This paper presents a quadratic numerical scheme for a class of fractional optimal control problems (FOCPs). The fractional derivative is described in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of fractional differential equations. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. For a linear system, this method results into a set of linear simultaneous equations, which can be solved using a direct or an iterative scheme. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs.

1.
Bagley
,
R. L.
, and
Torvik
,
P. J.
, 1983, “
A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
.”
J. Rheol.
0148-6055,
27
, pp.
201
210
.
2.
Mainardi
,
F.
, 1997. “
Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics
,”
Fractals and Fractional Calculus in Continuum Mechanics
,
A.
Carpinteri
and
F.
Mainardi
, eds.,
Springer-Verlag
,
New York
, pp.
291
348
.
3.
Oldham
,
K. B.
, and
Spanier
,
J.
, 1974,
The Fractional Calculus
,
,
New York
.
4.
Agrawal
,
O. P.
, 2004, “
Applications of Fractional Derivatives in Thermal Analysis of Disk Brakes
,”
Nonlinear Dyn.
0924-090X,
38
, pp.
191
206
.
5.
Agrawal
,
O. P.
, 2003, “
Response of a Diffusion-Wave System Subjected to Deterministic and Stochastic Fields
,”
ZAMM
0044-2267,
83
, pp.
265
274
.
6.
Oldham
,
K. B.
, and
Spanier
,
J.
, 1970, “
The Replacement of Ficks Law by a Formulation Involving Semidifferentiation
,”
J. Electroanal. Chem. Interfacial Electrochem.
0022-0728,
26
, pp.
331
341
.
7.
Magin
,
R. L.
, 2004, “
Fractional Calculus in Bioengineering
,”
Crit. Rev. Biomed. Eng.
0278-940X,
32
, pp.
1
104
.
8.
Magin
,
R. L.
, 2004, “
Fractional Calculus in Bioengineering, Part 2
,”
Crit. Rev. Biomed. Eng.
0278-940X,
32
, pp.
105
193
.
9.
Magin
,
R. L.
, 2004, “
Fractional Calculus in Bioengineering, Part 3
,”
Crit. Rev. Biomed. Eng.
0278-940X,
32
, pp.
194
377
.
10.
Oustaloup
,
A.
,
Levron
,
F. F. N.
, and
Mathieu
,
B.
, 2000, “
Frequency Band Complex Non Integer Differentiator: Characterization and Synthesis
,”
IEEE Trans. Circuits Syst., I: Fundam. Theory Appl.
1057-7122,
47
, pp.
25
40
.
11.
Xue
,
D.
, and
Chen
,
Y. Q.
, 2002. “
A Comparative Introduction of Four Fractional Order Controllers
,”
Proceedings of the Fourth IEEE World Congress on Intelligent Control and Automation (WCICA02)
,
IEEE
, pp.
3228
3235
.
12.
Manabe
,
S.
, 2003, “
Early Development of Fractional Order Control
,”
Proceedings of DETC2003
,
ASME
, ASME Paper No. DETC2003/VIB-48370.
13.
Carpinteri
,
A.
, and
Mainardi
,
F.
, 1997,
Fractals and Fractional Calculus in Continuum Mechanics
,
Springer-Verlag
,
New York
.
14.
Podlubny
,
I.
, 1999,
Fractional Differential Equations
,
,
San Diego, CA
.
15.
Hilfer
,
R.
, 2000,
Applications of Fractional Calculus in Physics
,
World Scientific
,
River Edge, NJ
.
16.
Miller
,
K. S.
, and
Ross
,
B.
, 1993,
An Introduction to the Fractional Calculus and Fractional Differential Equations
,
Wiley
,
New York
.
17.
Bryson
Jr.,
A. E.
, and
Ho
,
Y.
, 1975,
Applied Optimal Control: Optimization, Estimation, and Control
,
Blaisdell
,
Waltham, MA
.
18.
Sage
,
A. P.
, and
White
III,
C. C.
, 1977,
Optimum Systems Control
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
19.
Agrawal
,
O. P.
, 1989, “
General Formulation for the Numerical Solution of Optimal Control Problems
,”
Int. J. Control
0020-7179,
50
, pp.
627
638
.
20.
Gregory
,
J.
, and
Lin
,
C.
, 1992,
Constrained Optimization in the Calculus of Variations and Optimal Control Theory
,
Van Nostrand-Reinhold
,
New York
.
21.
Bode
,
H. W.
, 1945,
Network Analysis and Feedback Amplifier Design
,
Van Nostrand
,
New York
.
22.
Agrawal
,
O. P.
, 2004, “
A General Formulation and Solution Scheme for Fractional Optimal Control Problems
,”
Nonlinear Dyn.
0924-090X,
38
, pp.
323
337
.
23.
Agrawal
,
O. P.
, 2002, “
Formulation of Euler–Lagrange Equations for Fractional Variational Problems
,”
J. Math. Anal. Appl.
0022-247X,
272
, pp.
368
379
.
24.
Agrawal
,
O. P.
, and
Baleanu
,
D.
, 2007, “
A Hamiltonian Formulation and a Direct Numerical Scheme for Fractional Optimal Control
,”
J. Vib. Control
1077-5463,
13
, pp.
1269
1281
.
25.
Agrawal
,
O. P.
, 2004, “
Block-by-Block Method for Numerical Solution of Fractional Differential Equations
,”
Proceedings of the First IFAC Workshop on Fractional Differentiation and its Applications
,
Bordeaux
, July 19–21.
26.
Deithelm
,
K.
,
Ford
,
N. J.
, and
Freed
,
A. D.
, 2002, “
A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
,”
Nonlinear Dyn.
0924-090X,
29
, pp.
3
22
.
27.
Kumar
,
P.
, and
Agrawal
,
O. P.
, 2006, “
A Numerical Scheme for the Solution of Fractional Differential Equations of Order Greater Than 1
,”
ASME J. Comput. Nonlinear Dyn.
1555-1423,
1
, pp.
178
185
.