Application Of Pontryagin Research Articles

Mathematical modeling and optimal control of SARS-CoV-2 and tuberculosis co-infection: a case study of Indonesia.

A new mathematical model incorporating epidemiological features of the co-dynamics of tuberculosis (TB) and SARS-CoV-2 is analyzed. Local asymptotic stability of the disease-free and endemic equilibria are shown for the sub-models when the respective reproduction numbers are below unity. Bifurcation analysis is carried out for the TB only sub-model, where it was shown that the sub-model undergoes forward bifurcation. The model is fitted to the cumulative confirmed daily SARS-CoV-2 cases for Indonesia from February 11, 2021 to August 26, 2021. The fitting was carried out using the fmincon optimization toolbox in MATLAB. Relevant parameters in the model are estimated from the fitting. The necessary conditions for the existence of optimal control and the optimality system for the co-infection model is established through the application of Pontryagin’s Principle. Different control strategies: face-mask usage and SARS-CoV-2 vaccination, TB prevention as well as treatment controls for both diseases are considered. Simulations results show that: (1) the strategy against incident SARS-CoV-2 infection averts about 27,878,840 new TB cases; (2) also, TB prevention and treatment controls could avert 5,397,795 new SARS-CoV-2 cases. (3) In addition, either SARS-CoV-2 or TB only control strategy greatly mitigates a significant number of new co-infection cases.

Application of Pontryagin's maximum principle to quantum metrology in dissipative systems

Optimal control theory, also known as Pontryagin's Maximum Principle, is applied to the quantum parameter estimation in the presence of decoherence. An efficient procedure is devised to compute the gradient of quantum Fisher information with respect to the control parameters and is used to construct the optimal control protocol. The proposed procedure keeps the control problem in the time-invariant form so that both first-order and second-order optimality conditions derived from Pontryagin's Maximum Principle apply; the second-order condition turns out to be crucial when the optimal control contains singular arcs. Concretely we look for the optimal control that maximizes quantum Fisher information for "twist and turn" problem. We find that the optimal control is singular without dissipation but can become unbounded once the quantum decoherence is introduced. An amplitude constraint is needed to guarantee a bounded solution. With quantum decoherence, the maximum quantum Fisher information happens at a finite time due to the decoherence, and the asymptotic value depends on the specific decoherence channel and the control of consideration.

Modeling student engagement using optimal control theory

<p style='text-indent:20px;'>Student engagement in learning a prescribed body of knowledge can be modeled using optimal control theory, with a scalar state variable representing mastery, or self-perceived mastery, of the material and control representing the instantaneous cognitive effort devoted to the learning task. The relevant costs include emotional and external penalties for incomplete mastery, reduced availability of cognitive resources for other activities, and psychological stresses related to engagement with the learning task. Application of Pontryagin's maximum principle to some simple models of engagement yields solutions of the synthesis problem mimicking familiar behaviors including avoidance, procrastination, and increasing commitment in response to increasing mastery.</p>

Implementation and acceleration of optimal control for systems biology.

Optimal control theory provides insight into complex resource allocation decisions. The forward–backward sweep method (FBSM) is an iterative technique commonly implemented to solve two-point boundary value problems arising from the application of Pontryagin’s maximum principle (PMP) in optimal control. The FBSM is popular in systems biology as it scales well with system size and is straightforward to implement. In this review, we discuss the PMP approach to optimal control and the implementation of the FBSM. By conceptualizing the FBSM as a fixed point iteration process, we leverage and adapt existing acceleration techniques to improve its rate of convergence. We show that convergence improvement is attainable without prohibitively costly tuning of the acceleration techniques. Furthermore, we demonstrate that these methods can induce convergence where the underlying FBSM fails to converge. All code used in this work to implement the FBSM and acceleration techniques is available on GitHub at https://github.com/Jesse-Sharp/Sharp2021.

Dynamic compensation and contingent sourcing strategies for supply disruption

Alternative measures to deal with supply disruptions exist. We consider a make-to-order (MTO) supply chain with one manufacturer who sources from a single supplier. When a supply disruption occurs, the manufacturer can choose to satisfy some demand by either maintaining production through safety stocks or through a secondary contingent source, and turn some unmet demand into backorders on the basis of compensation. An optimal control model under consideration of the customers’ dynamic reactions to the joint implementation of these strategies is formulated with the objective of minimising the cost of disruption. Through the application of Pontryagin's Maximum Principle, optimal mitigation strategies are established in closed form. They provide analytical guidance on how to dynamically and jointly adapt the quantity of contingent sourcing, the price of compensation, and the speed of safety inventory consumption. The results indicate how cost and time-related factors impact these strategies. We also demonstrate that pure strategies are only effective in tackling short supply shortages. For long disruptions, it is superior to adopt combined strategies that simultaneously incorporate two countermeasures in certain periods.

Optimal Control of Shigellosis with Cost-Effective Strategies.

In this paper, we apply optimal control theory to the model for shigellosis. It is assumed that education campaign, sanitation, and treatment are the main controls for this disease. The aim is to minimize the number of infections resulting from contact with careers, infectious population, and contaminated environments while keeping the cost of associated controls minimum. We achieve this aim through the application of Pontryagin's Maximum Principle. Numerical simulations are carried out by using both forward and backward in time fourth-order Runge-Kutta schemes. We simulate the model under different strategies to investigate which option could yield the best results. The findings show that the strategy combining all three control efforts (treatment, sanitation, and education campaign) proves to be more beneficial in containing shigellosis than the rest. On the other hand, cost-effectiveness analysis is performed via incremental cost-effectiveness ratio (ICER). The findings from the ICER show that a strategy incorporating all three controls (treatment, sanitation, and education campaign) is the most cost-effective of all strategies considered in the study.

Symmetry Reduction in Optimal Control of Multiagent Systems on Lie Groups

We study the reduction of degrees of freedom for the equations that determine necessary optimality conditions for extrema in an optimal control problem for a multiagent system by exploiting the physical symmetries of agents, where the kinematics of each agent is given by a left-invariant control system. Reduced optimality conditions are obtained using techniques from variational calculus and Lagrangian mechanics. A Hamiltonian formalism is also studied, where the problem is explored through an application of Pontryagin's maximum principle for left-invariant systems, and the optimality conditions are obtained as integral curves of a reduced Hamiltonian vector field. We apply the results to an energy-minimum control problem for multiple unicycles.

Control Policy Mix in Measles Transmission Dynamics Using Vaccination, Therapy, and Treatment

This paper considers a deterministic model for the dynamics of measles transmission in a population divided into six classes with respect to the disease states: susceptible, vaccinated, exposed, infected, treated, and recovered. First, we investigate the dynamical properties of the SVEITR model such as its equilibrium points, their stability, and parameter sensitivity by applying constant controls. Criteria for determining the stability of disease-free and endemic equilibrium points are provided in terms of basic reproduction number. The model is then extended by incorporating vaccination, therapy, and treatment rates as time-dependent control variables representing the level of coverages. Application of Pontryagin’s maximum principle provides the necessary conditions that must be satisfied for the existence of optimal controls aiming at minimization of the number of exposed and infected individuals simultaneously with the control effort. Numerical simulations that were carried out using the backward sweep method and Runge–Kutta scheme suggest that optimal controls under moderate and high scenarios can effectively reduce the cases of measles. In particular, the moderate scenario that utilizes the existing coverage level of 86% for MCV1 and 69% for MCV2 can degrade the cost functional by 47% of the low scenario. Meanwhile, high scenario that takes the 2020 target of 96% as coverage only makes a slight difference in reducing the number of exposed and infected individuals.

Application of Pontryagin's minimum principle to Grover's quantum search problem

Grover's algorithm is one of the most famous algorithms which explicitly demonstrates how the quantum nature can be utilized to accelerate the searching process. In this work, Grover's quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagin's Minimum Principle we find that the time-optimal solution has the bang-singular-bang structure. This structure can be derived naturally, without integrating the differential equations, using the geometric control technique where Hamiltonians in the Schr\"odinger's equation are represented as vector fields. In view of optimal control, Grover's algorithm uses the bang-bang protocol to approximate the optimal protocol with a minimized number of bang-to-bang switchings to reduce the query complexity. Our work provides a concrete example how Pontryagin's Minimum Principle is connected to quantum computation, and offers insight into how a quantum algorithm can be designed.

Optimal control of epidemic size and duration with limited resources

The total number of infections (epidemic size) and the time needed for the infection to go extinct (epidemic duration) represent two of the main indicators for the severity of infectious disease epidemics in human and livestock. However, few attempts have been made to address the problem of minimizing at the same time the epidemic size and duration from a theoretical point of view by using optimal control theory. Here, we investigate the multi–objective optimal control problem aiming to minimize, through either vaccination or isolation, a suitable combination of epidemic size and duration when both maximum control effort and total amount of resources available during the entire epidemic period are limited. Application of Pontryagin’s Maximum Principle to a Susceptible–Infected–Removed epidemic model, shows that, when the resources are not sufficient to maintain the maximum control effort for the entire duration of the epidemic, the optimal vaccination control admits only bang–bang solutions with one or two switches, while the optimal isolation control admits only bang–bang solutions with one switch. We also find that, especially when the maximum control effort is low, there may exist a trade–off between the minimization of the two objectives. Consideration of this conflict among objectives can be crucial in successfully tackling real–world problems, where different stakeholders with potentially different objectives are involved. Finally, the particular case of the minimum time optimal control problem with limited resources is discussed.

Minimum-Time Trajectories for Steered Agent With Constraints on Speed, Lateral Acceleration, and Turning Rate

We present time-optimal trajectories for a steered agent with constraints on speed, lateral acceleration, and turning rate for the problem of reaching a point on the plane in minimum time with free terminal heading angle. Both open-loop and state-feedback forms of optimal controls are derived through application of Pontryagin's minimum principle. We apply our results for the single agent to solve a multi-agent coverage problem in which each agent has constraints on speed, lateral acceleration, and turning rate.

An Application of Pontryagin’s Principle to Brownian Particle Engineered Equilibration

We present a stylized model of controlled equilibration of a small system in a fluctuating environment. We derive the optimal control equations steering in finite-time the system between two equilibrium states. The corresponding thermodynamic transition is optimal in the sense that it occurs at minimum entropy if the set of admissible controls is restricted by certain bounds on the time derivatives of the protocols. We apply our equations to the engineered equilibration of an optical trap considered in a recent proof of principle experiment. We also analyze an elementary model of nucleation previously considered by Landauer to discuss the thermodynamic cost of one bit of information erasure. We expect our model to be a useful benchmark for experiment design as it exhibits the same integrability properties of well-known models of optimal mass transport by a compressible velocity field.

A cosmological inflationary model using optimal control

Cosmological inflationary models based on a self-interacting scalar field are considered. A slow-roll model is formulated as an optimal control problem. Application of Pontryagin’s maximum principle leads to an exact solution.

Real-time application of Pontryagin’s Minimum Principle to fuel cell hybrid buses based on driving characteristics of buses

The Pontryagin’s Minimum Principle (PMP)-based energy management strategy is regarded as one of the most promising strategies for hybrid vehicles, given that it instantaneously provides optimal power distribution solutions between power sources. The real-time application of the PMP, however, is still difficult due to the heavy computational burden and the uncertainty of the future vehicle driving cycle. The driving characteristics of city buses, including the bus dwell time at bus stops and comparatively specified driving routes, are very helpful when realizing the PMP to hybrid powertrains. An energy management approach of fuel cell hybrid buses for real-time applications is proposed in this research based on the driving characteristics of buses, in which a reference driving cycle (RDC) is defined for a bus driving route and the bus dwell time is sufficiently used to calculate the PMP-based power distribution solutions. In order to reflect the deviation of the real bus driving route from the RDC, the control parameter of the PMP is updated at every bus stop before calculating the solutions. Simulation results show that the power distribution result of the proposed energy management approach reaches that of the offline PMP application and the discrepancy is within 2.65% for the driving cycles studied.

High-accuracy optimal finite-thrust trajectories for Moon escape

The optimization problem of fuel-optimal trajectories from a low circular Moon orbit to a target hyperbolic excess velocity vector using finite-thrust propulsion is solved. The ability to obtain the most accurate satisfaction of necessary optimality conditions in a high-accuracy dynamic model is the main motivation of the current study. The solutions allow attaining anytime-return Earth-interface conditions from a low lunar orbit. Gravitational effects of the Sun, Earth, and Moon are included throughout the entire trajectory. Severe constraints on the fuel budget combined with high-accuracy demands on the endpoint conditions necessitate a high-fidelity solution to the trajectory optimization problem and JPL DE405 ephemeris model is used to determine the perturbing bodies' positions. The optimization problem is solved using an indirect method. The optimality of the solution is verified by an application of Pontryagin's maximum principle. More accurate and fuel-efficient trajectories are found for the same mission objectives and constraints published in other research, emphasizing the advantages of this technique. It is also shown that the thrust structure consists of three finite burns. In contrast to previous research, no singular arc is required in the optimal solutions, and all the controls appear bang-bang.

A multi‐regional epidemic model for controlling the spread of Ebola: awareness, treatment, and travel‐blocking optimal control approaches

Ebola virus disease (EVD) can rapidly cause death to animals and people, for less than 1month. In addition, EVD can emerge in one region and spread to its neighbors in unprecedented durations. Such cases were reported in Guinea, Sierra Leone, and Liberia. Thus, by blocking free travelers, traders, and transporters, EVD has had also impacts on economies of those countries. In order to find effective strategies that aim to increase public knowledge about EVD and access to possible treatment while restricting movements of people coming from regions at high risk of infection, we analyze three different optimal control approaches associated with awareness campaigns, treatment, and travel‐blocking operations that health policy‐makers could follow in the war on EVD. Our study is based on the application of Pontryagin's maximum principle, in a multi‐regional epidemic model we devise here for controlling the spread of EVD. The model is in the form of multi‐differential systems that describe dynamics of susceptible, infected, and removed populations belonging to p different geographical domains with three control functions incorporated. The forward–backward sweep method with integrated progressive‐regressive Runge–Kutta fourth‐order schemes is followed for resolving the multi‐points boundary value problems obtained. Copyright © 2016 John Wiley & Sons, Ltd.

Optimal Shielding Design for Minimum Materials Cost or Mass

The mathematical underpinnings of cost optimal radiation shielding designs based on an extension of optimal control theory are presented, a heuristic algorithm to iteratively solve the resulting optimal design equations is suggested, and computational results for a simple test case are discussed.A typical radiation shielding design problem can have infinitely many solutions, all satisfying the problem's specified set of radiation attenuation requirements. Each such design has its own total materials cost. For a design to be optimal, no admissible change in its deployment of shielding materials can result in a lower cost. This applies in particular to very small changes, which can be restated using the calculus of variations as the Euler-Lagrange equations. The associated Hamiltonian function and application of Pontryagin's theorem lead to conditions for a shield to be optimal.

Boundary value problem of Pontryagin’s maximum principle in a two-sector economy model with an integral utility function

An infinite-horizon two-sector economy model with a Cobb–Douglas production function and a utility function that is an integral functional with discounting and a logarithmic integrand is investigated. The application of Pontryagin’s maximum principle yields a boundary value problem with special conditions at infinity. The search for the solution of the maximum-principle boundary value problem is complicated by singular modes in its optimal solution. In the construction of the solution to the problem, they are described in analytical form. Additionally, a special version of the sweep method in continuous form is proposed, which is of interest from theoretical and computational points of view. An important result is the proof of the optimality of the extremal solution obtained by applying the maximum-principle boundary value problem.

Optimal strategies for biomass productivity maximization in a photobioreactor using natural light

We address the question of the optimization of the microalgal biomass long term productivity in the framework of production in photobioreactors under the influence of day/night cycles. For that, we propose a simple bioreactor model accounting for light attenuation in the reactor due to biomass density and we obtain the control law that optimizes productivity over a single day through the application of Pontryagin’s maximum principle. The dilution rate is the main control, the input concentration being only used as the secondary control to maintain the substrate concentration high. An important constraint on the obtained solution is that the biomass in the reactor should be at the same level at the beginning and at the end of the day so that the same control can be applied everyday and optimizes some form of long term productivity. Several scenarios are possible depending on the microalgae’s strain parameters and the maximal admissible value of the dilution rate: bang–bang or bang–singular–bang control or, if the growth rate of the algae is very strong in the presence of light, constant maximal dilution. A bifurcation diagram is presented to illustrate for which values of the parameters these different behaviors occur. Finally, a simple sub-optimal bang–bang strategy is proposed that numerically achieves productivity levels that almost match those of the optimal strategy.

SOLUTION OF INFINITE HORIZON NONLINEAR OPTIMAL CONTROL PROBLEMS BY PIECEWISE ADOMIAN DECOMPOSITION METHOD

In this paper, a Piecewise Adomian Decomposition Method (PADM) is used to obtain the analytical approximate solution for a class of infinite horizon nonlinear optimal control problems (OCPs). The method is a new modification of the standard ADM, in which it is treated as an algorithm in a sequence of small intervals (i.e. with small time step) for finding accurate approximate solutions to the corresponding OCPs. Applying the PADM, the nonlinear two-point boundary value problem (TPBVP), derived from the application of Pontryagin's maximum principle (PMP), is transformed into a sequence of linear time-invariant TPBVP's. Through the finite iterations of algorithm, a suboptimal control law is obtained for the nonlinear optimal control problem. Comparing the methodology with some known techniques shows that the present approach is powerful and reliable. It is remarkable accuracy properties are finally demonstrated by two examples.