Pontryagin Maximum Principle Research Articles

Low Computational Cost for Multiple Waypoints Trajectory Planning: A Time‐Optimal‐Based Approach

In the field of mobile robots, achieving minimum time in executing trajectories is crucial for applications like delivery, inspection, and search and rescue. In this article, a novel time‐optimal planner based on optimization methods is introduced. Despite the high computational cost associated with such methods, the solution calculates time‐optimal multi‐waypoint trajectories, achieving results in the order of milliseconds. The proposed method formulates a time‐optimal trajectory using the Pontryagin's maximum principle as a policy. By utilizing a point mass model, the planner generates trajectories that are adaptable to different robot models. The approach incorporates a definition of a search space to guarantee convergence while considering the system limits. Simulation and real‐world experiments are performed to validate the feasibility of our method with different configurations. Simulation results compared to a benchmark method demonstrate our approach's superior performance in terms of computational time, achieving near‐optimal solutions. In addition, in the real‐world experiments, the integration of the method into practical applications is validated.

Machine learning-based optimal temperature management model for safety and quality control of perishable food supply chain

The management of a food supply chain is difficult and complex because of the product's short shelf-life, time-sensitivity, and perishable nature which must be carefully considered to minimize food waste. Temperature-controlled perishable food supply chain provides the highly crucial facilities necessary to maintain the quality and safety of the product. The storage temperature is the most vital factor in maintaining both the quality and shelf-life of a perishable food. Adequate storage temperature control ensures that perishable foods are transported to the end-users in good quality and safe to consume. This paper presents perishable food storage temperature control through mathematical optimal control model where the storage temperature is regarded as the control variable and the deterioration of the perishable food’s quality follows the first-order reaction. The optimal storage temperature for a single perishable food is determined by applying the Pontryagin's maximum principle to solve the optimal control model problem. For multi-temperature commodities supply chain, an unsupervised machine learning (ML) method, called k-means clustering technique is used to determine the temperature clusters for a range of perishables. Based on descriptive analysis, it is observed that the k-means clustering technique is effective in identifying the best suitable storage temperature clusters for quality control of multi-commodity supply chain.

Dynamic pricing and inventory model for perishable products with reference price and reference quality

Reference price (quality) is a benchmark point used by consumers to make price (quality) judgements. Combining reference price and reference quality with dynamic pricing and inventory management, this paper applies differential equations theory to construct a optimal control model for perishable products when the quality of products declines exponentially. The demand of products depends on price, quality, reference price and reference quality, among which reference price and reference quality are influenced by consumers’ past memory. The aim is to maximize the retailer’s profit during the period. The conclusions are as follows. Firstly, a optimal dynamic pricing and inventory model for perishable products with reference price and reference quality is constructed, and the model is extended to dual decision variables, stochastic demand, time-dependent effects, competitive and infinite planing horizon setups. Secondly, through the Pontryagin maximum principle, the analytical expression of the optimal dynamic price is derived. Thirdly, a linear search method to solve the optimal static price is proposed. Finally, the sensitivity of the main parameters is analyzed and the corresponding management enlightenments are given. By comparison of dynamic and static pricing, we find that dynamic pricing can achieve more profits and take a shorter selling period. In addition, for the sales problem of perishable products affected by both reference price and reference quality, retailers should adopt skimming pricing strategy (the optimal price decreases with time). Furthermore, to obtain more profit, retails should strive to increase the sensitivity coefficients of two types of reference, and the reference quality memory coefficient, while to decrease the reference price memory coefficient.

Time Parametrized Motion Planning

Time can be treated as a free parameter to isotropically stretch the tangent space. A trajectory, which matches the boundary conditions on its configuration, is adjusted so that velocity conditions are met. The modified trajectory is found by substitution, without the computational cost of re-integrating the velocity function. This concept is extended to stretch the tangent space anisotropically. This method of time parametrization especially applies to Geometric Control, where the Pontryagin Maximum Principle minimizes some cost function and matches the boundary configuration constraints but not the velocity constraints. The optimal trajectory is modified by the parametrization so that the cost function is minimized if the stretching is stopped at any time. This is a theoretical contribution, using a wheeled robot example to illustrate the modification of an optimal velocity under multiple parametrizations.

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.

Investigation of an optimal control strategy for a cholera disease transmission model with programs

Cholera is a disease of poverty affecting people with inadequate access to safe water and basic sanitation. Conflict, unplanned urbanization and climate change all increase the risk of cholera. In this article, an optimal control deterministic mathematical model of cholera disease with cost-effectiveness analysis is developed and analyzed considering both direct and indirect contact transmission pathways. The model qualitative behaviors, such as the invariant region, the existence of a positive invariant solution, the two equilibrium points (disease-free and endemic equilibrium), and their stabilities (local as well as global stability) of the model are studied. Moreover, the basic reproduction number of the model is obtained. We also performed sensitivity analysis of the basic parameters of the model. Then an optimal control problem is designed with a control functional having five controls: vaccination, treatment, environment sanitation and personal hygiene, and water quality improvement program. We examined the existence and uniqueness of the optimal controls of the system. Through the implementation of Pontryagin's maximum principle, the characterization of the optimal controls optimality system is established. The numerical simulation results the integrated control strategies demonstrated that strategy 2, 7, and 12 are effective programs to combat cholera disease from the community. Based on the local circumstances, available funds, and resources, it is recommended to the government stakeholders and policymakers to execute any one of the three integrated intervention programs.

Ecological‐economic modeling of nutrient enrichment in multispecies fisheries

AbstractIn fisheries economics, multispecies or ecosystem models typically focus on interactions among fish species, often overlooking the flow of nutrients. However, biologists widely acknowledge that nutrients–Âİsuch as nitrogen, phosphorus, potassium, and calcium–Âİare vital for organisms and play crucial roles in ecosystem dynamics. This paper delves into a bioeconomic predator‐prey model incorporating nutrient enrichment through Holling Type II and Michaelis–Menten functional forms. Initially, we examine the system's dynamics before formulating an optimal time‐dependent harvest policy using the Pontryagin maximum principle. Our findings reveal that heightened nutrient enrichment can lead to the breakdown of a positive stable equilibrium, resulting in the extinction of both prey and predator. The stability of the ecosystem may also be influenced by the interplay of nutrient enrichment and predator harvest.

Optimal quantum controls robust against detuning error

Abstract Precise control of quantum systems is one of the most important milestones for achieving practical quantum technologies, such as computation, sensing, and communication. Several factors deteriorate the control precision and thus their suppression is strongly demanded. One of the dominant factors is systematic errors, which are caused by discord between an expected parameter in control and its actual value. Error-robust control sequences, known as composite pulses, have been invented in the field of nuclear magnetic resonance (NMR). These sequences mainly focus on the suppression of errors in one-qubit control. The one-qubit control, which is the most fundamental in a wide range of quantum technologies, often suffers from detuning error. As there are many possible control sequences robust against the detuning error, it will practically be important to find “optimal” robust controls with respect to several cost functions such as time required for operation, and pulse-area during the operation, which corresponds to the energy necessary for control. In this paper, we utilize the Pontryagin’s maximum principle (PMP), a tool for solving optimization problems under inequality constraints, to solve the time and pulse-area optimization problems. We analytically obtain pulse-area optimal controls robust against the detuning error. Moreover, we found that short-CORPSE, which is the shortest known composite pulse so far, is a probable candidate of the time optimal solution according to the PMP. We evaluate the performance of the pulse-area optimal robust control and the short-CORPSE, comparing with that of the direct operation.

Optimal resource allocation in micro-organisms under periodic nutrient fluctuations

Although microorganisms often live in dynamic environments, most studies, both experimental and theoretical, are carried out under static conditions. In this work, we investigate the issue of optimal resource allocation in bacteria growing in periodic environments. We consider a dynamic model describing the microbial metabolism under varying conditions, involving a control variable quantifying the protein precursors allocation. Our objective is to determine the optimal strategies maximizing the long-term growth of cells under a piecewise-constant periodic environment. Firstly, we perform a theoretical analysis of the resulting optimal control problem (OCP), based on the application the Pontryagin’s Maximum Principle (PMP). We determine that the structure of the optimal control must be bang–bang, with possibly some singular arcs corresponding to optimal equilibria of the system. If the control presents singular arcs, then these can only be reached and left through chattering arcs. We also use a direct optimization method, implemented in the BOCOP software, to solve the studied OCP. Our study reveals that the optimal solution over a large time horizon is related to the one over a single period of the varying environment with periodic constraints. Moreover, we observe that the maximal average growth rate attainable under periodic conditions can be higher than the one under a constant environment. We further extend our analysis to conduct a qualitative comparison between the predictions from our model and some recent biological experiments on E. coli. This analysis particularly highlights the mechanisms of action of the ppGpp signaling molecule, thus providing relevant explanations of the experimental observations. In conclusion, our study corroborates previous research indicating that this molecule plays a crucial role in the regulation of resource allocation of protein precursors in E. coli.

A risk based approach to the principal–agent problem

PurposeWe aim to generalize the continuous-time principal–agent problem to incorporate time-inconsistent utility functions, such as those of mean-variance type, which are prevalent in risk management and finance.Design/methodology/approachWe use recent advancements of the Pontryagin maximum principle for forward-backward stochastic differential equations (FBSDEs) to develop a method for characterizing optimal contracts in such models. This approach addresses the challenges posed by the non-applicability of the classical Hamilton–Jacobi–Bellman equation due to time inconsistency.FindingsWe provide a framework for deriving optimal contracts in the principal–agent problem under hidden action, specifically tailored for time-inconsistent utilities. This is illustrated through a fully solved example in the linear-quadratic setting, demonstrating the practical applicability of the method.Originality/valueThe work contributes to the existing literature by presenting a novel mathematical approach to a class of continuous time principal–agent problems, particularly under hidden action with time-inconsistent utilities, a scenario not previously addressed. The results offer potential insights for both theoretical development and practical applications in finance and economics.

Concurrent learning for adaptive pontryagin's maximum principle of nonlinear systems with inequality constraints

AbstractIn this article, a finite‐horizon adaptive Pontryagin's maximum principle is presented for nonlinear systems with state inequality constraints. Concurrent learning (CL) technique is adopted to identify the unknown parameters of the dynamic systems. Based on the identification model, a novel adaptive iterative algorithm under the Pontryagin's framework is introduced to learn the finite‐horizon optimal control solution. Convergence analysis of the algorithm is provided by showing that the cost function sequence is monotonically decreasing. Furthermore, we extend the adaptive iterative algorithm to time‐varying nonlinear systems. The new algorithm overcomes the technical obstacles of the existing adaptive/approximate dynamic programming (ADP) approaches to deal with the time‐varying characteristic of Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE), especially when state constraints exist. Simulation examples are carried out to validate the effectiveness of the theoretical results.

Optimal quantized feedback control for linear quandratic Gaussian systems with input delay

AbstractThis paper is concerned with the optimal quantized feedback linear quadratic Gaussian (LQG) control problem for a discrete‐time stochastic system with input delay as well as the measurements to be quantized before transmitted to the controller. In this scenario, the system is presented with several choices of quantizers, along with the cost of using each quantizer. The objective is to jointly select the quantizers and synthesize the controller to maintain an optimal balance between control performance and quantization cost. It is shown that this problem can be decoupled into two optimization problems when the innovation signal is quantized instead of state: one for optimal controller synthesis and the other for optimal quantizer selection. More specifically, a necessary and sufficient condition is derived for the optimal control problem based on Pontryagin's maximum principle. On the other hand, the optimal quantizer selection policy is established by dealing with a certain Markov decision process (MDP).

Escape from compact sets of normal curves in Carnot groups

In the setting of subFinsler Carnot groups, we consider curves that satisfy the normal equation coming from the Pontryagin Maximum Principle. We show that, unless it is constant, each such a curve leaves every compact set, quantitatively. Namely, the distance between the points at time 0 and time t grows at least of the order t^(1/s), where s denotes the step of the Carnot group. In particular, in subFinsler Carnot groups there are no periodic normal geodesics.

Nonlinear dynamics in a fear‐driven predator–prey system: Bistability, bifurcations, hydra effect, and optimal harvesting

The impact of predator‐driven fear on ecosystems is significant and can encompass both trophic (direct) and nontrophic (indirect) effects. Previous studies have shown that nontrophic fear effects have an important role in predator–prey dynamics. This study investigates the nontrophic fear effect on prey caused by generalist predators and explores optimal harvesting. We assume that the reproduction rate of prey is reduced due to the fear effect, and generalist predator follows Holling type II foraging strategy for predation. Additionally, we assume that predators are commercially valuable and harvested proportionately to their density. We demonstrate the existence of equilibrium points, their local and global stability, and bifurcation analysis. We observe that the model system undergoes a sequence of codimension one and codimension two bifurcations. Our results show that in the absence of predator harvesting, increasing levels of fear destabilize the predator–prey system, and controlled harvesting is beneficial for the coexistence of both populations. Also, the harvesting of predators may produce hydra and multiple hydra effects. We identify different types of bistability phenomena, which emphasize the importance of initial population size. Further, an optimal harvesting policy is also explored by formulating an optimal control problem (OCP). The harvesting cost‐functional is designed by incorporating the bionomic equilibrium state. We use Pontryagin's maximum principle and solve the OCP numerically. It is observed that implementing optimal harvesting not only contributes to the ecological benefits by maintaining a sustainable balance of predator–prey evolution but also plays a significant role in maximizing the economic benefits.

The Hybrid Maximum Principle for Optimal Control Problems with Spatially Heterogeneous Dynamics is a Consequence of a Pontryagin Maximum Principle for \(\boldsymbol{\textrm{L}}^{\boldsymbol{1}}_{\boldsymbol{\square }}\)-Local Solutions

The Hybrid Maximum Principle for Optimal Control Problems with Spatially Heterogeneous Dynamics is a Consequence of a Pontryagin Maximum Principle for \(\boldsymbol{\textrm{L}}^{\boldsymbol{1}}_{\boldsymbol{\square }}\)-Local Solutions

Discrete‐time Pontryagin maximum principle under rate constraints: Necessary conditions for optimality

Discrete‐time Pontryagin maximum principle under rate constraints: Necessary conditions for optimality

Bang-bang optimal control in coherent spin dynamics of radical pairs in quantum biology

Optimal control of the external electromagnetic field input for the maximization of the quantum triplet-singlet yield of the radical pairs in biochemical reactions modeled by Schrödinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms are analyzed. Fréchet differentiability and Pontryagin Maximum Principle in Hilbert space is proved and the bang-bang structure of the optimal control is established. A closed optimality system of nonlinear differential equations for the identification of the bang-bang optimal control is revealed. Numerical methods for the identification of the bang-bang optimal control based on the Pontryagin maximum principle are developed. Numerical simulations are pursued, and the convergence and stability of the numerical methods are demonstrated. The results contribute towards understanding the structure-function relationship of the putative magnetoreceptor to manipulate and enhance quantum coherences at room temperature and leveraging biofidelic function to inspire novel quantum devices.

A recursive framework of vehicle trajectory planning at mixed‐traffic signalized intersections

AbstractThis study aims to introduce a new strategy for anticipating the behaviour of human‐driven vehicles (HDVs) and designing trajectories for connected and automated vehicles (CAVs) at signalized intersections under mixed traffic scenarios. To tackle the challenge of unreliable HDV trajectory predictions stemming from driving unpredictability, a recursive framework is developed. This framework integrates real‐time tracking data from both traffic detectors and CAVs, continuously updating HDV predictions. The proposed approach employs the updated predictions to formulate optimal control problems recursively to optimize or adjust CAV trajectories, enhancing travel and energy efficiency. Besides, the recomputing of CAV trajectories will only be conducted when the variation in predictions rises to a certain threshold, balancing efficiency and computing consumption, inspired and modified based on MPC methods. The application of the Pontryagin maximum principle aids in finding solutions efficiently by transforming necessary conditions into a system of equations and consolidating elementary unconstrained and constrained arcs. Numerical simulations were carried out to evaluate the performance of the proposed recursive framework, revealing its superiority over the one‐time approach, particularly in isolated intersections with high traffic demands. Additionally, the recursive framework exhibited more robust and effective enhancements throughout the road network.

Discrete-time optimal control problems with time delay argument: New discrete-time Euler–Lagrange equations with delay

Discrete-time optimal control problems are a crucial type of control problems that deal with a dynamic system evolving in discrete time-steps. This paper introduces a new technique for solving linear discrete-time optimal control problems with state delays, applicable to both finite and infinite time horizons. Our method employs a Riccati matrix equation, optimizing control strategies and ensuring system stability through bounded control inputs. We adopt a Bolza problem for the performance index, which guides the classification of control issues. The technique simplifies problems into manageable Riccati matrix equations using the Euler–Lagrange equations and Pontryagin maximum principle, ensuring stability and necessary condition compliance. The paper validates the approach with numerical examples from quantum mechanics and classical physics, demonstrating its practicality and potential for broader application.

A Cortical-Inspired Contour Completion Model Based on Contour Orientation and Thickness.

An extended four-dimensional version of the traditional Petitot-Citti-Sarti model on contour completion in the visual cortex is examined. The neural configuration space is considered as the group of similarity transformations, denoted as M=SIM(2). The left-invariant subbundle of the tangent bundle models possible directions for establishing neural communication. The sub-Riemannian distance is proportional to the energy expended in interneuron activation between two excited border neurons. According to the model, the damaged image contours are restored via sub-Riemannian geodesics in the space M of positions, orientations and thicknesses (scales). We study the geodesic problem in M using geometric control theory techniques. We prove the existence of a minimal geodesic between arbitrary specified boundary conditions. We apply the Pontryagin maximum principle and derive the geodesic equations. In the special cases, we find explicit solutions. In the general case, we provide a qualitative analysis. Finally, we support our model with a simulation of the association field.