Necessary Conditions For Optimal Control Research Articles

Model‐free distributed optimal control for general discrete‐time linear systems using reinforcement learning

AbstractThis article proposes a novel data‐driven framework of distributed optimal consensus for discrete‐time linear multi‐agent systems under general digraphs. A fully distributed control protocol is proposed by using linear quadratic regulator approach, which is proved to be a sufficient and necessary condition for optimal control of multi‐agent systems through dynamic programming and minimum principle. Moreover, the control protocol can be constructed by using local information with the aid of the solution of the algebraic Riccati equation (ARE). Based on the Q‐learning method, a reinforcement learning framework is presented to find the solution of the ARE in a data‐driven way, in which we only need to collect information from an arbitrary follower to learn the feedback gain matrix. Thus, the multi‐agent system can achieve distributed optimal consensus when system dynamics and global information are completely unavailable. For output feedback cases, accurate state information estimation is established such that optimal consensus control is realized. Moreover, the data‐driven optimal consensus method designed in this article is applicable to general digraph that contains a directed spanning tree. Finally, numerical simulations verify the validity of the proposed optimal control protocols and data‐driven framework.

Modelling the impacts of media campaign and double dose vaccination in controlling COVID-19 in Nigeria

Corona virus disease (COVID-19) is a lethal disease that poses public health challenge in both developed and developing countries of the world. Owing to the recent ongoing clinical use of COVID-19 vaccines and non-compliance to COVID-19 health protocols, this study presents a deterministic model with an optimal control problem for assessing the community-level impact of media campaign and double-dose vaccination on the transmission and control of COVID-19. Detailed analysis of the model shows that, using the Lyapunov function theory and the theory of centre manifold, the dynamics of the model is determined essentially by the control reproduction number (Rmv). Consequently, the model undergoes the phenomenon of forward bifurcation in the absence of the double dose vaccination effects, where the global disease-free equilibrium is obtained whenever Rmv≤1. Numerical simulations of the model using data relevant to the transmission dynamics of the disease in Nigeria, show that, certain values of the basic reproduction number ((R0≥7)) may not prevent the spread of the pandemic even if 100% media compliance is achieved. Nevertheless, with assumed 75% (at R0=4)) media efficacy of double dose vaccination, the community herd immunity to the disease can be attained. Furthermore, Pontryagin's maximum principle was used for the analysis of the optimized model by which necessary conditions for optimal controls were obtained. In addition, the optimal simulation results reveal that, for situations where the cost of implementing the controls (media campaign and double dose vaccination) considered in this study is low, allocating resources to media campaign-only strategy is more effective than allocating them to a first-dose vaccination strategy. More so, as expected, the combined media campaign-double dose vaccination strategy yields a higher population-level impact than the media campaign-only strategy, double-dose vaccination strategy or media campaign-first dose vaccination strategy.

Optimal Combinations of Control Strategies and Cost-Effectiveness Analysis of Dynamics of Endemic Malaria Transmission Model.

In this study, we propose and analyze a determinastic nonlinear system of ordinary differential equation model for endemic malaria disease transmission and optimal combinations of control strategies with cost effective analysis. Basic properties of the model, existence of disease-free and endemic equilibrium points, and basic reproduction number of the model are derived and analyzed. From this analysis, we conclude that if the basic reproduction number is less than unity, then the disease-free equilibrium point is both locally and globally asymptotically stable. The endemic equilibrium will also exist if the basic reproduction number is greater than unity. Moreover, existence and necessary condition for forward bifurcation is derived and established. Furthermore, optimal combinations of time-dependent control measures are incorporated to the model. By using Pontryagin's maximum principal theory, we derived the necessary conditions of optimal control. Numerical simulations were conducted to confirm our analytical results. Our findings were that malaria disease may be controlled well with strict application of the combination of prevention of drug resistance, insecticide-treated net (ITN), indoor residual spray (IRS), and active treatment. The use of a combination of insecticide-treated net, indoor residual spray, and active treatment is the most optimal cost-effective and efficacious strategy.

Optimal Control Strategies to Cope with Unemployment During the Covid-19 Pandemic

In the present paper, a mathematical model using the non-linear differential equations depicting the impact of Covid-19 on unemployment is discussed. The stability of the system is studied and model is reformulated as an optimal control problem. To assess the impact of unemployment on human population, two time-dependent controls are used. Providing education and training of job-oriented persons act as first control and campaigning about the awareness of coronavirus disease and self-employment business is the second control. Necessary conditions for optimal control are derived by Pontryagins maximum principle. Further, the results are illustrated by numerical simulation.

Dynamic analysis of HIV model with a general incidence, CTLs immune response and intracellular delays

An HIV model with a general incidence, CTLs immune response and intracellular delays is considered. By using Lyapunov functions and LaSalle’s invariance principle, it is shown that when time delay is equal to zero or not, the reproductive number R0 is a threshold determining the global dynamics. These theoretical results are supported with numerical simulations. In order to effectively control the spread of HIV, A multi-pathways HIV-1 infection model without time delay is analyzed with three controls. The necessary conditions for optimal control are given by Pontryagin’s Maximum Principle. The optimality system is derived and solved numerically using the Euler procedure. Finally, numerical simulation shows the role of the best treatment in controlling the severity of HIV.

Optimal control and cost-effectiveness analysis of age-structured malaria model with asymptomatic carrier and temperature variability

This paper presents an age-structured mathematical model for malaria transmission dynamics with asymptomatic carrier and temperature variability. The temperature variability function is fitted to the temperature data, and the malaria model is then fitted to the malaria cases and validated to check its suitability. Time-dependent controls were considered, including Long Lasting Insecticide Nets, treatment of symptomatic, screening and treatment of asymptomatic carriers and spray of insecticides. Pontryagin's Maximum Principle is used to derive the necessary conditions for optimal control of the disease. The numerical simulations of the optimal control problem reveal that the strategy involving the combination of all four controls is the most effective in reducing the number of infected individuals. Furthermore, the cost-effectiveness analysis shows that treatment of symptomatic, screening and treatment of asymptomatic carriers and insecticide spraying is the most cost-effective strategy to implement to control malaria transmission when available resources are limited.

Optimal control for co-infection with COVID-19-Associated Pulmonary Aspergillosis in ICU patients with environmental contamination.

In this paper, we propose a mathematical model for COVID-19-Associated Pulmonary Aspergillosis (CAPA) co-infection, that enables the study of relationship between prevention and treatment. The next generation matrix is employed to find the reproduction number. We enhanced the co-infection model by incorporating time-dependent controls as interventions based on Pontryagin's maximum principle in obtaining the necessary conditions for optimal control. Finally, we perform numerical experiments with different control groups to assess the elimination of infection. In numerical results, transmission prevention control, treatment controls, and environmental disinfection control provide the best chance of preventing the spread of diseases more rapidly than any other combination of controls.

Second-order necessary conditions in optimal control of evolution systems

Second-order necessary conditions in optimal control of evolution systems

Generating Nesterov’s accelerated gradient algorithm by using optimal control theory for optimization

Nesterov’s accelerated gradient algorithm is derived from first principles. The first principles are founded on the recently-developed optimal control theory for optimization. This theory frames a generic optimization problem as an optimal control problem. The necessary conditions for optimal control produce a controllable dynamical system. Stabilizing this system by dissipating a simple, quadratic “control” Lyapunov function generates an equation that produces Nesterov’s algorithm. In this context, the proposed theory solves the long-standing mystery surrounding the algorithm.

Stability Analysis and Optimal Control of Endemic Malaria Disease Transmission Model with Cost-Effective Strategies

In this study, a non-linear system of ordinary differential equation model that describes the dynamics of malaria disease transmission is formulated and analyzed. Conditions are derived from the existence of disease-free and endemic equilibria. The basic reproduction number R0 of the model is obtained, and we investigated that it is the threshold parameter between the extinction and persistence of the disease. If R0 is less than unity, then the disease-free equilibrium point is both locally and globally asymptotically stable resulting in the disease removing out of the host populations. The disease can persist whenever R0 is greater than unity and the conditions for the existence of both forward and backward bifurcation at R0 is equal to unity are derived. Sensitivity analysis is also performed and the important parameter that derive the disease dynamics is identified. Furthermore, optimal combinations of time dependent control measures are incorporated to the model, and we derived the necessary conditions of optimal control using Pontryagins’s maximum principal theory. Numerical simulations were conducted using MATLAB to confirm our analytical results. Our findings were that, malaria may be controlled by reducing the requirement rate of mosquito populations and the use of a combination of vaccination, insecticide treated net ITN, indoor residual spray IRS and active treatment or strategy d can also help to reduce the number of populations with malaria symptoms to zero. We also find that the same strategy that is, strategy d proves to be efficacious and cost-effective.

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.

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.

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.

Malaria and COVID-19 co-dynamics: A mathematical model and optimal control

Malaria, one of the longest-known vector-borne diseases, poses a major health problem in tropical and subtropical regions of the world. Its complexity is currently being exacerbated by the emerging COVID-19 pandemic and the threats of its second wave and looming third wave. We formulate and analyze a mathematical model incorporating some epidemiological features of the co-dynamics of both malaria and COVID-19. Sufficient conditions for the stability of the malaria only and COVID-19 only sub-models’ equilibria are derived. The COVID-19 only sub-model has globally asymptotically stable equilibria while under certain condition, the malaria-only could undergo the phenomenon of backward bifurcation whenever the sub-model reproduction number is less than unity. The equilibria of the dual malaria-COVID19 model are locally asymptotically stable as global stability is precluded owing to the possible occurrence of backward bifurcation. Optimal control of the full model to mitigate the spread of both diseases and their co-infection are derived. Pontryagin’s Maximum Principle is applied to establish the existence of the optimal control problem and to derive the necessary conditions for optimal control of the diseases. Though this is not a case study, simulation results to support theoretical analysis of the optimal control suggests that concurrently applying malaria and COVID-19 protective measures could help mitigate their spread compared to applying each preventive control measure singly as the world continues to deal with this unprecedented and unparalleled COVID-19 pandemic.

Multi-Objective Optimization of Covariance and Energy for Asteroid Transfers

This paper presents a multi-objective optimization approach to minimize both error covariance and energy using indirect methods. Linear covariance dynamics and a continuous thrust model with control-dependent noise are implemented. Two formulations are considered: 1) a weighted-sum cost function with covariance and energy terms, and 2) a covariance cost function with an energy constraint (the epsilon-constraint method). In general, the former is simpler to implement, whereas the latter is more rigorous. However, a theorem is presented to prove that the necessary conditions for optimal control are equivalent in both formulations, and the first can be used to locate Pareto-optimal trajectories without sacrificing optimality. To provide an explicit example, this approach is applied to orbital maneuvers about an asteroid. Terminator orbit transfers, phasing maneuvers, and “proactive station-keeping” maneuvers are optimized, and the results indicate that significant uncertainty reduction is possible with small penalties in energy cost. Finally, Monte Carlo results verify that a linear covariance propagation was sufficient for these applications.

First-Order Necessary Conditions in Optimal Control

In an earlier analysis of strong variation algorithms for optimal control problems with endpoint inequality constraints, Mayne and Polak provided conditions under which accumulation points satisfy a condition requiring a certain optimality function, used in the algorithms to generate search directions, to be nonnegative for all controls. The aim of this paper is to clarify the nature of this optimality condition, which we call the first-order minimax condition, and of a related integrated form of the condition, which, also, is implicit in past algorithm convergence analysis. We consider these conditions, separately, when a pathwise state constraint is, and is not, included in the problem formulation. When there are no pathwise state constraints, we show that the integrated first-order minimax condition is equivalent to the minimum principle and that the minimum principle (and equivalent integrated first-order minimax condition) is strictly stronger than the first-order minimax condition. For problems with state constraints, we establish that the integrated first-order minimax condition and the minimum principle are, once again, equivalent. But, in the state constrained context, it is no longer the case that the minimum principle is stronger than the first-order minimax condition, or vice versa. An example confirms the perhaps surprising fact that the first-order minimax condition is a distinct optimality condition that can provide information, for problems with state constraints, in some circumstances when the minimum principle fails to do so.

OPTIMAL CONTROL ANALYSIS OF A HUMAN–BOVINE SCHISTOSOMIASIS MODEL

We formulate and analyze a deterministic mathematical model for the transmission dynamics of schistosomiasis with treatment of both humans and bovines and mollu-sciciding of the contaminated environment. The model effective reproduction number is derived and analytical results show that the disease-free and endemic equilibria are both locally and globally asymptotically stable. Pontryagin’s Maximum Principle which uses both Lagrangian and Hamiltonian principles with respect to a time-dependent constant is used to establish the existence of the optimal control problem and to derive the necessary conditions for optimal control of the disease. Mollusciciding of the contaminated environment has a major impact on disease control. However, combining it with treatment could help mitigate the spread of the disease compared to applying each control measure individually. Numerical simulations are performed to support theoretical results.

OPTIMAL CONTROL OF LASSA FEVER QUARANTINE MODEL

In this paper, Optimal control is applied to a Quarantine Lassa fever model developed. The controls w1, w2 and w3 which represent Public education on lassa fever mode of transfer, use of Standard precautions in treatment and Environmental sanitation respectively. The existence and necessary condition for Optimal control of the disease were verified. The effects of these controls on the model are dipitched in graphical representation showing with or without controls implementation.

On the modification and convergence of unconstrained optimal control using pseudospectral methods

AbstractA modified pseudospectral (PS) method is presented based on direct Legendre interpolation for unconstrained optimal control problems (OCP). The conditions for the convergence of the proposed PS method are provided and it is proved that the method possesses the spectral accuracy for solutions in appropriate Sobolev spaces. Different from existing convergence results in the literature, in this new analysis neither special cases of the problem nor relaxing the discretized dynamics is considered. Moreover, the proof does not use necessary conditions of optimal control. An iterative procedure is then proposed for nonlinear OCPs. The nonlinear problem is first replaced with a sequence of linear‐quadratic OCPs by utilizing the quasilinearization technique. Then, this sequence of problems are successively solved using the modified PS method. In some nonlinear OCPs where the resulting nonlinear programming problem is dense, the PS method becomes computationally intractable. In such a this problems, the iterative PS method reduces the computational complexity and saves work. In addition, a new efficient costate estimation procedure is derived by employing the necessary optimality conditions. This new costate estimation method is flexible in that it allows us to define three different schemes based on the structure of the problem. Several examples of varying complexity are included to demonstrate the efficiency and accuracy of the proposed methods.