Conditions For Optimal Control Research Articles

Optimal control for the complication of Type 2 diabetes: the role of awareness programs by media and treatment.

T2 diabetes is a silent killer and serious public health issue across the world, though awareness of diabetes allows understanding of the causes and prevention of the disease. With this inspiration, we formulate a deterministic model by incorporating awareness and saturated treatment function of the T2 diabetes model to study the dynamics of the disease. We have carried out thoroughly analysis of the model system, including positivity of solutions, boundedness, equilibrium, and stability analysis. Again, we consider the deterministic model system as an optimal control problem by taking awareness (M) and treatment (u) as time-depended control parameters. The sufficient conditions for optimal control for T2 diabetes are obtained utilizing the Pontryagin’s maximum principle in time-dependent controls to find optimal strategies for disease control. We intended to assess the efficacy and costs of several strategies to determine which is the best cost-effective strategy with the limited resources for treatment. The parameters incident rate (beta ), awareness coefficient (p), media (M), and treatment (u) highly influence the dynamics of T2 diabetes. Numerical simulations suggest that both awareness and treatment controls have a significant impact on the optimal system and are economically feasible to reduce the prevalence of T2 diabetes.

Co–pyrolysis of de–alkalized lignin and coconut shell via TG/DTG–FTIR and machine learning methods: pyrolysis characteristics, gas products, and thermo–kinetics

Co–pyrolysis characteristics and thermo–kinetics of de–alkalized lignin (DL) and coconut shell (CS) were investigated by using TG–FTIR and machine learning methods. The DL–CS samples mixed in different proportions were heated from ambient temperature to 1173.15 K at 10, 15, and 20 K·min−1. The gas functional groups in the pyrolysis process of the experimental samples were detected by TG–FTIR. The apparent activation energy (E) was estimated by Flynne–Walle–Ozawa (FWO), Kissinger–Akahira–Sunose (KAS), and Starink methods, and the R2 values of these three methods are greater than 0.97032, which means that the activation energy solved by these methods is feasible. An BP–NN model of 9×3×1 architecture was employed to predict the residual weight of DL–CS co–pyrolysis. The experimental values were in good agreement with the predicted values by BP–NN model (RMSE = 0.8606, R2 = 0.99888). A Stacked XGB–LGBM–MLP model was employed to improve the overall prediction performance of DL–CS co–pyrolysis, and this model succeeds to preform best (RMSE = 0.1888, R2 = 0.99995) of all individual models from test set result parameters. Our research results contribute to the optimal operating conditions for energy utilization, pollution control, and thermochemical conversion of lignin waste industry.

Necessary optimality conditions of fractional-order discrete uncertain optimal control problems

This paper is primarily dedicated to the necessary conditions of optimal control for uncertain discrete-time systems of fractional-order governed by left Riemann-Liouville-like fractional-order nabla difference equations. First, an existence result of solutions for backward right Riemann-Liouville-like fractional-order nabla difference equations is derived from the Banach contraction mapping theorem. Besides, necessary optimality conditions of these fractional-order uncertain optimal control problems are proposed by the variational method and Lagrange multiplier technique. Finally, a special LQ optimal control problem for uncertain discrete-time systems of fractional-order is solved by necessary conditions of optimality derived.

Bayesian learning for optimal control of quantum many-body states in optical lattices

Strongly correlated quantum many-body states provide invaluable resources for state-of-the-art quantum information science, ranging from quantum simulation to quantum metrology. In this work, we propose a supervised machine learning algorithm for optimal control of quantum many-body atomic states in optical lattices by numerical simulations based on the Bayesian method of Gaussian process regression (GPR). We combine this method with the time evolving block decimation (TEBD) algorithm for the preparation of the Heisenberg antiferromagnetic state in a system of bosonic atoms confined with a one-dimensional optical lattice. The quantum many-body ground state of 80 atoms is efficiently optimized within a few hundred machine learning iterations, reaching a state fidelity above $96%$. With a multistep learning strategy, we find the machine-learning-based optimal control method is scalable to large systems owing to its transferability. Its robustness against noise is demonstrated by considering imperfections that are typically present in optical lattice experiments. In the application of the GPR method to the preparation of the two-dimensional (2D) antiferromagnetic state, a state fidelity of $94%$ is reached for a 2D array of $6\ifmmode\times\else\texttimes\fi{}6$ spins through the time dependent variational principle (TDVP) algorithm, confirming the generalizability of our method. We further optimize the Hamiltonian ramping sequence crossing a quantum phase transition of the quantum $XXZ$ spin chain with the exact diagonalization method, from which a control protocol for generating the long-sought atomic Greenberger-Horne-Zeilinger state is obtained. We believe the proposed optimal control method for quantum Hamiltonian ground-state preparation would benefit present ultracold-atom experiments in the study of strongly correlated many-body physics.

Optimal Control with Partially Observed Regime Switching: Discounted and Average Payoffs

We consider an optimal control problem with the discounted and average payoff. The reward rate (or cost rate) can be unbounded from above and below, and a Markovian switching stochastic differential equation gives the state variable dynamic. Markovian switching is represented by a hidden continuous-time Markov chain that can only be observed in Gaussian white noise. Our general aim is to give conditions for the existence of optimal Markov stationary controls. This fact generalizes the conditions that ensure the existence of optimal control policies for optimal control problems completely observed. We use standard dynamic programming techniques and the method of hidden Markov model filtering to achieve our goals. As applications of our results, we study the discounted linear quadratic regulator (LQR) problem, the ergodic LQR problem for the modeled quarter-car suspension, the average LQR problem for the modeled quarter-car suspension with damp, and an explicit application for an optimal pollution control.

Necessary and sufficient conditions in optimal control of mean-field stochastic differential equations with infinite horizon

Abstract We consider an infinite horizon optimal control of a system where the dynamics evolve according to a mean-field stochastic differential equation and the cost functional is also of mean-field type. These are systems where the coefficients depend not only on the state variable, but also on its marginal distribution via some linear functional. Under some concavity assumptions on the coefficients as well as on the Hamiltonian, we are able to prove a verification theorem, which gives a sufficient condition for optimality for a given admissible control. In the absence of concavity, we prove a necessary condition for optimality in the form of a weak Pontryagin maximum principle, given in terms of stationarity of the Hamiltonian.

On partially observed optimal singular control of McKean–Vlasov stochastic systems: Maximum principle approach

In this paper, we study partially observed optimal stochastic singular control problems of general Mckean–Vlasov type with correlated noises between the system and the observation. The control variable has two components, the first being absolutely continuous and the second is a bounded variation, nondecreasing continuous on the right with left limits. The dynamic system is governed by Itô‐type controlled stochastic differential equation. The coefficients of the dynamic depend on the state process and of its probability law and the continuous control variable. In terms of a classical convex variational techniques, we establish a set of necessary conditions of optimal singular control in the form of maximum principle. Our main result is proved by applying Girsanov's theorem and the derivatives with respect to probability law in Lions' sense. To illustrate our theoretical result, we study partially observed linear‐quadratic singular control problem of McKean–Vlasov type.

Optimal vaccination strategy for a mean-field stochastic susceptible-infected-vaccinated system

The parameters of biological system may change under the influence of different states or state processes. The change of parameters can also affect the dynamic behaviors of epidemic disease. This paper presents a mean field stochastic susceptible-infected-vaccinated (SIV) epidemic model which parameters depend on the state process. The sufficient and necessary conditions of optimal control of the novel epidemic model are obtained by maximum principle. Finally, we perform representative numerical simulations to illustrate the theoretical results and further discuss the feasibility based on the hand, foot and mouth disease (HFMD) data in China.

Turnpike in optimal control of PDEs, ResNets, and beyond

Theturnpike propertyin contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states are close (often exponentially) most of the time, except near the initial and final times, to the optimal control and the corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade – the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal – and present several novel applications, including, among many others, the characterization of Hamilton–Jacobi–Bellman asymptotics, and stability estimates in deep learning via residual neural networks.

Mathematical analysis of a model of Ebola disease with control measures

The re-emergence of the Ebola virus disease has pushed researchers to investigate more on this highly deadly disease in order to better understand and control the outbreak and recurrence of epidemics. It is in this perspective that we formulate a realistic mathematical model for the dynamic transmission of Ebola virus disease, incorporating relevant control measures and factors such as ban on eating bush-meat, social distancing, observance of hygiene rules and containment, waning of the vaccine-induced, imperfect contact tracing and vaccine efficacy, quarantine, hospitalization and screening to fight against the spread of the disease. First, by considering the constant control parameters case, we thoroughly compute the control reproduction number [Formula: see text] from which the dynamics of the model is analyzed. The existence and stability of steady states are established under appropriate assumptions on [Formula: see text]. Also, the effect of all the control measures is investigated and the global sensitivity analysis of the control reproduction number is performed in order to determine the impact of parameters and their relative importance to disease transmission and prevalence. Second, in the time-dependent control parameters case, an optimal control problem is formulated to design optimal control strategies for eradicating the disease transmission. Using Pontryagin’s Maximum Principle, we derive necessary conditions for optimal control of the disease. The cost-effectiveness analysis of all combinations of the control measures is made by calculating the infection averted ratio and the incremental cost-effectiveness ratio. This reveals that combining the four restrictive measures conveyed through educational campaigns, screening, safe burial and the care of patients in health centers for better isolation is the most cost-effective among the strategies considered. Numerical simulations are performed to illustrate the theoretical results obtained.

Optimal control of TB transmission based on an age structured HIV-TB co-infection model

The World Health Organization’s 2021 global tuberculosis report points out that people living with human immunodeficiency virus are more likely than others to become sick with tuberculosis. Therefore, it is necessary to consider the factor of human immunodeficiency virus infection when studying how to control the tuberculosis epidemic. In this paper, we establish an age structured human immunodeficiency virus and tuberculosis co-infection model and prove well-posedness of the model. Then we study the least cost-deviation problem. In this problem, detection and treatment of latent tuberculosis cases and public education campaigns are use as the control variables. Through rigorous theoretical derivation and detailed mathematical calculation, we obtain the necessary condition of optimal control. We also perform numerical simulation based on the forward-backward sweep method. Finally, combining the theoretical analysis with numerical simulation of the model, we present optimal strategies that may help China achieve the End Tuberculosis Strategy of the World Health Organization as much as possible (the new tuberculosis cases reduce by 90% by 2035 compared to 2015). In addition, in the process of numerical simulation, we also study the effect of available funds on control effectiveness and find that the control effect is better in the case of sufficiency of available funds, which shows that available funds are crucial to tuberculosis control.

Attitude Control of Rigid Bodies: An Energy-Optimal Geometric Switching Control Approach

In this article, the attitude control problem for a class of rigid body systems with switching submodes is investigated. To address such issue, an energy-optimal geometric switching control method is proposed for the first time. First, a switching Lie group method and a switching Lie group discrete variational integrator method based on the variational principle are developed. Then, continuous-time and discrete-time attitude switching dynamics models for rigid bodies are globally expressed on a special orthogonal matrix group (i.e., SO(3) group) to avoid the locality, singularity, and ambiguity caused by using the traditional Euler angle method, minimum representation method, or quaternion method. Second, the switching process of rigid bodies from initial attitude and angular velocity to desired attitude and angular velocity within a fixed maneuver time and the minimum energy consumption of rigid bodies with control saturation are solved. Furthermore, by solving the energy-optimal geometric control problem of attitude switching dynamics models, the global optimal geometric switching control and optimal switching time conditions under continuous-time and discrete-time are first derived. The obtained criteria ensure that the intrinsic geometric properties of attitude switching dynamics models will not be lost during the optimization process. Finally, some simulation results are presented to demonstrate the feasibility of the proposed techniques.

Constrained optimization problems governed by PDE models of grain boundary motions

AbstractIn this article, we consider a class of optimal control problems governed by state equations of Kobayashi-Warren-Carter-type. The control is given by physical temperature. The focus is on problems in dimensions less than or equal to 4. The results are divided into four Main Theorems, concerned with: solvability and parameter dependence of state equations and optimal control problems; the first-order necessary optimality conditions for these regularized optimal control problems. Subsequently, we derive the limiting systems and optimality conditions and study their well-posedness.

Global analysis of tuberculosis dynamical model and optimal control strategies based on case data in the United States

Tuberculosis is a chronic infectious disease with a high death rate, which has attracted worldwide attention. In this paper, we focus on the annual data of tuberculosis in the United States from 1988 to 2019. Firstly, we propose an SVEITR dynamical model with vaccination, fast and slow progression, incomplete treatment, and relapse. Mathematical analyses show that the disease-free equilibrium is globally asymptotically stable (unstable) if R0≤1 (if R0>1). There exists a unique endemic equilibrium that is globally asymptotically stable when R0>1. Next, we use the Markov-chain Monte-Carlo (MCMC) method to acquire the model’s optimal parameters. From R0=1.3763, tuberculosis will not be eliminated without control measures taken in the United States. Through partial rank correlation analysis, the most sensitive parameters to the basic reproduction number are found. Based on this, we take the actual epidemic situation of tuberculosis in the United States into considerations to get the required conditions for optimal control using Pontryagin’s maximum principle and cost-effectiveness analysis. Implementation of three public health measures: tuberculosis prevention and control education, timely treatment, and enhanced efficacy can reduce the prevalence of tuberculosis in the United States. But at present, it is challenging for us to achieve the goal of eliminating tuberculosis by 2030 as called for by WHO.

Optimal Control for a Class of Riemann-Liouville Fractional Evolution Inclusions

In this paper, under symmetric properties of multivalued operators, the existence of mild solutions as well as optimal control for the nonlocal problem of fractional semilinear evolution inclusions are investigated in abstract spaces. At first, the existence results are proved by applying the theory of operator semigroups and the fixed-point theorem of multivalued mapping. Then the existence theorem on the optimal state-control pair is proved by constructing the minimizing sequence twice. An example is given in the last section as an application of the obtained conclusions.

Optimal control for non-cooperative systems involving fractional Laplace operator

This study investigates the optimal control problem associated with n ? n non-cooperative systems, including the fractional Laplace operator. The non-local issue is recast as a local problem. As a consequence, in these systems, the existence and uniqueness of the weak solution are established. A system?s existence and optimal control conditions are also established.

Unravelling the dynamics of Lassa fever transmission with differential infectivity: Modeling analysis and control strategies.

Epidemic models have been broadly used to comprehend the dynamic behaviour of emerging and re-emerging infectious diseases, predict future trends, and assess intervention strategies. The symptomatic and asymptomatic features and environmental factors for Lassa fever (LF) transmission illustrate the need for sophisticated epidemic models to capture more vital dynamics and forecast trends of LF outbreaks within countries or sub-regions on various geographic scales. This study proposes a dynamic model to examine the transmission of LF infection, a deadly disease transmitted mainly by rodents through environment. We extend prior LF models by including an infectious stage to mild and severe as well as incorporating environmental contributions from infected humans and rodents. For model calibration and prediction, we show that the model fits well with the LF scenario in Nigeria and yields remarkable prediction results. Rigorous mathematical computation divulges that the model comprises two equilibria. That is disease-free equilibrium, which is locally-asymptotically stable (LAS) when the basic reproduction number, $ {\mathcal{R}}_{0} $, is $ < 1 $; and endemic equilibrium, which is globally-asymptotically stable (GAS) when $ {\mathcal{R}}_{0} $ is $ > 1 $. We use time-dependent control strategy by employing Pontryagin's Maximum Principle to derive conditions for optimal LF control. Furthermore, a partial rank correlation coefficient is adopted for the sensitivity analysis to obtain the model's top rank parameters requiring precise attention for efficacious LF prevention and control.

On regularization of classical optimality conditions in convex optimal control

On regularization of classical optimality conditions in convex optimal control

Ergodic risk-sensitive control of Markov processes on countable state space revisited

We consider a large family of discrete and continuous time controlled Markov processes and study an ergodic risk-sensitive minimization problem. Under a blanket stability assumption, we provide a complete analysis to this problem. In particular, we establish uniqueness of the value function and verification result for optimal stationary Markov controls, in addition to the existence results. We also revisit this problem under a near-monotonicity condition but without any stability hypothesis. Our results also include policy improvement algorithms both in discrete and continuous time frameworks.

Estimation of shoulder and elbow joint angle from linear quadratic tracking based on six muscle force vectors

Two dimensional human’s arm is one of the optimal control application. This arm model consists of joint angles consisting of shoulder joint and elbow joint. Beside that, there are also muscle force vectors consisting of pectoralis major, posterior deltoid, brachialis, lateral head of triceps brachii, biceps brachii, and longhead of triceps. Linear Quadratic Tracking (LQT) is constructed to obtain solution of state and optimal control. From state solution and optimal control obtained, they will be used estimation by Backpropagation and ANFIS. From data resulted from LQT simulation, the input used are muscle force vectors. They will be used for estimation the output i.e. angle of shoulder joint and angle of elbow joint. From the LQT simulation, the angle position can follow the given reference function. Then Backpropagation and ANFIS can make estimation in angle of shoulder joint and angle of elbow joint based on six muscle force vectors with small RMSE. In Backpropagation, the estimation result of angle of shoulder joint between target and output with RMSE is 0.0954 and the estimation result of angle elbow joint between target and output with RMSE is 0.2528. In ANFIS, the estimation result of angle of shoulder joint between target and output with RMSE is 0.0062 and the estimation result of angle elbow joint between target and output with RMSE is 0.0526.