Introduction optimal control problems occur in engineering, science and many other fields. An optimal control problem is a problem of optimization of an objective functional on a set of state and control variables, which is called the performance index, subject to dynamic constraints on the states and controls. In the case that the dynamic constraints include delay fractional differential equation, the problem is called a delay fractional optimal control problem. In this paper, we consider the following optimal control problem ....

An efficient direct and numerical method has been proposed to approx-imate a solution of time-delay fractional optimal control problems. First, a class of discrete orthogonal polynomials, called Hahn polynomials, has been introduced and their properties are investigated. These properties are em-ployed to derive a general formulation of their operational matrix of fractional integration, in the Riemann{Liouville sense. Then, the fractional derivative of the state function in the dynamic constraint of time-delay fractional op-timal control problems is approximated by the Hahn polynomials with un-known coefficients. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration time-delay fractional optimal control prob-lems into an algebraic system. Some illustrative examples are given and the obtained numerical results are compared with those previously published in the literature.

In this paper, a new approach based on fuzzy systems is used for solving variable-order fractional delay differentialalgebraic equations. The fractional derivatives are considered in the Atangana-Baleanu sense that is a new derivativewith the non-singular and non-local kernel. By relying on the ability of fuzzy systems in function approximation,the fuzzy solutions of variables are substituted in variable-order fractional delay differential algebraic equations. Theobtained algebraic equations system is then transformed into an error function minimization problem. A learningalgorithm is used to achieve the adjustable parameters of fuzzy solutions. It is shown that the variable-order fractionaldelay optimal control problems can be reformulated as variable-order fractional delay differential algebraic equationsand solved by the proposed method. The efficiency and accuracy of the presented approach are assessed through someillustrative examples of the variable-order fractional delay differential algebraic equations

In this paper, a new numerical method for solving the fractional optimal control problem of the time delay is presented. The fractional integral and the fractional derivative are the Riemann-Liouville type and the Caputo type, respectively. In this method, the cardinal Hermite functions are used as a basis to approximate functions. Moreover, we obtain the fractional and delay integral operational matrices and use them to solve this optimal control problem. Using the collocation method, the problem leads to a system of algebraic equations, that is solved by Newton's iterative method. Finally, numerical examples are presented to investigate the efficiency of this method.

In this paper, we present a new method for solving fractional optimal control problems with delays in state and control. This method is based upon Bernstein polynomials basis and feedback control. The main advantage of feedback or closed-loop control is that one can monitor the effect of such control on the system and modify the output accordingly. In this work, we use Bernstein polynomials to transform the fractional time-varying multi-dimensional optimal control system with both state and control delays, into an algabric system in terms of the Bernstein coefficients approximating state and control functions. We use Caputo derivative of degree 0<a£1 as the fractional derivative in our work. Finally, some numerical examples are given to illustrate the effectiveness of this method.

This paper proposes and analyzes an applicable approach for numerically computing the solution of fractional optimal control-affine problems. The fractional derivative in the problem is considered in the sense of Caputo. The approach is based on a fractional-order hybrid of block-pulse functions and Jacobi polynomials. ‎First‎, ‎the corresponding Riemann-Liouville fractional integral operator of the introduced basis functions is calculated‎. ‎ Then, an approximation of the fractional derivative of the unknown state function is obtained by considering an approximation in terms of these basis functions‎. ‎ Next, ‎using the dynamical system and applying the fractional integral operator‎, ‎an approximation of the unknown control function is obtained based on the given approximations of the state function and its derivatives‎. ‎ Subsequently‎, ‎all the given approximations are substituted into the performance index‎. ‎Finally‎, ‎the optimality conditions transform the problem into a system of algebraic equations‎. ‎An error upper bound of the approximation of a function based on the fractional hybrid functions is provided‎. ‎The method is applied to several numerical examples‎, and ‎the experimental results confirm the efficiency and capability of the method.  Furthermore, they demonstrate a good agreement between the approximate and exact solutions‎. ‎

In this paper, we solve a class of fractional optimal control problems in the sense of Caputo derivative using Genocchi polynomials. At , rst we present some properties of these polynomials and we make the Genocchi operational matrix for Caputo fractional derivatives. Then using them, we solve the problem by converting it to a system of algebraic equations. Some examples are presented to show the e, ciency and accuracy of the method.

In this paper, we present an efficient method to solve linear time-delay optimal control problems with a quadratic cost function. In this regard, first, by employing the Pontryagin maximum principle to time-delay systems, the original problem is converted into a sequence of two-point boundary value problems (TPBVPs) that have both advance and delay terms. Then, using the continuous Runge–Kutta (CRK) method, the resulting sequences are recursively solved by the shooting method to obtain an optimal control law. This obtained optimal control consists of a linear feedback term, which is obtained by solving a Riccati matrix differential equation, and a forward term, which is an infinite sum of adjoint vectors, that can be obtained by solving sequences of delay TPBVPs by the shooting CRK method. Finally, numerical results and their comparison with other available results illustrate the high accuracy and efficiency of our proposed method.

The aim of this paper is to propose a new method for solving a calss of stochasticfractional optimal control problems. To this end, we introduce an equivalent form for the presented stochastic-fractional optimal control problem and prove that these problems have the same solution. Therefore, the corresponding Hamilton– Jacobi–Bellman (HJB) equation to the equivalent stochastic-fractional optimal control problem is presented and then the Hamiltonian of the system is obtained. Finally, by considering Sharpe ratio as a performance index, Merton’s portfolio selection problem is solved by the presented stochastic-fractional optimal control method. Moreover, for indicating the advantages of the proposed method, optimal pairs trading problem is simulated.