•Fractional order SEIR epidemic model with two infectious 'stages' is proposed.•Global dynamics of the model is performed.•Numerical simulations using Adam-Beshforth-Moulton method are conducted to ...support our results.•To present fractional order derivative as tool for the description of memory effects.•Fractional optimality condition for the proposed model is formulated.•Euler-Lagrange necessary conditions for the optimality of fractional optimal controls are obtained.
In this paper, a nonlinear fractional order epidemic model for HIV transmission is proposed and analyzed by including extra compartment namely exposed class to the basic SIR epidemic model. Also, the infected class of female sex workers is divided into unaware infectives and the aware infectives. The focus is on the spread of HIV by female sex workers through prostitution, because in the present world sexual transmission is the major cause of the HIV transmission. The exposed class contains those susceptible males in the population who have sexual contact with the female sex workers and are exposed to the infection directly or indirectly. The Caputo type fractional derivative is involved and generalized Adams-Bashforth-Moulton method is employed to numerically solve the proposed model. Model equilibria are determined and their stability analysis is considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle invariant principle. Analysis of the model demonstrates that the population is free from the disease if R0<1 and disease spreads in the population if R0>1. Meanwhile, by using Lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed. Furthermore, for the fractional optimal control problem associated with the control strategies such as condom use for exposed class, treatment for aware infectives, awareness about disease among unaware infectives and behavioral change for susceptibles, we formulated a fractional optimality condition for the proposed model. The existence of fractional optimal control is analyzed and the Euler-Lagrange necessary conditions for the optimality of fractional optimal control are obtained. The effectiveness of control strategies is shown through numerical simulations and it can be seen through simulation, that the control measures effectively increase the quality of life and age limit of the HIV patients. It significantly reduces the number of HIV/AIDS patients during the whole epidemic.
The aim of this paper is to investigate a nonlinear optimal control problem governed by a complicated dynamic variational-hemivariational inequality (DVHVI, for short) with history-dependent ...operators in the framework of an evolution triple of spaces. First, we prove a generalized existence and uniqueness theorem to a class of dynamic variational-hemivariational inequalities, which extends the recent results established by Han-Migórski-Sofonea 12, Theorem 9 and Migórski-Bai 23, Theorem 5. Then, an optimal control problem driven by a (DVHVI) is studied and an existence theorem is delivered. Finally, we explore the sensitivity properties of the optimal control problem under the consideration depending on the initial data ξ∈H and some further parameter λ∈E.
Thanks to their road safety potential, automated vehicles are rapidly becoming a reality. In the last decade, automated driving has been the focus of intensive automotive engineering research, with ...the support of industry and governmental organisations. In automated driving systems, the path tracking layer defines the actuator commands to follow the reference path and speed profile. Model predictive control (MPC) is widely used for trajectory tracking because of its capability of managing multi-variable problems, and systematically considering constraints on states and control actions, as well as accounting for the expected future behaviour of the system. Despite the very large number of publications of the last few years, the literature lacks a comprehensive and updated survey on MPC for path tracking. To cover the gap, this literature review deals with the research conducted from 2015 until 2021 on model predictive path tracking control. Firstly, the survey highlights the significance of MPC in the recent path tracking control literature, with respect to alternative control structures. After classifying the different typologies of MPC for path tracking control, the adopted prediction models are critically analysed, together with typical optimal control problem formulations. This is followed by a summary of the most relevant results, which provides practical design indications, e.g., in terms of selection of prediction and control horizons. Finally, the most recent development trends are analysed, together with likely areas of further investigations, and the main conclusions are drawn.
•A new mathematical model of transmission of the COVID-19 disease is proposed.•Two disease control policies namely, contact tracing and isolation are considered.•An optimal control problem is framed ...to measure the efficacy of contact tracing.•The model is calibrated with the data from the six most affected states of India.•A short-term prediction is given with possible variation in contact tracing level.
The ongoing pandemic situation due to COVID-19 originated from the Wuhan city, China affects the world in an unprecedented scale. Unavailability of totally effective vaccination and proper treatment regimen forces to employ a non-pharmaceutical way of disease mitigation. The world is in desperate demand of useful control intervention to combat the deadly virus. This manuscript introduces a new mathematical model that addresses two different diagnosis efforts and isolation of confirmed cases. The basic reproductive number, R0, is inspected, and the model’s dynamical characteristics are also studied. We found that with the condition R0<1, the disease can be eliminated from the system. Further, we fit our proposed model system with cumulative confirmed cases of six Indian states, namely, Maharashtra, Tamil Nadu, Andhra Pradesh, Karnataka, Delhi and West Bengal. Sensitivity analysis carried out to scale the impact of different parameters in determining the size of the epidemic threshold of R0. It reveals that unidentified symptomatic cases result in an underestimation of R0 whereas, diagnosis based on new contact made by confirmed cases can gradually reduce the size of R0 and hence helps to mitigate the ongoing disease. An optimal control problem is framed using a control variable u(t), projecting the effectiveness of diagnosis based on traced contacts made by a confirmed COVID patient. It is noticed that optimal contact tracing effort reduces R0 effectively over time.
In this paper, the bang–bang property of time optimal controls for heat equation is established. Compared with the existing results on these problems, the bound of control variables is not a constant ...but a time-varying function. As an application of the bang–bang property, some kind of relation between the time optimal control problem and its corresponding target optimal control problem is considered.
In this paper, we study the mixed virtual element approximation to an elliptic optimal control problem with boundary observations. The objective functional of this type of optimal control problem ...contains the outward normal derivatives of the state variable on the boundary, which reduces the regularity of solutions to the optimal control problems. We construct the mixed virtual element discrete scheme and derive an a priori error estimate for the optimal control problem based on the variational discretization for the control variable. Numerical experiments are carried out on different meshes to support our theoretical findings.
•A Pareto-based Stackelberg stochastic differential game is solved.•A feedback representation of Pareto-based Stackelberg equilibrium is obtained.•The solvability of Riccati equations is studied.
...This paper is concerned about a Pareto-based Stackelberg stochastic differential game with multi-followers, in which followers are cooperative relations. First of all, necessary and sufficient conditions for the existence of Pareto-based Stackelberg equilibrium are given for the general Pareto-based Stackelberg stochastic differential game problem. Next, we study the above game problem in the linear-quadratic (LQ) case and obtain a feedback form of Pareto-based Stackelberg equilibrium. The solvability of Riccati equations is guaranteed under some assumptions. Finally, the theoretical results are used to solve a national debt management problem.
In this paper we investigate virtual element discretization of optimal control problem governed by Stokes equations. Based on the strategy of first-discretize-then-optimize we build up the virtual ...element discrete scheme and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L2 and H1 norms are derived. Numerical experiments are presented to verify the theoretical findings.
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