( )

.

λ

∗

.

4. Numerical Examples

Here, we use our approach to obtain approximate optimal

solutions of the following three nonlinear optimal control

problems by solving linear programming (LP) problem

(8), via simplex method [20]. All the problems are pro-

grammed in MATLAB and run on a PC with 1.8 GHz

and 1GB RAM. Moreover, comparisons of our solutions

with the method that argued in [11] are included in Tables

1, 2 and 3 respectively for each example.

Example 4.1: Consider the following nonlinear op-

timal control problem:

()()

( )()()( )

( )

[ ]

()( )

1

0

3

minsin 3d

subject tocos2tan,

8

01, 0,1

01, 10.

txt t

xttxtu tt

ut t

xx

π

π

=π− +

≤≤ ∈

= =

∫

(10)

Here,

( )( )()

3

,tan,sin3

8

htuu t ctt

π

=− +=π

and

( )

At

( )

cos 2 t=π

for

( )

[][]

,0,1 0,1tu∈×

. Thus by (4)

and (5) for a ll

[ ]

0,1t∈

( )

3

[0,1]

min tantan,

88

u

gtu tt

∈

π π

= −+=−+

( )( )

3

[0,1]

maxtantan .

8

u

wtu tt

∈

π

= −+=−

Hence

( )( )( )( )

tan tan.

8

twt gttt

β

π

=−=−+ +

M. H. NOORI SKANDARI ET AL.

Copyright © 2011 SciRes. AM

649

Let

100.N=

Then

0.01

δ

=

and

100

j

j

s=

for

0,1,2, ,100.j=

The optimal solutions

j

x

∗

and

j

λ

∗

,

0,1,2, ,100.j=

Of problem (10) is obtained by solving

problem (8) which is illustrated in Figures 1 and 2 re-

spectively. Here, the value of optimal solution of objec-

tive function is 0.0977. In addition, the corresponding

Equation (9) of this example is

( )( )

3

tan0,1,2, ,100

8

j jjjj

u ssgsj

βλ

∗∗

π

−+= +=

Theref ore for

0,1,2, ,100j=

( )( )

( )

( )

1/3

1

8tan ,

jjjj j

usgs s

βλ

∗− ∗

=− −−

π

The optimal control

*

j

u

,

0,1,2, ,100j=

of prob-

lem (10) is showed in F igure 3.

Example 4.2: Consider the following nonlinear op-

timal control problem:

( )

( )

( )()( )

( )

( )

[][]

()( )

1

0

1

min2 d

2

subject toln3,

1,1,0,1

00,1 0.8

t

etxtt

xttxtut t

ut t

xx

−

−

=− +++

∈− ∈

= =

∫

(11)

Figure 1. Optimal state

( )

*.x

of Ex. 4.1.

Figure 2. Corresponding optimal control

( )

*

.λ

of Ex. 4.1.

Figure 3. Optimal control

( )

*.u

of Ex. 4.1.

By relations (4) and (5) for

[ ]

0,1t∈

( )()

{ }

( )

[ 1,1]

minln3 ln 2,

u

gtu tt

∈−

=++ =+

( )()

{ }

( )

[ 1,1]

maxln3ln 4.

u

wtu tt

∈−

=++=+

Hence

( )( )( )() ()

ln 4ln 2twt gttt

β

=−=+− +

Let

100.N=

Then

0.01

δ

=

and

100

j

j

s=

for all

0,j=

1,2, ,100

. We obtain the optimal solutions

j

x

∗

and

j

λ

∗

,

0,1,2, ,100j=

of this problem by solving

corresponding problem (8) which is illustrated in Fig-

ures 4 and 5 respectively. In addition, by relation (9) the

corresponding

( )

.u

∗

of this example is

() ()

3,0,1,2, ,100

jj j

s gs

jj

ues j

βλ

∗

+

∗

=−− =

The optimal controls

*

j

u

,

0,1,2, ,100j=

of prob-

lem (11) is shown in Figure 6. Here, The value of op-

timal solution of objective function is –0.1829.

Example 4.3: Consider the following nonlinear op-

timal control problem:

( )

( )

( )

( )

( )

( )

( )

[][ ]

()( )

1

0

3

52sin(2 )

minsin 2d

subject to()e,

1,1 ,0,1

00.9,10.4

t

t

text t

xtt ttxtut

ut t

xx

−

π

π−

= −+−

∈− ∈

= =

∫

(12)

Since

()( )

3sin(2π)

,e

t

htu ut= −

is a non-smooth func-

tion, the methods that discussed in [2,6] cannot solve the

problem (14) correctly. However, by relations (4) and (5),

we have for all

[ ]

0,1t∈

:

( )( )

{ }

3sin(2 )sin(2 )

[ 1,1]

minee,

tt

u

gt ut

ππ

∈−

=−=−

M. H. NOORI SKANDARI ET AL.

Copyright © 2011 SciRes. AM

650

Figure 4. Optimal state

( )

*

.x

of Ex. 4.2.

Figure 5. Corresponding optimal control

( )

*.λ

of Ex. 4.2.

Figure 6. Optimal control

( )

*.u

of Ex. 4.2.

( )()

{ }

3sin(2 )

[ 1,1]

maxe 0,

t

u

wt ut

π

∈−

=−=

thus

( )( )( )

sin(2 )

e.

t

twtgt

β

π

=−=

Let

100N=

. Then

0.01

δ

=

and

100

j

j

s=

for all

Figure 7. Optimal state

( )

*

.x

of Ex. 4.3.

Figure 8. Corresponding optimal control

( )

*.λ

of Ex. 4.3.

Figure 9. Optimal control

( )

*

.u

of Ex. 4.3.

0,1,2, ,100j=

. We obtain the optimal solutions

j

x∗

and

,

j

λ

∗

0,1,2, ,100j=

of this problem by solving

corresponding problem (8), which is illustrated in Fig-

ures 7 and 8 respectively. In addition, by relation (9) the

corresponding

( )

.u

∗

of this example is

( )()

( )

( )

1

3

sin(2 )

e,0,1,2, ,100

j

s

jjjj

usgsj

βλ

−π

∗∗

=−+ =

M. H. NOORI SKANDARI ET AL.

Copyright © 2011 SciRes. AM

651

Table 1. Solutions comparison of the Ex. 10.

N =100

Discretization

method [11]

Presented

approach

Objective value

0.1180

0.0980

CPU Times (Sec)

5.281

0.047

Table 2. Solutions comparison of the Ex. 11.

N =100

Discretization method [11]

Presented approach

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 [11] Presented ap p ro ach

Objective value –0.0261 –0.0434

CPU Times (Sec)

6.680 0.078

The optimal controls

*

j

u

,

0,1,2, ,100j=

of problem

(12) is shown in Figure 9. Here, the value of optimal

solution of objective function is –0.0435.

5. Conclusions

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

problems.

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