Numerical Solution of 1D and 2D Fractional Optimal Control of System via Bernoulli Polynomials

Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems

“…After these pioneer works, many other extensive researches have been done on the development of numerical methods for FOCPs. For instance, we can refer to Oustaloup recursive approximation [22], direct methods based on pseudo-state-space formulations of FOCP [36], spectral methods based on orthogonal polynomials and fractional operational matrices [37,38,39,40,41,42,43], Legendre multiwavelet collocation methods [44], direct methods based on Bernstein polynomials [45,46,47], nonstandard finite difference methods [48], linear programming approaches [49], integral fractional pseudospectral methods [50], direct methods based on Ritz's techniques [51,52], the epsilon-Ritz method [53], direct methods based on hybrid block-pulse with other basis functions [54,55], pseudospectral methods based on Legendre Müntz basis functions [56], dynamic Hamilton-Jacobi-Bellman methods [57], penalty and variational methods [58], control parameterization methods [59], differential and integral fractional pseudospectral methods [60], as well as other numerical techniques [61,62,63]. Efforts were also done to derive optimality conditions for special types of FOCPs, such as bang-bang FOCPs [64] and free final and terminal time problems [65,66].…”

Bernoulli Operational Matrix of Fractional Derivative for Solution of Fractional Differential Equations

“…Eqs. (31) and (32) generate (N − m + 1) and m linear equations, respectively. These linear equations can be solved for unknown coefficients of the vector C. Consequently, y(t) given in Eq.…”

“…In the present paper, we introduce a the operational matrix of derivative based on Bernoulli polynomials [31], for solving numerically linear and non-linear FDEs, using tau and collocation methods respectively.…”

A new operational matrix based on Boubaker wavelet for solving optimal control problems of arbitrary order

“…The application of fractional optimal control problems can be seen in engineering and physics and the aim of solving an optimal control problem is extremizing a cost function over an admissible set of control and state functions. Several numerical methods are applied to find an approximate solution to one-dimensional fractional optimal control problems, such as eigen functions method (Agrawal, 2008), rational approximation method (Tricaud and Chen, 2010), Legendre orthonormal basis method (Lotfi et al, 2013), Legendre operational technique (Bhrawy and Ezz-Eldien, 2016), Bernoulli polynomials method (Rabiei et al, 2018b), Hybrid of block-pulse functions and Bernoulli polynomials (Mashayekhi and Razzaghi, 2018), hybrid Chelyshkov functions (Mohammadi et al, 2018), fractional order Lagrange polynomials (Sabermahani et al, 2019), Adomian decomposition method (Alizadeh and Effati, 2018), Chebysheve collocation method (Rabiei and Parand, 2019), Grunwald-Letnikov, trapezoidal and Simpson fractional integral formulas (Salati et al, 2019), low dimensional approximations (Peng et al, 2019) and Spectral Galerkin approximation (Zhang and Zhou, 2019). But there are few researches devoted to two-dimensional problem especially in fractional area; for example, the authors in Nemati and Yousefi (2017) used the Ritz method to solve a class of these problems.…”