samaneh soradi-zeid

In this paper, two new methods are presented to find the optimal control of systems described by linear differential equations that have interval coefficients. In these methods, the given interval optimal control problem is transformed into a non-interval (deterministic) optimal control problem so that it is possible to use the optimal control theory to solve it. By considering the variation limits for the control signal, the optimal solution of the main problem is obtained.

In this research, two methods are presented to solve the deterministic sub-models of linear interval optimal control problems and find a good approximation for the optimal solution of them. In the first method, by using the center of intervals, the given interval optimal control problem becomes a deterministic optimal control problem and it is possible to use the Pontriagin conditions to solve it. By considering a resilience variable for the resulting solution, the limits of the control signal changes and the interval optimal solution for the main problem are obtained. In the second method, we first solve the optimal control problem for the beginning and the end of the interval, and at any time we consider the lowest value as the lower bound and the highest value as the upper bound of the interval solution. Then, by applying the constraints of the problem, we calculate the resilience of the state variable and the control signal. We also used Hokuhara's difference to reduce the error in interval calculations.

Solving optimization problems using the interval optimal control approach, show that the optimal solution of the submodel that is solved by any of these two methods is within the range of the solution of the main problem. In addition, the solutions of these problems are also presented as interval values.

Originality/Value:

 To obtain the numerical results, MATLAB software and INTLAB software package of interval calculations have been used. These approaches are categorized into indirect methods for solving optimal control problems, while it is far from their defects, e.g. curse of burdensome computational load, so that interval approach is applied to simply solve the problems.

The main aim of this paper is to find an optimal interval control law for quadratic linear problems under interval uncertainty. For this purpose, using Bellman's optimality principle and interval Hamilton-Jacobi-Bellman inequalities, the interval optimal control problem is transformed into a system of interval differential inequalities. These inequalities are called Riccati's differential inequalities, which is a result of the dynamic programming method. To solve this system of inequalities, we use inclusion relations and interval arithmetic. By this method, we can obtain the upper and lower bounds of the solutions. We also use Hokuhara's generalized difference to reduce errors in the interval arithmetic. We apply the presented method for solving some interval quadratic linear optimal control problems by using MATLAB software. The obtained results show the efficiency of the proposed method.

In this paper, we introduce a new direct scheme based on Dickson polynomials and collocation points to solve a class of optimal control problems (OCPs) governed by Volterra integro-differential equations namely Volterra integro-OCPs (VI-OCPs). This topic requires to calculating the corresponding operational matrices for expanding the solution of this problem in terms of Dickson polynomials. Further, the highlighted method allows us to transform the VI-OCP into a system of algebraic equations for choosing the coefficients and control parameters optimally. The error estimation of this technique is also investigated which given the high efficiency of the Dickson polynomials to deal with these problems. Finally, some examples are brought to confirm the validity and applicability of this approach in comparison with those obtained from other methods.

This paper is devoted to solve a class of differential equation with simultaneously combining variable coeffcients and variable delays namely variable-delay dierential equations (VDDEs). For this purpose, a numerical method is proposed in which the unknown function and its derivative are approximated with the basis of interpolating Multiquadric radial basis functions (MQRBFs) at arbitrary collocation points. According to the existing mechanism, the synchronization problem is recast to a system of algebraic equations. In the other hand, the proposed method provides a very adjustable framework for approximation according to the discretization and due to a board range of arbitrary nodes. Finally, some illustrative examples are given to verify the validity and applicability of the new technique and also a comparison betweenour results and the existing studies is performed.

The present paper aims to get through a class of fractional optimal control problems (FOCPs). Furthermore, the fractional derivative portrayed in the Caputo sense through the dynamics of the system as fractional differential equation (FDE). Getting through the solution, firstly the FOCP is transformed into a functional optimization problem. Then, by using known formulas for computing fractional derivatives of Legendre wavelets (LWs), this problem has been reduce to an equivalent system of algebraic equations. In the next step, we can simply solved this algebraic system. In the end, some examples are given to bring about the validity and applicability of this technique and the convergence accuracy.

This paper is devoted to solve a set of non-linear optimal control problems which are touched with time-delay  Fredholm integro-differential equations. The serious objective of this work  is to contribute  an appropriate   direct scheme for solving these problems. The technique used  in this paper  is based upon the Dickson polynomials and collocation points. Getting through the solutions, the states and controls variables can be approximated with  Dickson polynomials. Therefore, the  optimal control problem with  time-delay   integro-differential equation   transforms  into a system of algebraic equations that by solving it, we can obtain the unknown coefficients of the main problem. The residual  error estimation of this technique is also investigated. Accuracy amount of the absolute errors have been studied for the performance of this method by solving several non-trivial examples.

This paper introduces an efficient numerical scheme  for solving a significant class of  nonlinear parabolic integro-differential equations (PIDEs). The major contributions made in this paper are applying a direct approach based on a combination of   group preserving scheme (GPS) and  spectral meshless radial point interpolation (SMRPI) method   to transcribe the partial differential problem under study into a system of ordinary differential equations (ODEs). The resulting problem is then solved by employing the numerical method of lines, which is also a well-developed numerical method.   Two numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

Due to the easy adaption of radial basis functions (RBFs), a directRBF collocation method is considered to develop an approximate scheme to solvefractional delay differential equations (FDDEs). The method of RBFs is a method of scattered data interpolation that has many application in different fields. In spite of easy implementation of other high-order methods and finite difference schemes for solving a problem of fractional order derivatives, the challenge of these methods is their limited accuracy, locality, complexity and high cost of computing in discretization of the fractional terms, which suggest that global scheme such as RBFs that are more accurate way for discretizing fractional calculus and would allow us to remove the ill-conditioning of the system of discrete equations. Applications to a variety ofproblems confirm that the proposed method is slightly more efficient than thoseintroduced in other literature and the convergence rate of our approach is high.

In this research, orthoferrite neodymium nanoparticles (NdFeO3) synthesized via the Co-precipitation method and the oleic acid )C17H33COOH) was added as surfactant to control the particles size. The spherical NdFeO3 nanoparticles with an average particles size of 28 nm were obtained in calcination temperature of 800 °C for 4h. Co-precipitation method is a simple and inexpensive method for the production of mixed oxides with high purity. The morphology, lattice parameters and size of nanoparticles of product have been studied and characterized by Fourier transform infrared spectroscopy (FT-IR), X-ray diffraction (XRD), scanning electron microscopy (SEM) techniques. The XRD analysis shows only the pattern corresponding to perovskite-type NdFeO3 which crystallizes in the orthorhombic system. The average particles size of the nanoparticles was measured both by XRD and SEM and the results were in very good agreement with each other. Also, the results indicate that the amount of surfactant, pH and rate of stirring of solution has an important role in the homogeneity and particles size of product.

In this paper, the optimal conditions for fractional optimal control problems (FOCPs) were derived in which the fractional differential operators defined in terms of Caputo sense and reduces this problem to a system of fractional differential equations (FDEs) that is called twopoint boundary value (TPBV) problem. An approximate solution of this problem is constructed by using the Legendre-Gauss collocation method such that the exact boundary conditions are satisfied. Several example are given and the optimal errors are obtained for the sake of comparison. The obtained results are shown that the technique introduced here is accurate and easily applied to solve the FOCPs.