A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives

We propose a direct numerical method for the solution of an optimal control problem governed by a two-side space-fractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi-Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer-order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the Legendre-Gauss-Radau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided to demonstrate the efficiency and validity of the presented method. The results show that our method reaches the solutions with good accuracy and a low CPU time.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Vibration and Control', available from [http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised 03-Sept-2018; Accepted 12-Oct-201

A fractional B-spline collocation method for the numerical solution of fractional predator-prey models

We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost

Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control

This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating the fractional problem either by discretizing the fractional term or expanding the fractional derivatives as a series involving integer order derivatives. The former method, as a subclass of direct methods in the theory of calculus of variations, uses finite differences, Grunwald-Letnikov definition in this case, to discretize the fractional term. Any quadrature rule for integration, regarding the desired accuracy, is then used to discretize the whole problem including constraints. The final task in this method is to solve a static optimization problem to reach approximated values of the unknown functions on some mesh points. The latter method, however, approximates fractional problems by classical ones in which only derivatives of integer order are present. Precisely, two continuous approximations for fractional derivatives by series involving ordinary derivatives are introduced. Local upper bounds for truncation errors are provided and, through some test functions, the accuracy of the approximations are justified. Then we substitute the fractional term in the original problem with these series and transform the fractional problem to an ordinary one. Hereafter, we use indirect methods of classical theory, e.g. Euler-Lagrange equations, to solve the approximated problem. The methods are mainly developed through some concrete examples which either have obvious solutions or the solution is computed using the fractional Euler-Lagrange equation.Comment: This is a preprint of a paper whose final and definite form appeared in: Chapter V, Fractional Calculus in Analysis, Dynamics and Optimal Control (Editor: Jacky Cresson), Series: Mathematics Research Developments, Nova Science Publishers, New York, 2014. (See http://www.novapublishers.com/catalog/product_info.php?products_id=46851). Consists of 39 page

Energy efficiency optimization in MIMO interference channels: A successive pseudoconvex approximation approach

In this paper, we consider the (global and sum) energy efficiency optimization problem in downlink multi-input multi-output multi-cell systems, where all users suffer from multi-user interference. This is a challenging problem due to several reasons: 1) it is a nonconvex fractional programming problem, 2) the transmission rate functions are characterized by (complex-valued) transmit covariance matrices, and 3) the processing-related power consumption may depend on the transmission rate. We tackle this problem by the successive pseudoconvex approximation approach, and we argue that pseudoconvex optimization plays a fundamental role in designing novel iterative algorithms, not only because every locally optimal point of a pseudoconvex optimization problem is also globally optimal, but also because a descent direction is easily obtained from every optimal point of a pseudoconvex optimization problem. The proposed algorithms have the following advantages: 1) fast convergence as the structure of the original optimization problem is preserved as much as possible in the approximate problem solved in each iteration, 2) easy implementation as each approximate problem is suitable for parallel computation and its solution has a closed-form expression, and 3) guaranteed convergence to a stationary point or a Karush-Kuhn-Tucker point. The advantages of the proposed algorithm are also illustrated numerically.Comment: submitted to IEEE Transactions on Signal Processin