Control
Farzaneh Soufivand; Fahimeh Soltanian; Kamal Mamehrashi
Abstract
One of the most important classes of fractional calculus is the fractional optimal control problem (FOCP), which arises in engineering. This study presents a direct and efficient numerical method for solving a class of (FOCPs) in which the fractional derivative is in the Caputo sense and the dynamic ...
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One of the most important classes of fractional calculus is the fractional optimal control problem (FOCP), which arises in engineering. This study presents a direct and efficient numerical method for solving a class of (FOCPs) in which the fractional derivative is in the Caputo sense and the dynamic system includes the fractional- and integer-order derivatives. For this purpose, we use the operational matrix of fractional Riemann-Liouville integration based on the shifted Gegenbauer polynomials. First, the fractional- and integer-order derivatives in the given problem are approximated based on the shifted Gegenbauer polynomials with unknown coefficients. Then by substituting these approximations and the equation derived from the dynamic constraint into the cost functional, an unconstrained optimization problem is obtained. The main advantage of this approach is that it reduces the FOCP given to an unconstrained optimization problem and using the necessary optimality conditions yields a system of algebraic equations which can be easily solved by Newton’s iterative method. In addition, the convergence of the method is proved via several theorems. Finally, some numerical examples are presented to illustrate the validity and applicability of the proposed technique.
Control
Maryam Alipour; Pooneh Omidiniya
Abstract
This paper, proposes an approximate analytical method to solvea class of optimal control problems. This method is an enhancementof the variational iteration method (VIM) which is called modifi ed variational iteration method (MVIM) and eliminates all additional calculations in VIM, thus requires less ...
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This paper, proposes an approximate analytical method to solvea class of optimal control problems. This method is an enhancementof the variational iteration method (VIM) which is called modifi ed variational iteration method (MVIM) and eliminates all additional calculations in VIM, thus requires less time to do the calculations. In thisapproach, first, the optimal control problem is converted into a non-linear two-point boundary value problem via the Pontryagins maximum principle, and then we applied the MVIM method to solve this boundary value problem. This suggested method is suitable for a large class of non-linear optimal control problems that for the non-linear part of the problem, we used the Taylor series expansion. In the end, three examples are provided to demonstrate the simplicity and efficiency of the method. The numerical results of the proposed method versus other methods are presented in tables. All calculations were carried out using Mathematica software.
Control
Zahra Shabani; Haleh Tajadodi
Abstract
In this paper, a numerical technique is proposed to solve optimal control problems (OPCs) of Volterra integral equations (VIEs). We apply the linear B-spline polynomials to solve OPCs by VIEs. The B-spline function divides the interval into sub-intervals and then built a different approximating polynomial ...
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In this paper, a numerical technique is proposed to solve optimal control problems (OPCs) of Volterra integral equations (VIEs). We apply the linear B-spline polynomials to solve OPCs by VIEs. The B-spline function divides the interval into sub-intervals and then built a different approximating polynomial on each sub-interval. In this method, optimal trajectory and control functions are expanded in terms of B-spline functions. The linear B-spline operational matrix of integration and multiplication are utilized in the proposed method.The main characteristic this method is that by using the suggested numerical technique and the related operational matrices, optimal control problem governed by Volterra integral equations is converted to a system of equations. Suffice it to say that this scheme simplifies The main problems and also makes to obtain a good approximate solution for them. In the end, there are two illustrative examples which numerical results show the validity and applicability of our method.
