Optimal Control Problem Research Articles

Numerical Simulation for Fractional Optimal Control Problems via Euler Wavelets

Abstract The study suggests employing Euler wavelets to provide a precise and effective computational approach for certain fractional optimal control problems. The primary objective of the approach is to transform the fractional optimal control problem defined by the dynamical system and performance index into algebraic equation systems, which then can be readily solved using matrix techniques. Since Euler polynomials are employed to build Euler wavelets and since Euler polynomials have less number of terms than most of the other common polynomials to build wavelets, employing them results in sparser matrices, which are called operational matrices. The speed of the proposed numerical method is increased and the required computational load is decreased due to the fewer terms in Euler wavelet operational matrices. Incorporating operational matrices of Euler wavelets yields the corresponding systems of algebraic equations for state variable, control variable, and Lagrange multipliers (used to determine essential conditions of optimality). After transformation, those systems of algebraic equations are solved to obtain the numerical solution. Suggested method's efficiency is demonstrated using a few typical examples. The suggested procedure is accurate and efficient, according to the results.

A recursive framework of vehicle trajectory planning at mixed‐traffic signalized intersections

AbstractThis study aims to introduce a new strategy for anticipating the behaviour of human‐driven vehicles (HDVs) and designing trajectories for connected and automated vehicles (CAVs) at signalized intersections under mixed traffic scenarios. To tackle the challenge of unreliable HDV trajectory predictions stemming from driving unpredictability, a recursive framework is developed. This framework integrates real‐time tracking data from both traffic detectors and CAVs, continuously updating HDV predictions. The proposed approach employs the updated predictions to formulate optimal control problems recursively to optimize or adjust CAV trajectories, enhancing travel and energy efficiency. Besides, the recomputing of CAV trajectories will only be conducted when the variation in predictions rises to a certain threshold, balancing efficiency and computing consumption, inspired and modified based on MPC methods. The application of the Pontryagin maximum principle aids in finding solutions efficiently by transforming necessary conditions into a system of equations and consolidating elementary unconstrained and constrained arcs. Numerical simulations were carried out to evaluate the performance of the proposed recursive framework, revealing its superiority over the one‐time approach, particularly in isolated intersections with high traffic demands. Additionally, the recursive framework exhibited more robust and effective enhancements throughout the road network.

Stochastic optimal control problem of consumption and pension insurance with uncertain lifetime and its application to real-life data

We consider a continuous-time model of optimal consumption and pension insurance for a consumer with an uncertain lifetime. In the model, the consumer earns a stochastic wage income during her working life and optimally allocates her income between personal consumption, pension insurance, and securities with a deterministic dynamic return. Due to the weak development of the stock market in developing countries, employees' income comes mainly from wages and interest on savings from banks, that are discussed in this paper. The consumer's utility and bequest functions are constant absolute risk aversion (CARA). By characterizing the optimality condition of the consumer's problem using the Hamilton-Jacobi-Bellman equation, we find the optimal consumption and pension insurance as a function of wealth in closed form. We consider an application of the model while estimating its key elements using real-life data on age-specific population size, labor income, and interest rates. We show that as the absolute risk aversion for consumption increases, consumption and wealth move in the opposite direction. We also present a novel finding that wealth and consumption can be negatively related across consumers with different levels of consumption risk aversion.

Optimal control for nonlinear time-fractional Schrodinger equation: an application to quantum optics

Abstract In this research article, a fractional optimal control problem (FOCP) is applied to a nonlinear time-fractional Schrodinger equation (NTFSE) incorporating a trapping potential. The NTFSE is an innovative mathematical advancement in the field of quantum optics, bridging fractional calculus with nonlinear quantum mechanics and addressing the intricacies of systems involving memory and nonlinearity. This exploration helps with potential technological advancements in quantum optics and related domains. Examining the FOCP within this system allows one to design quantum optical systems with enhanced performance, improved precision stability, and robustness against disturbances. In this work, the performance index for the problem is constructed, and then it is reformulated using the fractional variational principle and the Lagrange multiplier method. Additionally, the Jacobi collocation numerical method is employed to solve the FOCP and numerical simulations are demonstrated across various parameters which offer valuable insights into the implemented methodology.

Numerical solution of different kinds of fractional‐order optimal control problems using generalized Lucas wavelets and the least squares method

Numerical solution of different kinds of fractional‐order optimal control problems using generalized Lucas wavelets and the least squares method

An efficient ADAM-type algorithm with finite elements discretization technique for random elliptic optimal control problems

We consider an optimal control problem governed by an elliptic partial differential equation (PDE) with random coefficient, and introduce an efficient numerical method for the problem under concern. Based on a finite element discretization of the constraint PDE and objective functional, an ADAM-type iteration scheme is proposed to compute the approximated optimal control, which equips the stochastic gradient iteration with modified momentum acceleration and adaptive step size. We provide theoretical bounds on both the discretization and iteration error of the proposed numerical method. Our method is tested on two examples, and the numerical results show the efficiency and performance of the method.

Multiobjective optimal control problems. Stationary Navier–Stokes equations

This paper deals with the solution of some multi-objective optimal control problems for stationary Navier–Stokes equations. More precisely, we look for Pareto and Nash equilibria associated to standard cost functionals. First, we prove the existence of equilibria and we deduce appropriate optimality systems. Then, we analyse the existence and characterization of Pareto and Nash equilibria for the Navier–Stokes equations. Here, we use the formalism of Dubovitskii and Milyoutin., see [Girsanov FV. Lectures on mathematical theory of extremum problems. Berlin: Springer-Verlag; 1972. (Notes in economics and mathematical systems; vol. 67)]. Finally, we also present a finite element approximation of the bi-objective problem, we illustrate the techniques with several numerical experiments and we compare the Pareto and Nash equilibria.

A gradient-based optimization algorithm to solve optimal control problems with conformable fractional-order derivatives

In this paper, we solve numerically a class of fractional optimal control problems in the conformable sense. We first propose a nonlinear fractional optimal control problem subject to general equality and inequality constraints. Then, based on a novel numerical integration scheme, we present a discretization method for the governed fractional-order system as well as the cost and constraint functionals, which yields a discretized approximate problem. We further derive the gradients of the cost and constraint functionals in regard to the discretized controls. On the basis of this result, a numerical optimization approach to solve the approximate problem is developed. Finally, three non-trivial example problems are solved to illustrate the applicability and effectiveness of the developed approach.

Verification of Taylor's theorem

Multivariate function calculus is an important part of mathematical analysis courses, and most conclusions can be found and generalized in univariate calculus. However, the biggest difficulty in teaching multivariate calculus lies in its abstraction, such as Taylors theorem, multiple integral regions drawing, and integral variable transformation. At the same time, ordinary differential equations are also one of the basic courses of the profession, and dynamic systems based on ordinary differential equations have extensive applications in mathematical models of continuity problems and optimal control problems. Software such as Mathematica, Python, Matlab, etc. can solve similar problems. Therefore, this article will use the visualization and computational capabilities of Mathematica to validate important definitions and conclusions in multivariate calculus, and compare the differences among the three software in solving approximate numerical solutions of dynamic systems of ordinary differential equations from different perspectives.

A variable-barrier-function-based prescribed finite-time bounded-[formula omitted] reinforcement learning optimal tracking control strategy without dependence on initial condition

The prescribed finite-time bounded-H∞ optimal tracking control (OTC) problem is investigated for a class of nonlinear systems with unknown initial tracking condition and external disturbances in this paper. A novel variable barrier function is designed to develop a new approach of prescribed performance control (PPC) which is irrelevant to the initial condition of the constrained variable in the system. Meanwhile, the reinforcement learning (RL) method with the actor–critic architecture is used to optimize the control input of the system in order to obtain the least energy consumption. By means of a newly proposed prescribed finite-time performance function (PFTPF), a prescribed finite-time optimal tracking controller is obtained regardless of initial tracking condition of the system. At the same time, the bounded-H∞ control problem is considered for the external disturbances in the system. The designed controller not only ensures the boundedness of all signals in the closed-loop system, but also has a finite-time control performance and an H∞ anti-interference performance. Finally, the effectiveness and the superiority of this proposed method are verified by the simulations.

Distributed and Multi-Agent Reinforcement Learning Framework for Optimal Electric Vehicle Charging Scheduling

The increasing number of electric vehicles (EVs) necessitates the installation of more charging stations. The challenge of managing these grid-connected charging stations leads to a multi-objective optimal control problem where station profitability, user preferences, grid requirements and stability should be optimized. However, it is challenging to determine the optimal charging/discharging EV schedule, since the controller should exploit fluctuations in the electricity prices, available renewable resources and available stored energy of other vehicles and cope with the uncertainty of EV arrival/departure scheduling. In addition, the growing number of connected vehicles results in a complex state and action vectors, making it difficult for centralized and single-agent controllers to handle the problem. In this paper, we propose a novel Multi-Agent and distributed Reinforcement Learning (MARL) framework that tackles the challenges mentioned above, producing controllers that achieve high performance levels under diverse conditions. In the proposed distributed framework, each charging spot makes its own charging/discharging decisions toward a cumulative cost reduction without sharing any type of private information, such as the arrival/departure time of a vehicle and its state of charge, addressing the problem of cost minimization and user satisfaction. The framework significantly improves the scalability and sample efficiency of the underlying Deep Deterministic Policy Gradient (DDPG) algorithm. Extensive numerical studies and simulations demonstrate the efficacy of the proposed approach compared with Rule-Based Controllers (RBCs) and well-established, state-of-the-art centralized RL (Reinforcement Learning) algorithms, offering performance improvements of up to 25% and 20% in reducing the energy cost and increasing user satisfaction, respectively.

Predictive functional control for chamber pressure: A quantum simultaneous whale optimization approach

This study introduces a Constrained Predictive Functional Controller (CPFC) designed for regulating chamber pressure in a coke furnace. While the traditional controllers face challenges posed by the system’s complex multi-input multi-output (MIMO) structure, cross-coupling, time delays, physical constraints, and uncertainties, this article’s approach realized set-point tracking and disturbance rejection, addressing model mismatches while considering constraints on system inputs, input variations, and outputs. To tackle the constrained Model Predictive Control (MPC) optimization problem and minimize the cost function, the Quantum Simultaneous Whale Optimization Algorithm (QSWOA) is employed. This meta-heuristic optimization method integrates quantum coding and simultaneous search with the whale optimization algorithm, enhancing convergence speed and precision. Furthermore, the performance of the proposed CPFC-QSWOA controller is assessed through simulations of the chamber pressure control of a Coke furnace. Results demonstrate the superiority of the CPFC-QSWOA approach, showcasing high precision and robustness, particularly in the face of uncertainties and disturbances.

Symmetry reduction and recovery of trajectories of optimal control problems via measure relaxations

We address the problem of symmetry reduction of optimal control problems under the action of a finite group from a measure relaxation viewpoint. We propose a method based on the moment-SOS (Sum of Squares) aka Lasserre hierarchy which allows one to significantly reduce the computation time and memory requirements compared to the case without symmetry reduction. We show that the recovery of optimal trajectories boils down to solving a symmetric parametric polynomial system. Then we illustrate our method on the symmetric integrator and the time-optimal inversion of qubits.

Optimal Relaxed Control for a Decoupled G-FBSDE

AbstractIn this paper we study a system of decoupled forward-backward stochastic differential equations driven by a G-Brownian motion (G-FBSDEs) with non-degenerate diffusion. Our objective is to establish the existence of a relaxed optimal control for a non-smooth stochastic optimal control problem. The latter is given in terms of a decoupled G-FBSDE. The cost functional is the solution of the backward stochastic differential equation at the initial time. The key idea to establish existence of a relaxed optimal control is to replace the original control problem by a suitably regularised problem with mollified coefficients, prove the existence of a relaxed control, and then pass to the limit.

Modelling the dynamics of syphilis infection with personal protection and treatment as optimal control strategies

Syphilis is a sexually transmitted infection which when left untreated wouldlead to major health problems. Syphilis can easily be contracted by direct contact withSyphilis sore during vaginal, anal, or oral sex. Syphilis can also be passed on froman infected mother to her unborn child. In this paper, a nonlinear deterministic modelof Syphilis disease was constructed to determine the dynamics of Syphilis infections.The study deduced model’s equilibria and analyzed the local and global stability ofthese equilibria. The model was extended to optimal control problem by adding timedependent controls that helped characterize a range of possible controls that minimizedthe disease. The control system was solved qualitatively and numerically to evaluatethe effectiveness of the considered controls using Pontryagin’s Maximum Principle.The analysis indicated that strategies B and C are considered most effective as theysubstantially minimized the exposed, asymptomatic and symptomatic infectious. Werecommend that stakeholders should consider strategy B and C in their effort to mitigate the disease from the population as they all have the same effect of substantiallyminimizing the exposed, symptomatic and asymptomatic populations.

The influence of prevention and isolation measures to control the infections of the fractional Chickenpox disease model

In this paper, we propose a mathematical model using the Caputo fractional derivative (CFD) and two control signals to study the transmission dynamics and control of Chickenpox (Varicella) outbreak. The model consists of six compartments representing susceptible, vaccinated, exposed, infected with complications, infected without complications, and recovered individuals. We analyze the theoretical properties of the model, including existence, uniqueness, and boundedness of solutions, and calculate the basic reproduction number (R0). We identify equilibrium points and establish conditions for their stability. Sensitivity analysis helps identify the most influential parameters on R0. We formulate a fractional optimal control problem (FOCP) by incorporating time-dependent prevention and isolation measures. The necessary optimality conditions are derived using Pontryagin’s maximum principle. Numerical simulations based on the Adams–Bashforth-Moulton (ABM) method illustrate the impact of control measures and fractional order on disease propagation. The results highlight the effectiveness of optimal controls and fractional order in understanding and managing epidemics, enhancing stability conditions. The study contributes to a better understanding of Chickenpox transmission dynamics and provides insights for disease control and management, aiding decision-makers and governments in taking preventive measures.

A new method for solving fractional optimal control problems

In this article, we present a novel approach that utilizes the operational matrix method to solve fractional optimal control problems. We address the challenge of missing initial conditions for the co-state function by incorporating the linear shooting method. Our objective is to provide a method that not only enhances computational efficiency and reduces cost but also improves accuracy and usability. With our new approach, we can calculate the coefficients of the expansion solution without the need to solve an algebraic system. These coefficients can be generated explicitly or through an iterative process. We provide a proof of the uniform convergence of the approximate series of functions to the unique solution of the optimal control problem. To demonstrate the effectiveness of our proposed method, we solve several numerical examples and compare the results with those obtained by other researchers. Additionally, we extend the application of this new method to solve problems with multiple states, thereby expanding its scope to a wider range of problem domains. This allows us to address diverse scenarios using the same efficient and accurate approach. Through our research, we contribute to the development of a powerful and versatile method for solving fractional optimal control problems, offering significant advantages over existing techniques.

Global stability and optimal control of an age-structured SVEIR epidemic model with waning immunity and relapses.

The efficacy of vaccination, incomplete treatment and disease relapse are critical challenges that must be faced to prevent and control the spread of infectious diseases. Age heterogeneity is also a crucial factor for this study. In this paper, we investigate a new age-structured SVEIR epidemic model with the nonlinear incidence rate, waning immunity, incomplete treatment and relapse. Next, the asymptotic smoothness, the uniform persistence and the existence of interior global attractor of the solution semi-flow generated by the system are given. We define the basic reproduction number and prove the existence of the equilibria of the model. And we study the global asymptotic stability of the equilibria. Then the parameters of the model are estimated using tuberculosis data in China. The sensitivity analysis of is derived by the Partial Rank Correlation Coefficient method. These main theoretical results are applied to analyze and predict the trend of tuberculosis prevalence in China. Finally, the optimal control problem of the model is discussed. We choose to take strengthening treatment and controlling relapse as the control parameters. The necessary condition for optimal control is established.

The implicit inversion method for calculating the forward dynamics input Jacobian

AbstractThis paper presents the implicit inversion method (IIM), a recursive method to evaluate the Jacobian of the forward dynamics w.r.t. the system inputs, using intermediate results obtained from an O(n) forward dynamics algorithm. The resulting coefficient matrix, called the inertia-weighted input matrix (IWIM), can be used to significantly improve the performance of solving optimal control problems that take into account system dynamics for only the current time step. As the relationship between inputs and accelerations appears fixed within a time step, this matrix can be evaluated in the initialization step of the optimization. This means that the forward dynamics only needs to be solved once at the initialization of the optimization, rather than having to solve the equations in every iteration of the optimization. The method presented in this paper especially targets a case where the forward dynamics are calculated using an O(n) method and takes advantage of variables that are already known through the evaluation of that method. These quantities allow us to obtain the inertia-weighted input matrix without having to convert the system to its generalized coordinate form. Exploiting the shape of the resulting equation, it is even possible to avoid an explicit inversion of any matrices in the process. Finally, runtime comparisons between three different methods to calculate the IWIM are made for several examples.

An Augmented Lagrangian Method for State Constrained Linear Parabolic Optimal Control Problems

An Augmented Lagrangian Method for State Constrained Linear Parabolic Optimal Control Problems