Numerical solution of time-delay optimal control problems by the operational matrix based on Hartley series

Abstract

In this paper, a numerical technique based on the Hartley series for solving a class of time-delayed optimal control problems (TDOCPs) is introduced. The main idea is converting such TDOCPs into a system of algebraic equations. Thus, we first expand the state and control variables in terms of the Hartley series with undetermined coefficients. The delay terms in the problem under consideration are expanded in terms of the Hartley series. Applying the operational matrices of the Hartley series including integration, differentiation, dual, product, delay, and substituting the estimated functions into the cost function, the given TDOCP is reduced to a system of algebraic equations to be solved. The convergence of the proposed method is extensively investigated. At last, the precision and applicability of the proposed method is studied through different types of numerical examples.

Optimal control of linear fractional-order delay systems with a piecewise constant order based on a generalized fractional Chebyshev basis

“…Many research works have been allocated to providing efficient numerical procedures for solving constant order fractional models. For example, a general formulation based on the Hamiltonian function associated with optimal control of fractional problems without delay (Agrawal 2004), a collocation method based on the Bessel functions (Tohidi and saberi 2015), two-dimensional Müntz-Legendre hybrid functions (Sabermahani et al 2020), Müntz-Legendre polynomials (Kheyrinataj and Nazemi 2020a), a hybrid of orthonormal Taylor polynomials (Marzban and Malakoutikhah 2019, a hybrid of the conventional Legendre polynomials (Marzban 2021a), combining fractional-order Legendre functions with the block-pulse functions (Marzban 2021b), fractional Chebyshev functions (Kheyrinataj and Nazemi 2020b), Genocchi polynimials (Chang et al 2018), a neural network scheme (Yavari and Nazemi 2019), Bernstein polynomials (Nemati 2018), two-dimensional Müntz-Legendre wavelets (Sabermahani 2020), Ritz’s method (Jahanshahi and Torres 2017), a hybrid method with the use of Hermite cubic spline multi-wavelets (Mohammadzadeh and Lakestani 2018), Bernoulli wavelets (Rahimkhani et al 2017), Euler–Lagrange equation (Rakhshan and Effati 2020), second Chebyshev wavelets technique (Baghani 2021), Legendre wavelet approach (Yuttanan et al 2021), Hartley series (Dadkhah and Mamehrashi 2021). The main idea and fundamental concepts of variable-order fractional operators have been extended and studied in the excellent works (Samko and Ross 1993) and (Lorenzo and Hartley 2002).…”

Operational matrix-based solution of optimal control problems using hybrid Chelyshkov polynomials

“…Understanding such growing needs, various numerical methods have been proposed to solve OCPs [6,15,16]. There are some types of OCPs that are considered in literature such as optimal control of time delay systems, fractional optimal control problems, two-dimensional optimal control problems, linear and nonlinear optimal control problems [9,1,18,10].…”