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Research Papers

# Fractional Optimal Control Problems With Specified Final Time

[+] Author and Article Information
Raj Kumar Biswas

Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721302, Indiarajkumar@ee.iitkgp.ernet.in

Siddhartha Sen

Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721302, India

J. Comput. Nonlinear Dynam 6(2), 021009 (Oct 28, 2010) (6 pages) doi:10.1115/1.4002508 History: Received December 17, 2009; Revised July 29, 2010; Published October 28, 2010; Online October 28, 2010

## Abstract

A constrained dynamic optimization problem of a fractional order system with fixed final time has been considered here. This paper presents a general formulation and solution scheme of a class of fractional optimal control problems. The dynamic constraint is described by a fractional differential equation of order less than 1, and the fractional derivative is defined in terms of Riemann–Liouville. The performance index includes the terminal cost function in addition to the integral cost function. A general transversility condition in addition to the optimal conditions has been obtained using the Hamiltonian approach. Both the specified and unspecified final state cases have been considered. A numerical technique using the Grünwald–Letnikov definition is used to solve the resulting equations obtained from the formulation. Numerical examples are provided to show the effectiveness of the formulation and solution scheme. It has been observed that the numerical solutions approach the analytical solutions as the order of the fractional derivatives approach 1.

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## Figures

Figure 1

State x(t) as a function of t for the fixed final state problem for different α(Δ:α=0.77,+:α=0.87,O:α=0.93,×:α=0.97,−:α=1),  N=100

Figure 2

Control u(t) as a function of t for the fixed final state problem for different α(Δ:α=0.77,+:α=0.87,O:α=0.93,×:α=0.97,−:α=1),  N=100

Figure 3

Convergence of x(t) for the fixed final state problem for α=0.85(Δ:N=20,+:N=40,O:N=60,×:N=80)

Figure 4

Convergence of u(t) for the fixed final state problem for α=0.85(Δ:N=20,+:N=40,O:N=60,×:N=80)

Figure 5

State x(t) as a function of t for the free final state problem for different α(×:α=0.5,O:α=0.75,+:α=0.88,∗:α=0.95,Δ:α=1,∇:α=1,analytical),  N=100

Figure 6

Control u(t) as a function of t for the free final state problem for different α(×:α=0.5,O:α=0.75,+:α=0.88,∗:α=0.95,Δ:α=1,∇:α=1,analytical),  N=100

Figure 7

State x(t) as a function of t for the free final state problem for different s(Δ:s=1,∗:s=5,O:s=20),  α=0.9

Figure 8

Control u(t) as a function of t for the free final state problem for different s(Δ:s=1,∗:s=5,O:s=20),  α=0.9

Figure 9

Convergence of x(1) for the free final state problem for different α(Δ:α=0.5,+:α=0.7,O:α=0.9,×:α=0.95)

Figure 10

Convergence of u(0) for the free final state problem for different α(Δ:α=0.5,+:α=0.7,O:α=0.9,×:α=0.95)

Figure 11

Convergence of x(t) for the free final state problem for α=0.8(Δ:N=20,+:N=40,O:N=60,×:N=80)

Figure 12

Convergence of u(t) for the free final state problem for α=0.8(Δ:N=20,+:N=40,O:N=60,×:N=80)

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