obta ining
( )
.
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
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.
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
,
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
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
,
0,1,2, ,100j=
of prob-
lem (11) is shown in Figure 6. Here, The value of op-
timal solution of objective function is0.1829.
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.
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
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.
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
,
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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