Solution of a specific class of nonlinear fractional optimal control problems including multiple delays

Abstract

SummaryThis research provides a new framework based on a hybrid of block‐pulse functions and Legendre polynomials for the numerical examination of a special class of scalar nonlinear fractional optimal control problems involving delay. The concepts of the fractional derivative and the fractional integral are employed in the Caputo sense and the Riemann‐Liouville sense, respectively. In accordance with the notion of the Riemann‐Liouville integral, we derive a new integral operator related to the proposed basis called the operational matrix of fractional integration. By employing two significant operators, namely, the delay operator and the integral operator connected to the hybrid basis, the system dynamics of the primal optimal control problem converts to another system involving algebraic equations. Consequently, the optimal control problem under study is reduced to a static optimization one that is solved by existing well‐established optimization procedures. Some new theoretical results regarding the new basis are obtained. Various kinds of fractional optimal control problems containing delay are examined to measure the accuracy of the new method. The simulation results justify the merits and superiority of the devised procedure over the existing optimization methods in the literature.

An accurate method for fractional optimal control problems governed by nonlinear multi-delay systems

“…A new delay fractional Euler–Lagrange equation has been developed in Rakhshan and Effati (2020). New efficient techniques have been introduced in Marzban and Malakoutikhah (2019); Marzban (2021a; 2021b). It is realized that it is either too hard or impossible to provide an analytic response for a nonlinear (DFOCP).…”

Suboptimal control synthesis for state and input delayed quadratic systems

“…Indeed, many researchers are actually proposing solution to some classes of Fredholm integro-differential equations by associating Legendre, 17 Chebychev, 18 or Bernstein polynomials 19 to BPFs basis. On the contrary, several mathematical problems related to fractional nonlinear differential equations had been tackled by the mean of Taylor series 20,21 or LPs 22 combined to the piecewise orthogonal basis.…”

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).…”