Delay Optimal Control Problems Research Articles

An iterative method to solve the time delay optimal control problem with bounded control variable

This paper presents an iterative approach for solving the optimal control problem (OCP) with time-delayed terms and bounded control variables. The projection function is used to construct the iterative method to approximate the control law. Employing the approximation of control law the approximation of state and the co-state variables are obtained. To the best of our knowledge, this paper is the first attempt to solve bounded and delayed OCPs with a projection function. The convergence theorem for the proposed method is also proven. Several numerical examples are added, showing the developed method's effectiveness to solve the time-delayed OCPs with bounded control variables.

Dynamics analysis and optimal control of delayed SEIR model in COVID-19 epidemic

The coronavirus disease 2019 (COVID-19) remains serious around the world and causes huge deaths and economic losses. Understanding the transmission dynamics of diseases and providing effective control strategies play important roles in the prevention of epidemic diseases. In this paper, to investigate the effect of delays on the transmission of COVID-19, we propose a delayed SEIR model to describe COVID-19 virus transmission, where two delays indicating the incubation and recovery periods are introduced. For this system, we prove its solutions are nonnegative and ultimately bounded with the nonnegative initial conditions. Furthermore, we calculate the disease-free and endemic equilibrium points and analyze the asymptotical stability and the existence of Hopf bifurcations at these equilibrium points. Then, by taking the weighted sum of the opposite number of recovered individuals at the terminal time, the number of exposed and infected individuals during the time horizon, and the system cost of control measures as the cost function, we present a delay optimal control problem, where two controls represent the social contact and the pharmaceutical intervention. Necessary optimality conditions of this optimal control problem are exploited to characterize the optimal control strategies. Finally, numerical simulations are performed to verify the theoretical analysis of the stability and Hopf bifurcations at the equilibrium points and to illustrate the effectiveness of the obtained optimal strategies in controlling the COVID-19 epidemic.

Operator theoretic approach in fractional‐order delay optimal control problems

This paper investigates the fractional optimal control problem with a single delay in the state by using the operator theoretic approach. In this approach, we first reduce the delay fractional dynamical system into an equivalent operator equation, and then, by providing sufficient conditions to the operators, the existence of an optimal pair is proved for the abstract system. The optimality system for the quadratic cost functional is derived by using the Frechet derivative. Then we relate the operator theoretic approach optimality system to a Hamiltonian system of Pontryagin's minimum principle. The primary goal of this article is to demonstrate the existence of an optimal pair for the fractional‐order delay dynamical system by using a functional minimization theorem in functional analysis. Likewise, the optimality systems for the fractional‐order delay dynamical system are derived by using the functional (either Gateaux or Frechet) derivatives. Finally, we provide two numerical examples that support our theoretical findings.

Shooting continuous Runge–Kutta method for delay optimal control problems

Shooting continuous Runge–Kutta method for delay optimal control problems

A novel high accurate numerical approach for the time-delay optimal control problems with delay on both state and control variables

<abstract><p>In this study, we intend to present a numerical method with highly accurate to solve the time-delay optimal control problems with delay on both the state and control variables. These problems can be seen in many sciences such as medicine, biology, chemistry, engineering, etc. Most of the methods used to work out time delay optimal control problems have high complexity and cost of computing. We extend a direct Legendre-Gauss-Lobatto spectral collocation method for numerically solving the issues mentioned above, which have some difficulties with other methods. The simple structure, convergence, and high accuracy of our approach are the advantages that distinguish it from different processes. At first, by replacing the delay functions of state and control variables in the dynamical method, we propose an equivalent system. Then discretizing the problem at the collocation points, we achieve a nonlinear programming problem. We can solve this discrete problem to obtain the approximate solutions for the main problem. Moreover, we prove the gained approximate solutions convergent to the exact optimal solutions when the number of collocation points increases. Finally, we show the capability and the superiority of the presented method by solving some numeral examples and comparing the results with those of others.</p></abstract>

Extended Chebyshev cardinal wavelets for nonlinear fractional delay optimal control problems

In this study, a new family of cardinal wavelets called the extended Chebyshev cardinal wavelets is introduced to investigate problems on arbitrary finite domains. These wavelets possess many beneficial properties, such as spectral (exponential) accuracy, cardinality and orthogonality. The classical and fractional derivative matrices together with the fractional integral matrix of these wavelets are obtained. Two formulations of delay optimal control problems with a dynamical system involved with fractional derivatives are introduced. Two direct approaches based on these wavelets, together with their fractional derivative and integral matrices, are adopted for solving such problems. The presented methods transform solving the problems under study into solving constrained minimisation problems by approximating the state and control variables via the expressed wavelets. Some numerical examples are provided to show the efficiency of the expressed wavelet methods.

Application of periodic Bernoulli polynomials in approximating the solution of linear time-delay optimal control problems

In this paper, a numerical method based on application of periodic Bernoulli polynomials for solving optimal control of linear time-delay systems with quadratic performance index is presented. Some important properties of Bernoulli polynomials are reviewed and a new operational matrix of delay based on these functions is introduced. Also an error bound for approximating a function with periodic Bernoulli polynomials is presented. Using operational matrices of differentiation, delay and the integration of the cross product of two Bernoulli polynomials, the linear time delay optimal control problem with quadratic objective function is converted to a nonlinear optimization problem. Finally, numerous illustrative examples are studied to show the efficiency and accuracy of the proposed method compared with some other methods.

Modified ADMM algorithm for solving proximal bound formulation of multi-delay optimal control problem with bounded control

This study presents an algorithm for solving Optimal Control Problems(OCPs) with objective function of the Lagrange - type and multiple delayson both the state and control variables of the constraints; with bounds on thecontrol variable. The full discretization of the objective functional and the multipledelay constraints was carried out using the Simpson numerical scheme.The discrete recurrence relations generated from the discretization of both theobjective functional and constraints were used to develop the matrix operatorswhich satisfy the basic spectral properties The primal-dual residuals of the algorithmwere derived in order to ascertain the rate of convergence of the algorithm;which performs faster when relaxed with an accelerator variant in the sense ofNesterov. The direct numerical approach for handling the multi-delay controlproblem was observed to obtain an accurate result at a faster rate of convergencewhen over-relaxed with an accelerator variant. This research problemis limited to linear constraints and objective functional of the Lagrange-typeand can address real-life models with multiple delays as applicable to quadraticoptimization of Intensity Modulated Radiation Theory (IMRT) planning. Thenovelty of this research paper lies in the method of discretization and its adaptationto handle linearly and proximal bound constrained program formulatedfrom the multiple delay optimal control problems.

A direct computational method for nonlinear variable‐order fractional delay optimal control problems

AbstractThis paper introduces a highly accurate operational matrix technique based upon the Chebyshev cardinal functions (CCFs) for solving a new category of nonlinear delay optimal control problems (OCPs) involving variable‐order (VO) fractional dynamical systems. The VO fractional derivatives are defined in the Caputo type. The main aim is converting such OCPs into systems of algebraic equations. Thus, we first expand the state and control variables in terms of the CCFs with undetermined coefficients. Then, by utilizing the cardinal property of these basis functions, the delay terms in the problem under consideration are expanded in terms of the CCFs. Thereafter, these expansions are substituted into the cost functional, dynamical system and delay conditions. Next, the cardinality of the CCFs together with their operational matrix (OM) of VO fractional derivative are employed to extract a nonlinear algebraic equation from the cost functional, and several algebraic equations from the dynamical system and delay conditions. Eventually, the method of the constrained extremum is employed by coupling the algebraic constraints yielded from the dynamical system and delay conditions with the algebraic equation extracted from the cost functional using a set of Lagrange multipliers. The precision of the method is studied through different types of numerical examples.

Fibonacci wavelets and their applications for solving two classes of time‐varying delay problems

SummaryIn this paper, a numerical method for solving time‐varying delay equations and optimal control problems with time‐varying delay systems is discussed. This method is based upon Fibonacci wavelets and Petrov‐Galerkin method. To solve these problems, first, the Fibonacci wavelets are presented. With the aid of operational matrices of integration and delay for Fibonacci wavelets and using Petrov‐Galerkin method and Newton's iterative method, we solve two classes of time‐varying delay problems, numerically. The approximate solutions achieved by this method satisfy all the initial conditions. In addition, an estimation of the error is given. Numerical results are included to demonstrate the accuracy and applicability of the present technique.

Müntz–Legendre neural network construction for solving delay optimal control problems of fractional order with equality and inequality constraints

In this paper, an artificial intelligence approach using neural networks is described to solve a class of delay optimal control problems of fractional order with equality and inequality constraints. In the proposed method, a functional link neural network based on the Muntz–Legendre polynomial is developed. The problem is first transformed into an equivalent problem with a fractional dynamical system without delay, using a Pade approximation. According to the Pontryagin’s minimum principle for optimal control problems of fractional order and by constructing an error function, the authors then define an unconstrained minimization problem. The authors use trial solutions for the states, Lagrange multipliers and control functions where these trial solutions are constructed by a single-layer Muntz–Legendre neural network model. The authors then exploit an unconstrained optimization scheme for adjusting the network parameters (weights and bias) and to minimize the computed error function. Some numerical examples are given to illustrate the effectiveness of the proposed method.

Linear optimal control of time delay systems via Hermite wavelet

To solve the time delay optimal control problem with quadratic performance index, a direct numerical method based on Hermite wavelet has been proposed in the present study. The idea is to convert the time delay optimal control problem into a quadratic programming problem. To do so, various time functions in the system are expanded as their truncated series and the properties of the operational matrices of integration, delay and product of two Hermite wavelet vectors are used as well. These matrices are utilized to reduce the solution of optimal control with time delay system, to the solution of a quadratic programming with linear constraints. Finally, three examples of time varying and time invariant coefficients are given to compare the results with some of the existing methods.

Fractional optimal control in transmission dynamics of West Nile virus model with state and control time delay: a numerical approach

In this paper, an optimal control for a novel fractional West Nile virus model with time delay is presented. The proposed model is governed by a system of fractional delay differential equations, where the fractional derivative is defined in the Grünwald–Letnikov sense. Stability analysis of fixed points is studied. Corresponding fractional optimal control problem, with time delays in both state and control variables, is formulated and studied. Two simple numerical methods are used to study the nonlinear fractional delay optimal control problem. The methods are standard finite difference method and nonstandard finite difference method. Comparative studies are implemented, it is found that the nonstandard finite difference method is better than the standard finite difference method.

Fractional power series neural network for solving delay fractional optimal control problems

ABSTRACTIn this paper, we develop a numerical method for solving the delay optimal control problems of fractional-order. The fractional derivatives are considered in the Caputo sense. The process begins with the assumption that the problem is first transformed into an equivalent problem with a fractional dynamical system without delay, using a Padé approximation. We then try to approximate the solution of the Hamiltonian conditions based on the Pontryagin minimum principle. The main feature is to implement nonlinear polynomial expansions in a neural network adaptive structure. The transfer functions of the employed neural network follow a fractional power series. The proposed technique does not use sigmoid or hyperbolic tangent nonlinear transfer functions commonly adopted in conventional neural networks at the output. Instead, linear transfer functions are employed which lead to explicit fractional power series formulae for the fractional optimal control problem. To do this, we use trial solutions for the states, Lagrange multipliers and control functions where these trial solutions are constructed by fractional power series neural network model. We then minimise the error function using an unconstrained optimisation scheme where weight parameters (or coefficients of the series) and biases associated with all neurons are unknown. Some numerical examples are given to illustrate the effectiveness of the proposed scheme.

A class of nonlinear optimal control problems governed by Fredholm integro-differential equations with delay

ABSTRACT This paper introduces an efficient numerical scheme for solving a significant class of nonlinear delay optimal control problem governed by Fredholm integro-differential equations. A direct approach based on a combination of block-pulse functions and orthonormal Taylor polynomials is utilised to transcribe the problem under study into a nonlinear programming one. The resulting optimisation problem is then solved by employing the method of Lagrange multipliers. An upper error bound for the hybrid functions with respect to the maximum norm is obtained. In addition, the convergence of the hybrid functions is established. Numerous numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

Optimal Control Problems with Time Delays: Constancy of the Hamiltonian

This paper concerns necessary conditions of optimality for optimal control problems with time delays in the state variable. It is well known that, when there are no time delays and the dynamics are autonomous, the standard necessary conditions in the form of a maximum principle can be supplemented by an extra condition, namely “constancy of the Hamiltonian" along optimal trajectories (and associated costate trajectories). This property, possibly supplemented by other invariance principles, has been used to investigate properties of optimal trajectories, such as solution regularity, without the need to solve the underlying extremal equations. In classical mechanics, for example, the constancy of the Hamiltonian condition can be used to derive a conservation of energy principle from Hamilton's principle of least action. While the maximum principle has been generalized to cover time delays, the validity of constancy of the Hamiltonian-type conditions has not been previously investigated. We provide the first “extra" optimality condition of this nature for autonomous, time delay optimal control problems. The new “constancy of the Hamiltonian" condition involves a correction term, without which the condition is not valid. We illustrate the significance of this condition by applications to minimizer regularity and conservation laws in nonclassical Hamiltonian mechanics.

Multiple-interval pseudospectral approximation for nonlinear optimal control problems with time-varying delays

• A multiple-interval pseudospectral scheme is developed for solving nonlinear optimal control problems with time-varying delays. • The size of mesh intervals in the multiple-interval approach can be various and the location of mesh points can be placed freely. • A novel and smart approach is presented for computing the values of state delay efficiently and stably. A multiple-interval pseudospectral scheme is developed for solving nonlinear optimal control problems with time-varying delays, which employs collocation at the shifted flipped Jacobi-Gauss–Radau points. The new pseudospectral scheme has the following distinctive features/abilities: (i) it can directly and flexibly solve nonlinear optimal control problems with time-varying delays without the tedious quasilinearization procedure and the uniform mesh restriction on time domain decomposition , and (ii) it provides a smart approach to compute the values of state delay efficiently and stably, and a unified framework for solving standard and delay optimal control problems. Numerical results on benchmark delay optimal control problems including challenging practical engineering problems demonstrate that the proposed pseudospectral scheme is highly accurate, efficient and flexible.

A class of infinite-horizon stochastic delay optimal control problems and a viscosity solution to the associated HJB equation

In this paper, we investigate a class of infinite-horizon optimal control problems for stochastic differential equations with delays for which the associated second order Hamilton−Jacobi−Bellman (HJB) equation is a nonlinear partial differential equation with delays. We propose a new concept for the viscosity solution including timetand identify the value function of the optimal control problems as a unique viscosity solution to the associated second order HJB equation.

Optimal control of mean-field jump-diffusion systems with noisy memory

ABSTRACTThis paper is concerned with an optimal control problem under mean-field jump-diffusion systems with delay and noisy memory. First, we derive necessary and sufficient maximum principles using Malliavin calculus technique. Meanwhile, we introduce a new mean-field backward stochastic differential equation as the adjoint equation which involves not only partial derivatives of the Hamiltonian function but also their Malliavin derivatives. Then, applying a reduction of the noisy memory dynamics to a two-dimensional discrete delay optimal control problem, we establish the second set of necessary and sufficient maximum principles under partial information. Moreover, a natural link between the above two approaches is established via the adjoint equations. Finally, we apply our theoretical results to study a mean-field linear-quadratic optimal control problem.

Numerical solution of time delay optimal control problems by hybrid of block-pulse functions and Bernstein polynomials

Numerical solution of time delay optimal control problems by hybrid of block-pulse functions and Bernstein polynomials