Optimal Control Problem Research Articles

Distributed model predictive control for asynchronous multiagent systems with self‐triggered coordinator

AbstractThis paper investigates a distributed model predictive control strategy for asynchronous nonlinear multiagent systems (MASs) with external disturbances via self‐triggered generators and prediction horizon regulators. First, a shrinking constraint related to the error between the actual state and the predicted state is introduced into the optimal control problem to enhance the robustness of the system. Then, the triggering interval and the corresponding prediction horizon are determined by altering the expression of the Lyapunov function, thus achieving a trade‐off between control performance and energy loss. By implementing the proposed algorithm, the coordination objective of the MASs is achieved under asynchronous communication. Finally, the recursive feasibility and closed‐loop stability are proven successively. An illustrative example is conducted to demonstrate the merits of the presented approach.

Maximum principle for partially observed risk-sensitive optimal control problem of McKean–Vlasov FBSDEs involving impulse controls

Maximum principle for partially observed risk-sensitive optimal control problem of McKean–Vlasov FBSDEs involving impulse controls

Optimal Control of Microcephaly Under Vertical Transmission of Zika

The Zika virus, known for its potential to induce neurological conditions such as microcephaly when transmitted vertically from infected mothers to infants, has sparked widespread concerns globally. Motivated by this, we propose an optimal control problem for the prevention of vertical Zika transmission. The novelty of this study lies in its consideration of time-dependent control functions, namely, insecticide spraying and personal protective measures taken to safeguard pregnant women from infected mosquitoes. New results provide a way to minimize the number of infected pregnant women through the implementation of control strategies while simultaneously reducing both the associated costs of control measures and the mosquito population, resulting in a decline in microcephaly cases.

Dynamic Contract Design in the Presence of Double Moral Hazard

We consider a stylized incentive management problem over an infinite time horizon, where the principal hires an agent to provide services to customers. Customers request service in one of two ways: either via an online or a traditional offline channel. The principal does not observe the offline customers’ arrivals, nor does she observe whether the agent exerts (costly) effort that can increase the arrival rate of customers. This creates an opportunity for the agent (i) to divert cash (that is, to under-report the number of offline customers and pocket respective revenues) and also (ii) to shirk (that is, not to exert effort), thus leading to a novel and thus far unexplored double moral hazard problem. To address this problem, we formulate a constrained, continuous-time, stochastic optimal control problem and derive an optimal contract with a simple intuitive structure that includes a payment scheme and a potential termination time of the agent. We enrich the model to allow the principal to either (i) dynamically adjust the prices for the services in both channels or (ii) monitor the agent. Both tools help the principal to alleviate the double moral hazard problem. We derive respective optimal strategies for using those tools that guarantee the highest profits. We show that the worse the agent’s past performance is, the lower the prices should be set and the more the principal should monitor the agent. This paper was accepted by Ilia Tsetlin, behavioral economics and decision analysis. Funding: The work of F. Tian is financially supported by the Hong Kong RGC Funding [RGC General Research Fund 2023/24 17502023], HKU Seed Fund for Basic Research for New Staff (2022), and the HKU Business School Shenzhen Research Institute [SZRI2023-BRF-04]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.01169 .

An approach for regime-switching stochastic control problems with memory and terminal conditions

In this research article, we focus on a stochastic optimal control problem with two types of terminal constraints. These specific conditions provide real-valued and stochastic Lagrange multipliers. Our model evolves according to a Markov regime-switching jump diffusion model with memory. In this context, the memory is represented by a Stochastic Differential Delay Equation. We present two theorems for each constraint within the general formulation of stochastic optimal control theory in a Lagrangian environment. We approach to this task from a theoretical perspective and provide mild technical assumptions, which make our theorems applicable for a broad class of stochastic control problems as well as for a wide range of disciplines such as engineering, biology, operations research, medicine, computer science and economics. In this work, we apply Stochastic Maximum Principle to demonstrate an optimal dividend policy corresponding to a time-delayed wealth process of a company. Moreover, we determine the real-valued Lagrange multiplier of this control problem explicitly.

A GUMBORO-SALMONELLA CO-INFECTION MATHEMATICAL MODEL WITH OPTIMAL CONTROL

Poultry production contributes immensely to the economic growth of a country. For instance in Kenya, 20tonnes of poultry meat worth 3.5 billion kenya shillings and 1.3 billion eggs worth 9.7 billion Kenya Shillingscome from this sector. However, the sector is greatly threatened by poultry diseases among them, Gum-boro(IBD) and Salmonella as inadequate knowledge exists of optimal control strategies for various poultryco-infections. In this research, a Gumboro-Salmonella co-infection mathematical model with optimal controlis developed using a system of ODEs to perform an optimal control analysis. The analysis was done by for-mulating an optimal control problem and using Pontryagin’s maximum principle to solve it. The Numericalsimulation results showed that the best Gumboro-Salmonella co-infection control strategy involved combiningall the interventions

Modified Shooting with Discretization Validation for Non-Standard Optimal Control Problem: Case Study for Four-Stage Royalty Payment Problem

A modified shooting method augmented by discretization validation is presented in this paper to address the challenges inherent in non-standard optimal control problems. Specifically, a specific case study involving four-stage royalty payment functions is focused on, with the aim of effectively optimizing these complex, non-differentiable functions. The shooting method is adapted and enhanced to compute optimal solutions, and its accuracy is rigorously confirmed through discretization techniques. The primary objectives involve maximizing the performance index and determining optimal control strategies for scenarios where the final state variable remains unknown. In this study, the royalty payment is defined as a four-stage piecewise function. The incorporation of piecewise royalty functions introduces non-differentiability at specific time intervals, necessitating innovative approaches for finding optimal solutions. This, in turn, leads to the utilization of the continuous hyperbolic tangent (tanh) function to address the non-differentiability issue. A hybrid shooting method, combining the Newton and Golden Section Search methods, is employed in the C++ programming language to compute the unknown final state value. A new natural boundary condition, based on established theory, is introduced to further facilitate the investigation. Discretization methods such as Euler, Runge-Kutta, Trapezoidal, and Hermite-Simpson approximations are employed for validation. The validation process entails the use of the AMPL programming language with the MINOS solver. Comparative analyses reveal that the modified shooting method yields more accurate optimal results than discretization methods, thus demonstrating the method’s effectiveness in addressing non-standard optimal control problems. The significance of fundamental theories in addressing real-world challenges is underscored by this research, providing valuable insights for future researchers exploring mathematical approaches in similar contexts. The study contributes to the continued relevance of the academic field, particularly in science and mathematics education.

Optimal control of a model for dynamic district heating networks

In the present paper an optimal control problem for a concrete system of differential-algebraic equations (DAEs) is considered. This problem arises in the dynamic optimization of unsteady district heating networks. Based on the Carathéodory theory existence of a unique solution to the specific DAE system is proved using specific properties of the considered district heating network model. Moreover, it is shown that the optimal control problem possesses optimal solutions. For the numerical experiments different networks are considered including also data from a real district heating network.

Transmission Dynamics of Typhoid Fever Outbreak: A Mathematical Modelling and Optimal Control Approach

In this paper, we propose and analyze a deterministic mathematical model for the control of typhoid fever. This model incorporates five effective control strategies, including vaccination, booster vaccination, personal hygiene, treatment, and bacteria sterilization, in order to effectively stop the spread of the disease in the population. The model consists of seven compartments, each representing a different stage in the disease’s progression: vaccinated, susceptible, carrier, infected, recovered, booster vaccinated, and bacterial. Using the next-generation matrix approach, we calculated the reproduction number , which measures the disease’s potential for spreading. Our stability analysis, using the Routh-Hurwitz criteria, revealed that the disease-free equilibrium point is locally asymptotically stable when is less than 1, among other conditions. This means that typhoid can be completely eradicated if the rate of secondary infection is kept below a certain threshold. Further sensitivity analyses were conducted to determine the key parameters that impact . Based on our findings, we formulated an optimal control problem and utilized Pontryagin’s Maximum Principle to determine the most effective strategy for controlling and eliminating the disease. Our results show that a combination of vaccination, booster vaccination, personal hygiene, treatment, and bacteria sterilization is the most optimal approach. With our comprehensive model and effective control strategies, we are one step closer to wiping out typhoid fever for good.

Understanding population dynamics and management strategies for a newly emerging pest Carea sp. in Eucalyptus plantations in Indonesia through a mathematical model

In this current study, we investigate the population dynamics of Carea sp. (Lepidoptera: Nolidae), a newly emerging pest in Indonesian Eucalyptus plantations, utilizing the MELP-SI-B mathematical model based on the first reported outbreak in 2022. The larval infestation of Carea sp. has been observed to stunt plant growth and can even lead to plant mortality. The proposed model incorporates pest management interventions, including chemical control with pesticides and biological control through the introduction of natural predators. Analysis of the deterministic compartment model revealed four locally asymptotically stable equilibrium points, validated using eigenvalue approach and application of the center manifold theorem. Bifurcation diagrams are presented to elucidate the relationships between all equilibrium points. An optimal control problem is explored within this model, considering three scenarios: chemical control without predators, simultaneous chemical control with predators, and chemical control followed by predator management. The results of the optimal control strategy analysis highlight the critical importance of timing the initial chemical control application for effective pest management. Notably, this study reveals that a combined approach involving chemical control (without predators) followed by predator management during the post-chemical control period yields the most favorable outcomes among the examined strategies.

The inverse problem of identifying complex hyperbolic equation source terms in electromagnetic propagation

Abstract This article investigates the inverse problem of determining the source term of the hyperbolic equation for electromagnetic propagation using terminal data. This study is an important method for identifying propagation sources in electromagnetics. Unlike wave equations, the complexity of the underlying equations can make theoretical analysis quite difficult. Firstly, the uniqueness of the inverse problem was proved using the energy method. Then, based on the optimal control framework, the inverse problem was transformed into an optimal control problem, and the existence of the optimal solution and its necessary conditions were established. Secondly, the global uniqueness and stability of the optimal solution have been proven, which is a completely new conclusion. This has laid a solid theoretical foundation for numerical algorithms. Finally, it is proposed to apply the Landweber iteration method and conjugate gradient method to this problem, and some numerical examples are provided to demonstrate the effectiveness and convergence speed of these two algorithms.

Dynamic programming principle for optimal control of uncertain random differential equations and its application to optimal portfolio selection

This study aimed to examine an uncertain stochastic optimal control problem premised on an uncertain stochastic process. The proposed approach is used to solve an optimal portfolio selection problem. This paper’s research is relevant because it outlines the procedure for solving optimal control problems in uncertain random environments. We implement Bellman’s principle of optimality method in dynamic programming to derive the principle of optimality. Then the resulting Hamilton-Jacobi-Bellman equation (the equation of optimality in uncertain stochastic optimal control) is used to solve a proposed portfolio selection problem. The results of this study show that the dynamic programming principle for optimal control of uncertain stochastic differential equations can be applied in optimal portfolio selection. Also, the study results indicate that the optimal fraction of investment is independent of wealth. The main conclusion of this study is that, in Itô-Liu financial markets, the dynamic programming principle for optimal control of uncertain stochastic differential equations can be applied in solving the optimal portfolio selection problem.

Bilinear optimal control for a fractional diffusive equation

We consider a bilinear optimal control for an evolution equation involving the fractional Laplace operator of order 0<s<1. We first give some existence and uniqueness results for the considered evolution equation. Next, we establish some weak maximum principle results allowing us to obtain more regularity of our state equation. Then, we consider an optimal control problem which consists to bring the state of the system at final time to a desired state. We show that this optimal control problem has a solution and we derive the first- and second-order optimality conditions.

Optimization-based trajectory planning for reconfigurable unmanned vehicle units with multi-steering modes during the reconfiguration process

Reconfigurable unmanned vehicle unit (RUVU) exhibits three distinct steering modes: Ackermann steering, Diagonal steering, and In-Situ steering. This arises from its capacity for All-Wheel-Independent-Steering (AWIS) and All-Wheel-Independent-Driving (AWID). Given its unique motion characteristics, there is currently a lack of established methods to assist a RUVU in autonomously planning trajectories. In response to this deficiency, this paper proposes the Multi-Steering Modes Trajectory Planning algorithm (MSMTP) based on the framework of optimal control problem (OCP). The approach innovatively formulates the kinematics and Steering-Mode-Switch theory for vehicles with multi-steering modes. Additionally, a safe traveling corridor (STC) is implemented to guarantee non-conflicting interaction with the surrounding environment. Ultimately, the Bonmin solver is utilized to derive optimal results. Our findings demonstrate the effectiveness of the proposed algorithm in RUVU’s trajectory planning. When comparing the trajectory of RUVU to that of a vehicle only with Ackermann steering, the results also reveal that RUVU has the advantage of reaching its destination in a shorter time, especially in narrow environments.

Research on trajectory planning based on optimal control modeling

With the advancement of autonomous driving technology, trajectory planning has emerged as one of the core technologies to achieve safe and efficient vehicle driving. An algorithm for single-vehicle trajectory planning based on the Optimal Control Problem (OCP) is proposed in this paper, which can not only plan a safe and efficient path for autonomous vehicles, but also realize the obstacle avoidance function. Firstly, the OCP is modeled for the cycle trajectory planning problem. Secondly, this paper transforms it into a Nonlinear Programming (NLP) with all discrete methods, and uses the interior point method to solve it. Then, the Y-type intersection is selected as the verification scene, and through a series of simulation experiments, the efficiency and safety of the algorithm in different driving environments are verified. The model and algorithm proposed in this paper are not only applicable to Y-type intersections, but also can be extended to other complex traffic roads, providing a new idea and method for trajectory planning in intelligent transportation systems.

A novel optimization framework for natural gas transportation pipeline networks based on deep reinforcement learning

Natural gas is an emerging and reliable energy source in transition to a low-carbon economy. The natural gas transportation pipeline network systems are crucial when transporting natural gas from the production endpoints to processing or consuming endpoints. Optimizing the operational efficiency of compressor stations within pipeline networks is an effective way to reduce energy consumption and carbon emissions during transportation. This paper proposes an optimization framework for natural gas transportation pipeline networks based on deep reinforcement learning (DRL). The mathematical simulation model is derived from mass balance, hydrodynamics principles of gas flow, and compressor characteristics. The optimization control problem in steady state is formulated into a one-step Markov decision process (MDP) and solved by DRL. The decision variables are selected as the discharge ratio of each compressor. By the comprehensive comparison with dynamic programming (DP) and genetic algorithm (GA) in three typical element topologies (a linear topology with gun-barrel structure, a linear topology with branch structure, and a tree topology), the proposed method can obtain 4.60% lower power consumption than GA, and the time consumption is reduced by 97.5% compared with DP. The proposed framework could be further utilized for future large-scale network optimization practices.

A Research on Single-Vehicle Trajectory Planning Algorithm in MultiObstacle Environments

This work aims to investigate the optimal trajectory planning for single-vehicle scenarios in obstacle-filled environments. The first section of the paper provides background information and an overview of autonomous driving trajectory planning, emphasizing its crucial role in ensuring safety and efficiency. A review of previous literature shows a range of approaches and constraints for trajectory optimization, especially in complicated and dynamic situations. Then, the single-vehicle trajectory planning optimum control problem (OCP) is stated and, in order to solve it practically, it is transformed into a nonlinear programming (NLP) issue. A classification of numerical solution methods is presented, including direct and indirect approaches. Extensive trials conducted with different initial and terminal circumstances and with single and multiple obstacles have demonstrated that the model can reliably produce the desired outcomes. As a result, the study demonstrates the robustness and efficiency of the proposed algorithm.

Cauchy discretization method for solve fractional linear optimal control problems

This paper presents a numerical scheme designed for solving fractional linear optimal control problems (FLOCPs) governed by Caputo fractional derivatives (CFDs) of order 0&lt;α≤1. The method transforms the original fractional control problem into an equivalent linear programming problem (LPP), enabling a more straightforward solution process. To evaluate the performance and accuracy of this approach, several numerical experiments were conducted using MATLAB. These experiments highlight the method’s computational efficiency and its simplicity in terms of implementation. The proposed approach offers accurate results, particularly in scenarios where traditional methods may fall short due to the complexities introduced by fractional derivatives. A detailed numerical example is presented to further illustrate the method’s effectiveness in addressing FLOCPs. The outcomes suggest that this approach is not only practical but also provides valuable insights for researchers dealing with fractional calculus in optimal control theory. This contribution is expected to be beneficial for applications in engineering and the physical sciences, where fractional-order systems play a significant role.

Stochastic optimal control of pre-exposure prophylaxis for HIV infection for a jump model.

We analyze a stochastic optimal control problem for the PReP vaccine in a model for the spread of HIV. To do so, we use a stochastic model for HIV/AIDS with PReP, where we include jumps in the model. This generalizes previous works in the field. First, we prove that there exists a positive, unique, global solution to the system of stochastic differential equations which makes up the model. Further, we introduce a stochastic control problem for dynamically choosing an optimal percentage of the population to receive PReP. By using the stochastic maximum principle, we derive an explicit expression for the stochastic optimal control. Furthermore, via a generalized Lagrange multiplier method in combination with the stochastic maximum principle, we study two types of budget constraints. We illustrate the results by numerical examples, both in the fixed control case and in the stochastic control case.

A dynamical optimal control theory and cost-effectiveness analyses of the HBV and HIV/AIDS co-infection model

Studies have shown that the co-infection of Human Immunodeficiency Virus (HIV) and Hepatitis B Virus (HBV) poses a major threat to the public health due to their combined negative impacts on health and increased risk of complications. Even though, some scholars formulated and analyzed the HBV and HIV co-infection model they did not consider the compartment that contains protected individuals against both HBV and HIV infections. They incorporated the optimal control theory and cost-effectiveness analysis simultaneously. With this in mind, we are motivated to formulate and analyze the HBV and HIV co-infection model, considering the protected group and incorporating optimal control theory and cost-effectiveness. In this study, we have theoretically computed all of the models disease-free equilibrium points, all the models effective reproduction numbers and unique endemic equilibrium points. The two sub-models disease-free equilibrium points are locally as well as globally asymptotically stable whenever their associated effective reproduction numbers are less than one. We reformulated the optimal control problem by incorporating five time-dependent control measures and conducted its theoretical analysis by utilizing the Pontryagin's maximum principle. Using the fourth order Runge–Kutta numerical method and MATLAB ODE45, we performed the numerical simulations with various combinations of control efforts to verify the theoretical results and investigate the impacts of the suggested protection and treatment control strategies for both the HBV and HIV diseases. Also, we carried out a cost-effectiveness analysis of the proposed control strategies. Eventually, we compared our model results with other researcher similar model results whenever cost-effectiveness analysis is not carried out the findings of this particular study suggest that implementing each of the proposed control strategies simultaneously has a high potential to reduce and control the spread of HBV and HIV co-infections in the community. According to the cost-effectiveness analysis, implementing the HBV treatment and the HIV and HBV co-infection treatment measures has a high potential effect on reducing and controlling the HBV and HIV co-infection transmission problem in the community.