccination rates, and quarantine level. The developed model is validated by proving the pivotal delicacies such as positivity, invariant region, reproduction number, stability, and sensitivity analysis. The legitimacy of the proposed model is delineated through the detailed sensitivity analysis along with the documentation of local and global features in a comprehensive manner. The maximum sensitivity index parameters are disease transmission and people moved from acute stages into chronic stages whose value is (0.439, 1) increase in parameter by 10 percent would increase the threshold quantity by (4.39, 1). Under the condition of a stable system, we witnessed an inverse relationship between susceptible class and time. Moreover, to assist the gain of the fundamental aim of this research, we take the control variables as time-dependent and obtain the optimal control strategy to minimize infected populations and to maximize the recovered population, simultaneously. The objectives are attained by the employment of the Pontryagin maximum principle. Furthermore, the efficacy of the usual health interventions such as quarantine, face mask usage, and hand sanitation are also noticed. The effectiveness of the suggested control plan is explained by using numerical evaluation. The advantages of the new strategy are highlighted in the article.

The role of antisymmetric functions in nonlocal equations

We use a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms, in combination with the method of moving planes, to prove symmetry for the semilinear fractional parallel surface problem. That is, we prove that non-negative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set Ω ⊂ R n \Omega \subset \mathbb {R}^n must be radially symmetric if one of their level surfaces is parallel to the boundary of Ω \Omega ; in turn, Ω \Omega must be a ball. Furthermore, we discuss maximum principles and the Harnack inequality for antisymmetric functions in the fractional setting and provide counter-examples to these theorems when only ‘local’ assumptions are imposed on the solutions. The construction of these counter-examples relies on an approximation result that states that ‘all antisymmetric functions are locally antisymmetric and s s -harmonic up to a small error’.

Mathematical Modeling Approach to the Optimization of Biomass Storage Park Management

This paper addresses the critical issue of managing biomass parks, a key component in the shift towards sustainable energy sources. The research problem centers on optimizing the management of these parks to enhance production and economic viability. Our aim was to bridge the gap in current research by developing and applying mathematical models tailored for biomass park management. The study commenced by constructing a basic model based on assumptions such as uniform biomass and steady input rates. Progressing from this initial model, we explored sophisticated control strategies, including Pontryagin’s maximum principle and dynamic programming, and employed numerical methods to tackle the nonlinearities and complexities inherent in biomass management. Our approach’s scope extended to predicting and managing biomass flow, highlighting each method’s distinct advantages. The simple model laid the groundwork for understanding, while optimal control techniques revealed the system’s intricate dynamics. The numerical methods provided practical solutions to complex equations. We found that while each method is beneficial on its own, their combined use can significantly improve decision-making in biomass park management. This research emphasizes the importance of aligning the chosen method with specific operational challenges and desired outcomes for optimal efficacy, offering both theoretical insights and practical applications in the field of renewable energy management.

Reconstructing parton distribution function based on maximum entropy method* *Supported by Key Project for Undergraduate Teaching Reform and Quality Enhancement Research Plan in Ordinary Colleges and Universities in Tianjin (A231005505)

A new method based on the maximum entropy principle for reconstructing the parton distribution function (PDF) from moments is proposed. Unlike traditional methods, the new method does not require any artificial assumptions. For the case of moments with errors, we introduce Gaussian functions to soften the constraints of moments. Through a series of tests, the effectiveness and reconstruction efficiency of this new method are evaluated comprehensively, demonstrating that this method is reasonable and can achieve high-quality reconstruction with at least the first six moments as input. Finally, we select a set of lattice quantum chromodynamics (QCD) results regarding moments as input and provide reasonable reconstruction results for the pion.

Research on satellite laser ranging observation task scheduling method

Satellite laser ranging (SLR) is a technology with the highest precision of single measurement of satellite radial distance, which is developing rapidly in the direction of long-distance, high-precision and automation. SLR autonomous observation task scheduling is an important step in realizing station automation, which needs to satisfy the principles of satellite tracking priority and maximization of observation revenue at the same time. In order to improve the automation and intelligence level of SLR system, based on the framework of ant colony optimization (ACO) algorithm, this paper combines the dynamic optimization characteristics of ACO algorithm and the local optimization characteristics of greedy algorithm, introduces the maximum-minimum ant mechanism, and puts forward a scheduling scheme for SLR observation task based on greedy ant colony algorithm (GACA). The results show that compared with the current scheduling method applied in practice, the number of observation satellites obtained from the GACA algorithm-based observation task planning for the SLR system has been improved by 37.4%, the total arc segment of satellite observation with higher priority has been extended by 36.47%, and the total observation gain has been increased by 42.39% in the same period of time. It effectively solves the problems of low efficiency, easy to miss satellites (being equipped with corner-cubes) and less satellites (being equipped with corner-cubes) stars in the observation process in manual scheduling, and provides a simple, practical, efficient and convenient observation task planning scheme for the establishment of an unmanned SLR system.

Second-order optimality conditions for the bilinear optimal control of a degenerate equation

The main purpose of this paper is the study of second-order optimality conditions for the bilinear control of a strongly degenerate parabolic equation. The equation is degenerate at the boundary of the spatial domain. The well-posedness of the state equation, as well as weak maximum principles are established. We prove some differentiability properties of the control-to-state operator and the existence of optimal solutions. Finally, we derive first- and second-order optimality conditions for the system.

Pneumonia and COVID-19 co-infection modeling with optimal control analysis

In this study, we present a nonlinear deterministic mathematical model for co-infection of pneumonia and COVID-19 transmission dynamics. To understand the dynamics of the co-infection of COVID-19 and pneumonia sickness, we developed and examined a compartmental based ordinary differential equation type mathematical model. Firstly, we showed the limited region and non-negativity of the solution, which demonstrate that the model is biologically relevant and mathematically well-posed. Secondly, the Jacobian matrix and the Lyapunov function are used to illustrate the local and global stability of the equilibrium locations. If the related reproduction numbers R0c, R0p, and R0 are smaller than unity, then pneumonia, COVID-19, and their co-infection have disease-free equilibrium points that are both locally and globally asymptotically stable otherwise the endemic equilibrium points are stable. Sensitivity analysis is used to determine how each parameter affects the spread or control of the illnesses. Moreover, we applied the optimal control theory to describe the optimal control model that incorporates four controls, namely, prevention of pneumonia, prevention of COVID-19, treatment of infected pneumonia and treatment of infected COVID-19. Then the Pontryagin's maximum principle is introduced to obtain the necessary condition for the optimal control problem. Finally, the numerical simulation of optimality system reveals that the combination of treatment and prevention is the most optimal to minimize the diseases.

Role of finite element analysis for selection of single point fixation in zygomaticomaxillary complex fracture

BackgroundOne-point fixation was superior to the two and three-points fixation in minimally displaced zygomaticomaxillary complex (ZMC) fracture regarding the cost, invasiveness, scaring, number of wounds, and operation time. Accordingly, this study aimed to predict which one-point fixation is the most stable in managing minimally displaced ZMC fracture.Material & methodsThis study simulated the different one-point fixation approaches on three ZMC models after fracture reduction and application of all forces exerted on the fractured area. The findings were represented as stress impact on the ZMC fracture and plating system as well as the inter-fragments micro-motion.ResultsThe von misses stresses of plates for the zygomaticofrontal, infra-orbital rim, and zygomaticomaxillary buttress model were (66.508, 1.285, and1.16 MPa) respectively. While the screws’ von misses for the infraorbital rim, zygomaticofrontal, and zygomaticomaxillary buttress models were (13.8, 4.05, and 1.60 MPa) respectively. Whereas, the maximum principles stress at zygomaticofrontal, zygomaticomaxillary buttress, and infraorbital rim models were (37.03, 37.01, and 34.46 MPa) respectively. In addition, the inter-fragment micro-motion for zygomaticomaxillary buttress, infraorbital rim, and zygomaticofrontal models were (0.26, 0.25, and 0.15 mm) respectively.ConclusionOne-point fixation at zygomaticomaxillary buttress is the preferred point because it is exposed to low stresses, and the inter-fragment micro-motion is within the approved limit with the elements in the same direction of fixation which indicates the rigid fixation. In addition, it is less palpable and scarless.Trial registrationclinical trial.gov (NCT05819372) at 19/04/2023.

Evaluation of bone-related mechanical properties in female patients with long-term remission of Cushing's syndrome using quantitative computed tomography-based finite element analysis.

Hypercortisolism in Cushing's syndrome (CS) is associated with bone loss, skeletal fragility, and altered bone quality. No studies evaluated bone geometric and strain-stress values in CS patients after remission thus far. Thirty-two women with CS in remission (mean age [±SD] 51 ± 11; body mass index [BMI], 27 ± 4 kg/m2; mean time of remission, 120 ± 90 months) and 32 age-, BMI-, and gonadal status-matched female controls. Quantitative computed tomography (QCT) was used to assess volumetric bone mineral density (vBMD) and buckling ratio, cross-sectional area, and average cortical thickness at the level of the proximal femur. Finite element (FE) models were generated from QCT to calculate strain and stress values (maximum principal strain [MPE], maximum strain energy density [SED], maximum Von Mises [VM], and maximum principal stress [MPS]). Areal BMD (aBMD) and trabecular bone score (TBS) were assessed by dual-energy X-ray absorptiometry (2D DXA). Trabecular vBMD at total hip and trochanter were lower in CS as compared with controls (P < .05). Average cortical thickness was lower, and buckling ratio was greater in CS vs controls (P < .01). All strain and stress values were higher in CS patients vs controls (P < .05). 2D DXA-derived measures were similar between patients and controls (P > .05). Prior hypercortisolism predicted both VM (β .30, P = .014) and MPS (β .30, P = .015), after adjusting for age, BMI, menopause, delay to diagnosis, and duration of remission. Women with prior hypercortisolism have reduced trabecular vBMD and impaired bone geometrical and mechanical properties, which may contribute to an elevated fracture risk despite long-term remission.

Ground state solutions for a kind of superlinear elliptic equations with variable exponent

AbstractIn this paper, we focus on the existence of ground state solutions for the $p(x)$ p ( x ) -Laplacian equation $$ \textstyle\begin{cases} -\Delta _{p(x)}u+\lambda \vert u \vert ^{p(x)-2}u=f(x,u)+h(x) \quad \text{in } \Omega , \\ u=0,\quad \text{on }\partial \Omega . \end{cases} $$ { − Δ p ( x ) u + λ | u | p ( x ) − 2 u = f ( x , u ) + h ( x ) in Ω , u = 0 , on ∂ Ω . Using the constraint variational method, quantitative deformation lemma, and strong maximum principle, we proved that the above problem admits three ground state solutions, especially speaking, one solution is sign-changing, one is positive, and one is negative. Our results improve on those existing in the literature.

Positive steady-state solutions for a vegetation–water model with saturated water absorption

In this paper, a vegetation–water model with saturated water absorption is considered. The global stability of boundary equilibrium is first obtained. Then the stability and Turing instability of positive equilibria are discussed and we find that the equilibrium with vegetation small is always unstable and if the vegetation diffusion is small or the water diffusion is large, the other positive equilibrium loses its stability and Turing instability occurs. In addition, a priori estimates of nonnegative steady-state solutions is obtained by the maximum principle. Moreover, some detailed qualitative analyses are carried out on the steady-state bifurcations at both simple and double eigenvalues. In particular, we establish the criterion to determine the bifurcation direction. Finally, some dynamics near the bifurcation point are showed numerically and we depict the evolution processes of vegetation patterns under different parameters. Our results show that the parameter 1/p, which represents the conversion of water absorption, has a great impact on the vegetation patterns formation: with the increase of p, the vegetation biomass decreases, and meanwhile it can induce the transition of different pattern structures.

The infinite-dimensional Pontryagin maximum principle for optimal control problems of fractional evolution equations with endpoint state constraints

&lt;abstract&gt;&lt;p&gt;In this paper, we study the infinite-dimensional endpoint state-constrained optimal control problem for fractional evolution equations. The state equation is modeled by the $ \mathsf{X} $-valued left Caputo fractional evolution equation with the analytic semigroup, where $ \mathsf{X} $ is a Banach space. The objective functional is formulated by the Bolza form, expressed in terms of the left Riemann-Liouville (RL) fractional integral running and initial/terminal costs. The endpoint state constraint is described by initial and terminal state values within convex subsets of $ \mathsf{X} $. Under this setting, we prove the Pontryagin maximum principle. Unlike the existing literature, we do not assume the strict convexity of $ \mathsf{X}^* $, the dual space of $ \mathsf{X} $. This assumption is particularly important, as it guarantees the differentiability of the distance function of the endpoint state constraint. In the proof, we relax this assumption via a separation argument and constructing a family of spike variations for the Ekeland variational principle. Subsequently, we prove the maximum principle, including nontriviality, adjoint equation, transversality, and Hamiltonian maximization conditions, by establishing variational and duality analysis under the finite codimensionality of initial- and end-point variational sets. Our variational and duality analysis requires new representation results on left Caputo and right RL linear fractional evolution equations in terms of (left and right RL) fractional state transition operators. Indeed, due to the inherent complex nature of the problem of this paper, our maximum principle and its proof technique are new in the optimal control context. As an illustrative example, we consider the state-constrained fractional diffusion PDE control problem, for which the optimality condition is derived by the maximum principle of this paper.&lt;/p&gt;&lt;/abstract&gt;

A Schwarz lemma of harmonic maps into metric spaces

&lt;p&gt;We established a Schwarz lemma for harmonic maps from Riemannian manifolds to metric spaces of curvature bounded above in the sense of Alexandrov. We adopted the gradient estimate technique which was based on Zhang-Zhu's maximum principle. In particular, when the domain manifold was a hyperbolic surface, the energy of any conformal harmonic maps into $ \operatorname{CAT}(-1) $ spaces were bounded from above uniformly.&lt;/p&gt;

Research on low-carbon closed-loop supply chain strategy based on differential games-dynamic optimization analysis of new and remanufactured products

&lt;p&gt;With the current increasing global demand for low-carbon and environmentally friendly products, promoting the sustainability of closed-loop supply chains has become one of the key measures. However, consumers often do not regard remanufactured products as equivalent to new products. Therefore, this paper proposes a dynamic closed-loop supply chain that incorporates consumers' purchasing preferences to model a long-term game with product differentiation. Moreover, to enhance consumer acceptance of remanufactured products and reduce manufacturers' costs, low-carbon technologies and cost-sharing mechanisms are introduced. In this way, we construct a differential game in which the manufacturer sells new and remanufactured products through a retailer and makes decisions about the level of low-carbon technology in the remanufacturing process. Based on the theory of differential games, this paper analyzes three different power structures: the manufacturer-dominated Stackelberg game, the Nash game, and the retailer-dominated Stackelberg game. The optimal low-carbon technology level and pricing strategy are obtained by applying Pontryagin's maximum principle. The study shows that the retailer-led Stackelberg game helps retailers maximize profits, while the Nash game enables the entire closed-loop supply chain system to achieve the highest overall profits. This paper innovatively integrates low-carbon technologies into the dynamic game model of the remanufacturing process and reveals how the game behavior of supply chain participants affects the application of low-carbon technologies and the overall profit of the supply chain by comparing the cost-sharing mechanisms under different power structures. The results provide important theoretical support and practical references for closed-loop supply chain management with product differentiation.&lt;/p&gt;

Geometric preconditioner for indirect shooting and application to hybrid vehicle

In this article, we are interested in the hybrid electric vehicle torque split and gear shift problem, which can be formulated as a classical Lagrange optimal control problem with fixed initial condition. The Pontryagin maximum principle gives necessary optimality conditions adjoining to the state a covector called costate. Thus, the optimal state trajectory has to be found among the projections of the lifted trajectories, called Pontryagin extremals, given by the maximum principle. The indirect simple shooting method aims to compute Pontryagin extremals reducing the resolution to the research of the initial costate. Classically, a Newtonlike solver is used to compute zeros of the so-called shooting equations. The main drawback of this method is its sensitivity to the initial guess. Indeed, a good one needs to be given to make the Newton solver converge, which is not an easy task in practice. We propose a preconditioning method of the shooting function based on a geometrical interpretation of the costate, in relation with the reachable set of the extended system and the underlying symplectic structure. We numerically show that this method reduces the number of iterations of our solver. Remarkably, in our experiments, it is better to use the preconditioner than a clever initial guess for the Newton solver.

On the role of vaccination, health education, and hygiene compliance in the elimination and control of Hepatitis A Virus: An optimal control approach

A deterministic mathematical model for Hepatitis A infection is established and subsequently examined to optimize control strategies. The model incorporates three time-dependent controls: vaccination, health education, and hygiene compliance, focusing on mitigating disease transmission in the community. The derivation of the basic reproduction number was conducted using the Next-Generation Matrix (NGM) technique, which was subsequently utilized to analyze the stability of the equilibria of the model. The optimal control problem is established and analyzed using Pontryagin’s Maximum principle. The numerical simulation of the optimal control problem is achieved via Runge–Kutta fourth-order schemes (forward and backward sweeps). The numerical findings demonstrate a significant reduction in Hepatitis A cases by implementing at least one control measure. Besides that, it has been established that coupling vaccination, health education and hygiene compliance results in the lowest number of cases, making it an optimal option for eradicating Hepatitis A in the community. However, applying this strategy could be more costlier. As such, the cost-effective analysis was carried out via an incremental cost-effectiveness ratio approach to ascertain the most cost-effective strategy. The findings confirmed that the vaccination strategy was the most cost-effective approach among the strategies under consideration because it offers the minimum number of cases at the minimum cost. This approach is particularly applicable in situations with constrained resources, a circumstance prevalent in many developing nations.

Inserting a Maximum-Mass Spacecraft into a Target Orbit Using a Limited-Thrust Engine with Releasing the Separable Part of Its Launch Vehicle into the Earth’s Atmosphere

—Space debris is an urgent problem of our time. This paper considers the idea of reducing near-Earth space debris by releasing the spent additional fuel tank (AFT) and the booster’s central block (CB) into the Earth’s atmosphere. The spacecraft transfer from a reference circular orbit of an artificial Earth satellite to a target elliptical orbit is optimized. The transition maneuvers are carried out using a booster with a high limited-thrust engine and the AFT. The second zonal harmonic of the Earth’s gravitational field is taken into account. The optimal control problem is solved based on Pontryagin’s maximum principle. Bulky derivatives are calculated using a specially developed numerical-analytical differentiation technique. The Pontryagin extremals obtained below are the next step in implementing the problem hierarchy methodology.

A Mathematical Model of Tumor‐Immune and Host Cells Interactions with Chemotherapy and Optimal Control

In this article, we propose the interaction of tumor cells with the immune system in the presence of chemotherapy. The existence, uniqueness, non‐negativity, and boundedness of the solutions have been established. The conditions for the existence and stability of equilibrium points have been presented in both drug‐free and treated systems. The local stability of the co‐existing equilibrium point is proved using the Routh–Hurwitz rule, and the global stability is proved using the Lyapunov function. We have used quadratic optimal control to minimize the number of tumor cells and the side effects of chemotherapy on the immune system and healthy cells. We have demonstrated the existence of optimal control and derived the corresponding optimality system using Pontryagin’s maximum principle. The optimal system is solved using the forward‐backward sweep method with fourth‐order Runge–Kutta approximation. Reduction in tumor cell growth has been observed due to the increase in recruitment of immune cells activated by tumor cell antigenicity and the rate of conversion of resting immune cells into active immune cells. Additionally, the impact of administering varying chemotherapy doses on reducing tumor cell growth has been noted. Finally, a comparison between controlled and uncontrolled dynamics has been conducted to comprehend the effect of optimal control.

The impact of rumors on the success of Covid-19 vaccination programs in a Coronavirus-infected environment: optimal control approach

In this paper, we propose a mathematical model that describes the effect of rumors on the success of vaccination programs against Covid-19 in an environment infected by the coronavirus. The aim of this study is to highlight the role of addressing the spread of rumors regarding vaccination risks and booster doses in the success of vaccination programs and in achieving herd immunity. Additionally, we formulate an optimal control problem by proposing several strategies, including awareness and anti-rumor programs, to assist country officials in achieving successful vaccination programs with optimal effort. The existence of optimal controls is investigated, and Pontryagin's maximum principle is used to characterize them. The optimality system is solved using an iterative method. Finally, we conduct numerical simulations to verify the theoretical analysis using Matlab.

Quantifying and eliminating the time delay in stabilization exponential time differencing Runge–Kutta schemes for the Allen–Cahn equation

Although the stabilization technique is favorable in designing unconditionally energy stable or maximum-principle-preserving schemes for gradient flow systems, the induced time delay is intractable in computations. In this paper, we propose a class of delay-free stabilization schemes for the Allen–Cahn gradient flow system. Considering the Fourier pseudo-spectral spatial discretization for the Allen–Cahn equation with either the polynomial or the logarithmic potential, we establish a semi-discrete, mesh-dependent maximum principle by adopting a stabilization technique. To unconditionally preserve the mesh-dependent maximum principle and energy stability, we investigate a family of exponential time differencing Runge–Kutta (ETDRK) integrators up to the second-order. After reformulating the ETDRK schemes as a class of parametric Runge–Kutta integrators, we quantify the lagging effect brought by stabilization, and eliminate delayed convergence using a relaxation technique. The temporal error estimate of the relaxation ETDRK integrators in the maximum norm topology is analyzed under a fixed spatial mesh. Numerical experiments demonstrate the delay-free and structure-preserving properties of the proposed schemes.