Collocation method for solving nonlinear fractional optimal control problems by using Hermite scaling function with error estimates

Abstract

Summary This article presents an efficient numerical method for solving fractional optimal control problems (FOCPs) by utilizing the Hermite scaling function operational matrix of fractional‐order integration. The proposed technique is applied to transform the state and control variables into nonlinear programming (NLP) parameters at collocation points. The NLP solver is then used to solve FOCP. Furthermore, the L2‐error estimates in the approximation of unknown variables and the approximation of block pulse operational matrix of fractional‐order integration are derived and illustrative examples are included to demonstrate the applicability of the proposed method. Moreover, the results are compared with the Haar wavelet collocation method, hybrid of block‐pulse and Taylor polynomials method, Bernstein polynomials method, and the Boubaker hybrid function method to show the superiority of the proposed method.

Optimal control of a fractional diffusion Sturm-Liouville problem on a star graph

“…Equation ( 49) is true for any φ satisfying (47). Therefore we may take φ ∈ C c (0, T ) and hence (49) gives…”

Necessary optimality conditions for two dimensional fractional optimal control problems and error estimates for the numerical approximation

“…A FOCP-1D with cost as a fractional integral, called fractional Bolza cost, has been solve in Kumar and Mehra [12,13] by using the indirect method. In Kumar and Mehra [14], FOCP-1D has been consid-ered with inequality constraints and problem has been solved by using a direct method. FOCP-1D with time-varying delay system has been considered in Sabermahani et al [15,16].…”

“…Section 6 is concerned with the convergence analysis and the 𝐿 2 -error estimates in the approximation of unknown function and its partial derivative by shifted Legendre polynomials. In Section 7, the numerical algorithm has been developed to solve the necessary optimality conditions ( 11)- (14). Numerical examples are considered in Section 8 to illustrate our method and the paper is concluded in Section 9.…”

Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph

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