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This paper deals with the Ritz spectral method to solve a class of fractional optimal control problems (FOCPs). The developed numerical procedure is based on the function approximation by the Bernstein polynomials along with fractional operational matrix usage. The approximation method is computationally consistent and moreover, has a good flexibility in the sense of satisfying the initial and boundary conditions of the optimal control problems. We construct a new fractional operational matrix applicable in the Ritz method to estimate the fractional and integer order derivatives of the basis. As a result, we achieve an unconstrained optimization problem. Next, by applying the necessary conditions of optimality, a system of algebraic equations is obtained. The resultant problem is solved via Newton's iterative method. Finally, the convergence of the proposed method is investigated and several illustrative examples are added to demonstrate the effectiveness of the new methodology.

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Purpose -The main purpose of this paper is to find convenient methods to solve the differential-algebraic equations which have great importance in various fields of science and engineering. Design/methodology/approach -The paper applies a semi-analytical approach, using both the homotopy analysis method (HAM) and the modified homotopy analysis method (MHAM) for finding the solution of linear and nonlinear DAEs. Findings -The results show that the new modification can effectively reduce computational costs and accelerates the rapid convergence of the series solution. Originality/value -Some high index DAEs are investigated to present a comparative study between the HAM and the MHAM.Keywords Linear and nonlinear differential-algebraic equations, Homotopy analysis method, Modified homotopy analysis method, Applications, Semi-analytical approach,

Fractional order differential equations accurately model dynamic systems and processes. In some of the fractional optimal control problems (FOCPs), due to the ambiguity in the initial conditions and the transfer of ambiguity to the solution, it is necessary to use fuzzy mathematics. In this paper, a numerical method is presented to approximate the solution for a class of Fuzzy Fractional Optimal Control Problems (FFOCPs) using the Legendre basis functions. The fuzzy fractional derivative is described in the Caputo sense. The performance index of an FFOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of Fuzzy Fractional Differential Equations (FFDEs). After obtaining Euler–Lagrange equations for FFOCPs and the necessary and sufficient conditions for the existence of solutions, using the definition of generalized Hukuhara differentiability (types I, II), the problem is considered in two cases. Then the distance function and an approach similar to the variational type along with the Lagrange multiplier method are used to formulate and solve the equations in a system. Time-invariant and time-varying examples are provided to assess the presented method. Numerical results show a similar trend for the state and control variables for various numbers of Legendre polynomials. Also, the convergence of state and control variables for the time-invariant system can be seen, and the same is true for control variables for the time-varying system.

A numerical method for solving a 2D optimal control problem (2DOCP) governed by a linear time-varying constraint is presented in this paper. The method is based upon the Bernstein polynomial basis. The properties of Bernstein polynomial functions are presented. These properties, together with the Ritz method, are then utilized to reduce the given 2DOCP to the solution of an algebraic system of equations. By solving this system, the solution of the proposed problem is achieved. The main advantage of this scheme is that the approximate solutions satisfy all initial and boundary conditions of the problem. We extensively discuss the convergence of the method. Finally, an illustrative example is included to demonstrate the validity and applicability of the new technique.

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