Uniqueness Of Optimal Control Research Articles

Dynamical analysis and optimal control of an age‐structured epidemic model with asymptomatic infection and multiple transmission pathways

The diversity of transmission modes and the heterogeneity of populations in the transmission of infectious diseases are issues that have to be faced in the current disease protection. In this paper, an infectious disease model incorporating age structure and horizontal and environmental spread, along with asymptomatic infection, is proposed to describe diversification of disease transmission routes and population heterogeneity. The expression of the basic reproduction number is derived by a linear approximation method, and it is concluded that if , the disease‐free steady state is globally asymptotically stable; if , the model has a unique endemic steady state which is locally asymptotically stable under certain conditions. Further, the uniform persistence of disease is proved using the persistence theory of the infinite‐dimensional dynamical system when . In additional, the issue of optimal control problem according to this model is investigated, and the representation of optimal control with respect to state variables and adjoint variables is obtained which also imply the existence and uniqueness of optimal control. Finally, numerical simulations are carried out to interpret the theoretical results and to discuss the impact of age and control measures in disease transmission.

Optimal control analysis on the impact of non-pharmaceutical interventions and vaccination on the dynamics of COVID-19

COVID-19 is a serious issue, claiming millions of lives across the globe. Starting from the beginning of the pandemic, different controlling strategies, such as non-pharmaceutical interventions and vaccination, have been adapted to mitigate the virus. In this paper, we developed an epidemiological mathematical model to assess the transmission dynamics of COVID-19 with interventions. The impact of non-pharmaceutical interventions and vaccination of vulnerable individuals on its transmission has been discussed in the developed model. We have proved the mathematical well-posedness and epidemiological meaningfulness of the model. The disease free and endemic equilibrium points are calculated, and the stability analysis of the equilibria have been proved by obtaining the control as well as the basic reproduction numbers of the model. Furthermore, an optimal control analysis is implemented to assess the impacts of employing control interventions. The controls are employed as a time dependent self protective measures, vaccinating individuals, and treating the infected ones. Then we proved the existence and uniqueness of optimal controls, and analyzed the problem using Pontryagin’s Maximum Principle. We estimated the parameter values of the proposed model with collected epidemiological data. Different simulation cases were performed to compare the analytical results and identify the most appropriate control intervention strategies. After implementing the proposed control measures, and performing the efficiency analysis of the control strategies, we obtained a promising result for reducing the number of infected individuals by implementing the control measures properly.

Optimal intelligent fuzzy TID controller for an uncertain level process with actuator and system faults: Population-based metaheuristic approach

For the Conical Tow-Tank Non-Interacting Level (CTTNL) prototype system, this article discusses the design of an unique optimal fractional order TID controller employing a type-1 fuzzy set. To generate a linearized model transfer function for the nonlinear CTTNL process, local linearization around the equilibrium state was performed. This work compares the performance of a fuzzy tilted-integral-derivative (FTID) controller for the CTTNL system to a traditional proportional–integral–derivative (PID) controller. The parameters of the FTID controller were optimized using the Chicken Search Optimization (CSO) algorithm in this study, and those values were used to create a PID controller. Thereafter, the FTID and PID controllers’ performance is compared. The investigation shows that the FTID controller beats the PID controller in terms of IAE, ISE, ITAE, and ITSE integral errors. The suggested controller’s fault-tolerant performance was also assessed using the fault recovery time (Frt) parameter. The FTID controller outperforms traditional PID controllers in terms of transient and steady-state performance. The fault tolerance capabilities of the FTID controller will be tested on the CTTNL system in the future, using additive and multiplicative failures. Further research will include a comprehensive examination of robustness in the context of model parameter uncertainties.

Optimal control for a nonlinear stochastic PDE model of cancer growth

ABSTRACT We study an optimal control problem for a stochastic model of tumour growth with drug application. This model consists of three stochastic hyperbolic equations describing the evolution of tumour cells. It also includes two stochastic parabolic equations describing the diffusions of nutrient and drug concentrations. Since all systems are subject to many uncertainties, we have added stochastic terms to the deterministic model to consider the random perturbations. Then, we have added control variables to the model according to the medical concepts to control the concentrations of drug and nutrient. In the optimal control problem, we have defined the stochastic and deterministic cost functions and we have proved the problems have unique optimal controls. For deriving the necessary conditions for optimal control variables, the stochastic adjoint equations are derived. We have proved the stochastic model of tumour growth and the stochastic adjoint equations have unique solutions. For proving the theoretical results, we have used a change of variable which changes the stochastic model and adjoint equations (a.s.) to deterministic equations. Then we have employed the techniques used for deterministic ones to prove the existence and uniqueness of optimal control.

Mean-Field Type FBSDEs under Domination-Monotonicity Conditions and Application to LQ Problems

This paper is concerned with a class of mean-field type coupled forward-backward stochastic differential equations (MF-FBSDEs), in which the coupling appears in integral terms, terminal terms, and initial terms. Inspired by various mean-field type linear-quadratic (MF-LQ) optimal control problems, we propose a type of randomized domination-monotonicity conditions, under which and the usual Lipschitz condition we obtain a well-posedness result on MF-FBSDEs in the sense of square integrability including the unique solvability, an estimate of the solution, and the related continuous dependence property of the solution on the coefficients. The result of MF-FBSDEs in turn extends MF-LQ problems in the literature to a general situation where the initial states or the terminal states are also controlled at the same time, and gives explicit expressions of the related unique optimal controls.

A linear quadratic tracking problem for impulsive controlled stochastic systems: The infinite horizon time case

We investigate a linear quadratic tracking problem for impulsive controlled stochastic systems. The main tool in the derivation of the optimal control is the stabilizing solution for a class of backward jump matrix linear differential equation. We introduce the concept of mean square stability by impulses and consider the properties of stabilizing solution of the associated backward jump matrix linear differential equation with Riccati‐type operators. Necessary and sufficient conditions that guarantee the existence of the unique optimal control for the linear quadratic tracking problem for impulsive controlled stochastic systems are proved.

OPTIMAL CONTROL ANALYSIS OF A TUBERCULOSIS MODEL

In this paper, an optimal control theory was applied to the tuberculosis (TB) model governed by system of nonlinear ordinary differential equations. The aim is to investigate the impact of treatment failure on the TB epidemic. An optimal control strategy is proposed to minimize the disease effect and cost incurred due to treatment failure. The existence and uniqueness of optimal controls are proved. The characterization of optimal paths is analytically derived using Pontryagin’s Minimum Principle. The control-induced model is then fitted using TB infected cases reported from the year 2010–2019 in East Shewa zone Oromia regional state, Ethiopia. Different simulation cases were performed to compare with analytical results. The simulation results show that the combined effect of awareness via various mass media and continuous supervision during the treatment period helps to reduce treatment failure and hence reduced the TB epidemic in the community.

Energy and Time Optimal Autopilot for Electric Vehicles Performing Ackerman Cornering

This paper studies energy and time optimality of electric vehicles during constant Ackerman steering along a quad-circle, and proposes an autopilot assimilating the optimal results. The energy and time optimal solutions satisfying the steering and battery limitations are generated and a Pareto-front analysis is carried out with multi-objective optimization using NSGA-II algorithm. In the autopilot design, the indicators for the energy and time optimality performances are merged in a vehicle status vector. At each control cycle of the optimal drives, the torque commands and the vehicle status vectors are stored and later clustered using the k-means algorithm. At each cluster centre, a pair of the vehicle status vector and the control command vector is acquired and these pairs designate the rules that generate the unique optimal control command associated with a particular vehicle status. A convex combination of these rules constitutes the autopilot design. The proposed autopilot is tested against the energy-time Pareto-front extracted, and it is observed that in terms of the performance measures, the optimality has been preserved. Additionally, for various weightings of time and energy optimality objectives, the performance of the autopilot is compared with that of optimal controllers with weighted objectives. The results verify the quality of the autopilot design using a decision-making process over a rule set inferred from the solutions acquired with optimal solutions.

Optimal control analysis for the coinfection of COVID-19 and TB

The COVID-19 pandemic continues to clame the lives of many people globally and controlling the disease has became the most challenging part of the modern health care system. Tuberculosis (TB) is also a major global health threat affecting millions of people every year. In this study, we extended the deterministic mathematical model to provide insight for the coinfection of COVID-19 and TB into an optimal control problem. The validity of the coinfection model is qualitatively studied by showing the well-posedness and positivity of the solutions. The analytical computations on the impacts of the disease revealed that an increase in infected individuals with TB has a positive impact on the spread of COVID-19 while under some conditions, an increase in the number of COVID-19 cases has a positive impact on the spread of TB disease. We add four control measures in the deterministic model such as: the prevention effort against TB, prevention techniques against COVID-19, treatments for TB infections and medical care for COVID-19 infection to optimally manage the diseases. The extended optimal control problem is analyzed with the help of Pontryagin’s Minimum Principle. The existence and uniqueness of optimal control are proved. We fitted the parameter values of our proposed model with collected epidemiological data using a modified combination of the Bayesian and least square estimation technique. Different simulation cases were performed to compare the analytical results and to identify the most appropriate control intervention strategies. The simulation results show that the prevalence of the coinfection reduced when all the four control measures were concurrently implemented.

FBSDEs involving time delays and advancements on infinite horizon and LQ problems with delays

This paper is concerned with a class of coupled forward–backward stochastic differential equations (FBSDEs, for short) involving time delays and time advancements on infinite horizon. By introducing a randomized Lipschitz condition and a randomized monotonicity condition, the unique solvability of FBSDEs is obtained. Then the theoretical result is applied to a linear–quadratic (LQ, for short) problem of a time-delayed system with random coefficients. An explicit expression of the unique optimal control is obtained.

Optimal harvesting for a periodic competing system with size structure in a polluted environment

<abstract><p>As a renewable resource, biological population not only has direct economic value to people's lives, but also has important ecological and environmental value. This study examines an optimal harvesting problem for a periodic, competing hybrid system of three species that is dependent on size structure in a polluted environment. The existence and uniqueness of the nonnegative solution are proved via an operator theory and fixed point theorem. The necessary optimality conditions are derived by constructing an adjoint system and using the tangent-normal cone technique. The existence of unique optimal control pair is verified by means of the Ekeland variational principle and a feedback form of the optimal policy is presented. The finite difference scheme and the chasing method are used to approximate the nonnegative T-periodic solution of the state system corresponding to a given initial datum. The objective functional represents the total profit obtained from harvesting three species. The results obtained in this work can be extended to a wide variety of fields.</p></abstract>

Optimal control in poroelasticity

In this paper we address optimal control problems subject to fluid flows through deformable, porous media. In particular, we focus on linear quadratic elliptic-parabolic control problems, with both distributed and boundary controls, and prove existence and uniqueness of optimal control. Furthermore, we derive the first order necessary optimality conditions. These problems have important biological and biomechanical applications. For example, optimizing the pressure of the flow, and investigating the influence and control of pertinent biological parameters are relevant in the case of the lamina cribrosa – a porous tissue at the base of the optic nerve head inside the eye – which is modeled by poroelasticity – where these factors are believed to be related to the development of ocular neurodegenerative diseases such as glaucoma. Moreover, the study and results will be applicable to other situations such as the poroelastic modeling of cartilages, bones, and engineered tissues.

Bilinear boundary optimal control with final observation for the heat equation

In this paper, we consider a bilinear optimal control with final observation for the heat equation. We first give the existence and uniqueness results of weak solutions under an integral constraints on the control and the state. We prove the existence and uniqueness of the solution to a quadratic boundary optimal control problem by means of compactness Sobolev space embedding. Then we provide a characterization of the optimal control via the Euler-Lagrange first-order optimality conditions. The unique optimal control is characterized in terms of the solution of the optimality system, which consists of the state system coupled with an adjoint system.

The Dynamics of a Spatial Economic Model with Bounded Population Growth

We investigate a spatial economic growth model with bounded population growth to obtain the asymptotic behavior of detrended capital in a continuous space. The formation of capital accumulation is expressed by a partial differential equation with corresponding boundary conditions. The capital accumulation interacts with the morphology to affect the optimal dynamics of economic growth. After redrafting the spatial growth model in the infinite dimensional Hilbert space, we identify the unique optimal control and value function when the bounded population growth is considered. With nonnegative initial distribution of capital, the explicit solution of the model is obtained. The time behavior of the explicit solution guarantees the convergence issue of the detrended capital level across space and time.

An extremum problem for a linear integro-differential system describing creeping flows of a viscoelastic fluid

We consider an optimal control problem for an integro-differential system (with a quadratic cost functional) modeling a three-dimensional creeping flow of an incompressible viscoelastic fluid in a bounded domain with impermeable solid walls. The fluid flow is controlled by the time-dependent external force. The concept of the control operator is proposed. We prove a theorem on the existence of a unique optimal control under the assumption that the set of admissible controls is convex and that it is closed in a suitable function space. Moreover, we obtain a variational ine-quality for the optimal control. The proof of this theorem is based on the application of the Faedo–Galerkin approximation scheme taking into account energy estimates of approximate solutions and using the lemma on the existence and uniqueness of the metric projection of a point onto a closed convex set in a real Hilbert space.

Discounted cost linear quadratic Gaussian control for descriptor systems

We consider the linear quadratic Gaussian control problem with a discounted cost functional for descriptor systems on the infinite time horizon. Based on recent results from the deterministic framework, we characterise the feasibility of this problem using a linear matrix inequality. In particular, conditions for existence and uniqueness of optimal controls are derived, which are weaker compared to the standard approaches in the literature. We further show that also for the stochastic problem, the optimal control is given in terms of the stabilising solution of the Lur'e equation, which generalises the algebraic Riccati equation. We conclude our paper by examining an application of our theory to fluid dynamics problems.

Optimal control for a nonlinear stochastic parabolic model of population competition

In this investigation, an optimal control problem for a stochastic mathematical model of population competition is studied. We have considered the stochastic model of population competition by adding the stochastic terms to the deterministic model to take into account the random perturbations and uncertainties caused by the environment to have more reliable model. The model has formulated the population densities competing against each other to be saved from destruction, attract more members and so on, using two nonlinear stochastic parabolic equations. Four factors including the status and the growth rates of the populations are considered as the control variables (which can be controlled by the members of the populations or policy makers who make decisions for the populations for particular purposes) to control the evolution of the population densities. Then, the optimal control problem for the stochastic model of population competition is studied. Employing the tangent-normal cone techniques, the Ekeland variational principle and other theorems proved throughout the paper, we have shown there exists unique stochastic optimal control. We have also presented the exact form of the optimal control in terms of stochastic adjoint states.

OPTIMAL DIVIDEND POLICY AND STOCK PRICES

We model a corporation dividend as an exchange option on stochastic cash flow and capital budge. Then we solve optimal dividend policy problem completely based on the dividend model under the assumption that the cash reservoir of a corporation follows a mean reverting process from empirical evidence and economic arguments. Our optimal dividend controls depend on explicitly with the cash flow and the capital budget of the corporation, and maximizes the HARA utility performance. We specify the unique optimal dividend control for the cash flow and the capital budge. Multiplicity or absence of optimal dividend policies are given. The stock price of the corporation is studied in terms of our stochastic dividend model. We find an explicit relation among the volatility of the stock price, the volatility of the cash flow and the volatility of the capital budget. The ex-dividend stock price is positively proportional to the stochastic cash flow and the probability of the dividend delta with respect to the cash flow, and negatively proportional to the capital budget and the probability of the dividend delta with respect to the capital budget. Hence, our approach provides another passage through which countercyclical volatility of the stock price can arise from the countercyclical cash flow and capital budget directly.

Optimal Control of the Double Integrator with Minimum Total Variation

We study the well-known minimum-energy control of the double integrator, along with the simultaneous minimization of the total variation in the control variable. We derive the optimality conditions and obtain the unique optimal solution to the combined problem, where the initial and terminal boundary points are specified. We study the problem from a multi-objective optimal control viewpoint, constructing the Pareto front. We show that the unique asymptotic optimal control function, for the minimization of the total variation alone, is piecewise constant with one switching at the midpoint of the time horizon. For any instance of the boundary conditions of the problem, we prove that the asymptotic optimal total variation is exactly 2/3 of the total variation of the minimum-energy control. We illustrate the results for a particular instance of the problem and include a link to a video which animates the solutions while moving along the Pareto front.

Constraints Optimal Control Governing by Triple Nonlinear Hyperbolic Boundary Value Problem

The focus of this work lies on proving the existence theorem of a unique state vector solution (Stvs) of the triple nonlinear hyperbolic boundary value problem (TNHBVP) when the classical continuous control vector (CCCVE) is fixed by using the Galerkin method (Galm), proving the existence theorem of a unique constraints classical continuous optimal control vector (CCCOCVE) with vector state constraints (equality EQVC and inequality INEQVC). Also, it consists of studying for the existence and uniqueness adjoint vector solution (Advs) of the triple adjoint vector equations (TAEqs) associated with the considered triple state equations (Tsteqs). The Fréchet Derivative (Frde.) of the Hamiltonian (HAM) is found. At the end, the theorems for the necessary conditions and the sufficient conditions of optimality (Necoop and Sucoop) are achieved.