A Central Difference Numerical Scheme for Fractional Optimal Control Problems

By Dumitru Băleanu, Ozlem Defterli, Om Prakash Agrawal. 2008

212 citations recorded by the source corpus.

This paper presents a modified numerical scheme for a class of fractional optimal control problems where a fractional derivative (FD) is defined in the Riemann—Liouville sense. In this scheme, the entire time domain is divided into several sub-domains, and a FD at a time node point is approximated using a modified Grünwald—Letnikov approach. For the first-order derivative, the proposed modified Grünwald— Letnikov definition leads to a central difference scheme. When the approximations are substituted into the fractional optimal control equations, it leads to a set of algebraic equations which are solved using a direct numerical technique. Two examples, one time-invariant and the other time-variant, are considered to study the performance of the numerical scheme. Results show that 1) as the order of the derivative approaches an integer value, these formulations lead to solutions for the integer-order system, and 2) as the sizes of the sub-domains are reduced, the solutions converge. It is hoped that the present scheme would lead to stable numerical methods for fractional differential equations and optimal control problems.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/81064a2fd930c29eec4c8dee045ff2d3a5f90c31

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

An approximate method for solving fractional optimal control problems by hybrid functions (2018, unknown): “Email: razzaghi@math.msstate.edu Received: 31 March 2016; accepted: 1 August 2016 Baleanu et al., 2009; Tricaud and Chen, 2010; Yousefi et al., 2011).”
NECESSARY OPTIMALITY CONDITIONS FOR FRACTIONAL DIFFERENCE PROBLEMS OF THE CALCULUS OF VARIATIONS (2010, background): “The study of fractional problems of the Calculus of Variations and respective Euler-Lagrange equations is a fairly recent issue – see [1, 2, 5, 7, 8, 11, 13, 15, 16, 21] and references therein – and include only the continuous case.”
Suboptimal control of fractional-order dynamic systems with delay argument (2018, background): “The OCP reduces to a fractional optimal control problem (FOCP) when either the performance index or the dynamic constraints or both include at least one fractional-order derivative term (Agrawal and Baleanu, 2007; Baleanu et al., 2009; Ezz–Eldien et al., 2015).”
An iterative approach for solving fractional optimal control problems (2018, unknown): “Some numerical methods for solving some types of FOCPs can be seen in Agrawal and Baleanu (2007), Agrawal (2008a, b), Baleanu et al. (2009), Agrawal et al. (2010, 2012), Tricaud and Chen (2010), Lotfi et al. (2011), Yousefi et al. (2011), Jarad et al. (2012), Akbarian and Keyanpour (2013), Alipour…”
Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices (2013, background): “Recent contributions in this field have been reported by several authors (Agrawal, 2004; Agrawal and Baleanu, 2007; Baleanu et al., 2009; Lotfi et al., 2011).”
Fractional Optimal Control Problems with Several State and Control Variables (2010, background): “This necessity motivated Agrawal and co-workers to develop formulations and numerical schemes for fractional optimal control problems (FOCPs) (Agrawal, 2004 Agrawal and Baleanu, 2007 Baleanu et al., 2009).”
Non standard finite difference method for solving variable order fractional optimal control problems (2017, methodology): “…better agreement with other techniques of solutions used for the constant order derivatives as in Agrawal (1989, 2002, 2004), Agrawal and Baleanu (2008), Baleanu et al. (2009), Biswas and Sen (2009), Jafari and Trajadodi (2014, Khadr and Hendy (2012) Lotfi et al. (2011) and Tricaud and Chen (2010).”
A numerical solution for fractional optimal control problems via Bernoulli polynomials (2016, methodology): “The Hamiltonian system of equations for FOCPs has been solved approximately by Agrawal (2004, 2008a,b), Agrawal and Baleanu (2004), Baleanu et al. (2009), Tricaud and Chen (2010a,b), and Yousefi et al. (2011) for Riemann– Liouville and Caputo fractional derivatives, respectively.”
Fractional optimal control problems: a pseudo-state-space approach (2011, background): “Baleanu et al. (2009) proposed a central difference numerical scheme for the solution of FOCPs.”