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

Sinc numerical methods are essential approaches for solving nonlinear PROBLEMs. In this work, based on this method, the sinc neural networks (SNNs) are designed and applied to solve the FRACTIONAL OPTIMAL CONTROL PROBLEM (FOCP) in the sense of the Riemann–Liouville (RL) derivative. To solve the FOCP, we first approximate the RL derivative using Grunwald–Letnikov operators. Then, according to Pontryagin’s minimum principle for FOCP and using an error function, we construct an unconstrained minimization PROBLEM. We approximate the solution of the ordinary differential equation obtained from the Hamiltonian condition using the SNN. Simulation results show the efficiencies of the proposed approach.

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 research, an efficient numerical method is presented for solving a class of nonlinear DELAY FRACTIONAL OPTIMAL CONTROL PROBLEMs with inequality constraints on the state and CONTROL variables. The proposed approach is based on the hybrid of block-pulse functions and FRACTIONAL-order Legendre functions. By using the operational matrices of DELAY and derivative associated with the hybrid functions, the original OPTIMAL CONTROL PROBLEM is transformed into a parameter optimization one. The numerical results, demonstrate the accuracy and validity of the suggested method.

This study presents an algorithm for solving OPTIMAL CONTROL PROBLEMs with the objective function of the Lagrange-type and multiple DELAYs on both the state and CONTROL variables of the constraints, with bounds on the CONTROL variable. The full discretization of the objective functional and the multiple DELAY constraints is carried out by using the Simpson numerical scheme. The discrete recurrence relations generated from the discretization of both the objective functional and constraints are used to develop the matrix operators, which satisfy the basic spectral properties. The primal-dual residuals of the algorithm are derived in order to ascertain the rate of convergence of the algorithm, which performs faster when relaxed with an accelerator variant in the sense of Nesterov. The direct numerical approach for handling the multi-DELAY CONTROL PROBLEM is observed to obtain an accurate result at a faster rate of convergence when over-relaxed with an accelerator variant. This research PROBLEM is limited to linear constraints and objective functional of the Lagrange-type and can address real-life models with multiple DELAYs as applicable to quadratic optimization of intensity modulated radiation theory planning. The novelty of this research paper lies in the method of discretization and its adaptation to handle linearly and proximal bound-constrained program formulated from the multiple DELAY OPTIMAL CONTROL PROBLEMs.

This paper considers an OPTIMAL sliding mode CONTROL based on the cost CONTROL guaranteed approach using the linear quadratic regulator method to stabilize DELAY FRACTIONAL under involved disturbance. We propose an approach to an open research PROBLEM in the design of an LMI-based sliding mode CONTROLler in which there are some constraints such as optimizing system performance. The sliding mode technique is well-known as an effective tool for calculating the transient response of the system and achieving robust system performance. LQR classic techniques are less effective for studying an OPTIMAL FRACTIONAL system in the presence of disturbance due to nonlinearity, so we use the OPTIMAL sliding mode approach CONTROL law designed for the nominal system and, then, combined it with a FRACTIONAL sliding mode CONTROLler. By using the Razumikhin theorem for the stability of FRACTIONAL order systems with DELAY and linear matrix inequality, conditions on asymptotically stabilization were obtained . The presented CONTROLler stabilizes the nominal system and guarantees an adequate level of system performance. The sliding mode CONTROLler presented in the article, in addition to eliminating the effect of disturbance in the system, is independent of the DELAY A numerical example was provided to illustrate the effectiveness of the main results.

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.