M. H. NOORI SKANDARI ET AL.
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Table 1. Solutions comparison of the Ex. 10.
Table 2. Solutions comparison of the Ex. 11.
Discretization method 
Objective value –0.1808 –0.1830
CPU Times (Sec) 95.734 0.125
Table 3. Solutions comparison of the Ex. 12.
N =100 Discretization method  Presented ap p ro ach
Objective value –0.0261 –0.0434
CPU Times (Sec)
The optimal controls
(12) is shown in Figure 9. Here, the value of optimal
solution of objective function is –0.0435.
In this paper, we proposed a different approach for solv-
ing a class of nonlinear optimal control problems which
have a linear functional and nonlinear dynamical control
system. In our approach, the linear combination property
of intervals is used to obtain the new corresponding pro-
ble m wh ich is a linear optimal control problem. The new
problem can be converted to an LP problem by discrete-
zation method. Finally, we obtain an approximate solu-
tion for the main problem. By the approach of this paper
we may solve a wide class of nonlinear optimal control
 M. Diehl, H. G. Bock and J. P. Schloder, “A Real-Time
Iteration Scheme for Nonlinear Optimization in Optimal
Feedback Control,” Siam Journal on Control and Opti-
mization, Vol. 43, No. 5, 2005, pp.1714-1736.
 M. Diehl, H. G. Bock, J. P. Schloder, R. Findeisen, Z.
Nagy c and F. Allgower, “Real-Time Optimization and
Nonlinear Model Predictive Control of Processes Go-
verned by Differential-Algebraic Equations,” Journal of
Process Control, Vol. 12, No. 4, 2002, pp. 577-585.
 M. Gerdts and H. J. Pesch, “Direct Shooting Method for
the Numerical Solution of Higher-Index DAE Optimal
Control Problems,” Journal of Optimization Theory and
Applications, Vol. 117, No. 2, 2003, pp. 267-294.
 H. J. Pesch, “A Practical Guide to the Solution of Real-
Life Optimal Control Problems,” 1994.
 J. A. Pietz, “Pseudospectral Collocation Methods for the
Direct Transcription of Optimal Control Problems,”
Master Thesis, Ric e University, Houston, 2003.
 O. V. Stryk, “Numerical Solution of Optimal Control
Problems by Direct Collocation,” International Series of
Numerical Mathematics, Vol. 111, No. 1, 1993, pp. 129-
 A. H. Borzabadi, A. V. Kamyad, M. H. Farahi and H. H.
Mehne, “Solving Some Optimal Path Planning Problems
Using an Approach Based on Measure Theory,” Applied
Mathematics and Computation, Vol. 170, N o. 2, 2005, pp.
 M. Gachpazan, A. H. Borzabadi and A. V. Kamyad, “A
Measure-Theoretical Approach for Solving Discrete
Optimal Control Problems,” Applied Mathematics and
Computation, Vol. 173, No. 2, 2006, pp. 736-752.
 A.V. Kamyad, M. Keyanpour and M. H. Farahi, “A New
Approach for Solving of Optimal Nonlinear Control
Problems,” Applied Mathematics and Computation, Vol.
187, No. 2, 2007, pp. 1461-1471.
 A. V. Kamyad, H. H. Mehne and A. H. Borzabadi, “The
Best Linear Approximation for Nonlinear Systems,” Ap-
plied Mathematics and Computation, Vol. 167, No. 2,
2005, pp. 1041-1061.
 K. P. Badakhshan and A. V. Kamyad, “Numerical Solu-
tion of Nonlinear Optimal Control Problems Using Non-
linear Programming,” Applied Mathematics and Compu-
tation, Vol. 187, No. 2, 2007, pp. 1511-1519.
 K. P. Badakhshan, A. V. Kamyad and A. Azemi, “Using
AVK Method to Solve Nonlinear Problems with Uncer-
tain Parameters,” Applied Mathematics and Computation,
Vol. 189, No. 1, 2007, pp. 27-34.
 W. Alt, “Approximation of Optimal Control Problems
with Bound Constraints by Control Parameterization,”
Control and Cybernetics, Vol. 32, No. 3, 2003, pp. 451-
 T. M. Gindy, H. M. El-Hawary, M. S. Salim and M.
El-Kady, “A Chebyshev Approximation for Solving Op-
timal Control Problems,” Computers & Mathematics with
Applications, Vol 29, No. 6, 1995, pp 35-45.
 H. Jaddu, “Direct Solution of Nonlinear Optimal Control
Using Quasilinearization and Chebyshev Polynomials
Problems,” Journal of the Franklin Institute, Vol. 339,
No. 4-5, 2002, pp. 479-498.
 G. N. Saridis, C. S. G. Lee, “An Approximation Theory
of Optimal Control for Trainable Manipulators,” IEEE
Transations on Systems, Vol. 9, No. 3, 1979, pp. 152-159.
 P. Balasubramaniam, J. A. Samath and N. Kumaresan,
“Optimal Control for Nonlinear Singular Systems with
Quadratic Performance Using Neural Networks,” Applied
Mathematics and Computation, Vol. 187, No. 2, 2007, pp.
 T. Cheng, F. L. Lewis, M. Abu-Khalaf, “A Neural Net-
work Solution for Fixed-Final Time Optimal Control of