Intelligent Control and Automation, 2011, 2, 121-125
doi:10.4236/ica.2011.22014 Published Online May 2011 (http://www.SciRP.org/journal/ica)
A New Approach for a Class of Optimal Control Problems
of Volterra Integral Equations
Mohammad Hadi Noori Skandari1, Hamid Reza Erfanian2, Ali Vahidian Kamyad1
1Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi
University of Mashhad, Mashhad, Iran
2Department of Mathematics & Statistics, University of Science and Culture, Tehran, Iran
Received January 30, 2011; revised March 23, 2011; accep t e d M ay 5, 2011
Abstract
In this paper, we propose a new approach for a class of optimal control problems governed by Volterra inte-
gral equations which is based on linear combination property of intervals. We convert the nonlinear terms in
constraints of problem to the corresponding linear terms. Discretization method is also applied to convert the
new problems to the discrete-time problem. In addition, some numerical examples are presented to illustrate
the effectiveness of the proposed approach.
Keywords: Volterra Integral Equations, Optimal Control, Linear Programming
1. Introduction
Consider the following optimal control problem gov-
erned by Volterra integral equation (OCV):
    
0
minimize, ,d
T
GyTFtyt utt

(1)
  
0
subject to,,,d,
0
t
y
tpt Ktsysuss
tT


 (2)
where and are the state and control func-
tions respectively on . The integral Equation (2) is
applied in a natural way in the study of economic prob-
lems, population dynamics and etc., see for instance Hri-
tonenko and Yatsenco [1], and Kamien and Schwartz [2].
The OCV problem (1)-(2) has been studied by many au-
thors, including Neustadt [3-5], Bakke [6], Carlson [7],
Vinokurov [8], Medhim [9], Schmidt [10-13], Wolfans-
dorf [14], Elnagar, Kazemi and Kim [15], Pan and Teo
[16], Angell [17,18], Belbas [19,20], Carlier and Tah-
raoui [21], and Burnap, and Kazemi [22]. The method
usually employed for OCV problem (1)-(2) is method of
necessary conditions of the type of Pontryagin maximum
principle. In the Recent works, Vega [23] gives the nec-
essary condition for optimal terminal time of OCV prob-
lem (1)-(2) and verifies the terminal time T and final state
(.)y(.)u
[0, ]T

y
T by conditions
 
,0T

Ty

 and

 
,TyT 0


.
Bonnens and Vega [24] discuss problem (1)-(2) with
running state on the initial and final states. Also, Belbas
[25] applied the ideas of dynamic programming to OCV
problem (1)-(2).
In this paper, we are interested to solving the follow-
ing class of the OCV problem (1)-(2) which we called it
COCV problem:
 
0
minimize d
Tctyt t
 (3)
   
0
subject to,d,d,
0
t
y
tptfsus tsyss
tT


 




(4)
where function is a continuous function. A con-
trolled Volterra integral equation similar to equation (4)
is discussed in [16]. We suppose that
(.,.)f


ut U

,
[0, ]tT

where U a compact and connected set. In addi-
tion, we let the final state

y
T is a given known num-
ber. Here, the linear combination property of intervals is
used to convert nonlinear controlled Volterra integral
Equation (4) to the equivalent linear equation. The new
optimal control problem with this linear Volterra integral
equation is transformed to a discrete-time problem that
could be solved by linear programming methods. This
paper organized as follows. Section 2, transforms the
nonlinear function to a corresponding function
that is linear respect to a new control function. Section 3,
converts the new problem to the discrete-time problem
via discretization. In Section 4, numerical examples are
presented to illustrated effectivness of this approach.
(.f,.)

M. H. N. SKANDARI ET AL. 122
Finally, the conclusion of this paper is given in Section
5.
2. Metamorphosis of the COCV Problem
In this section, COCV problem (1) is transformed to the
new equivalent problem. First, we state and prove the
following two theorems:
Theorem 2.1: Let be a continuous function on
where U is a compact and connected subset of
, then for any arbitrary (but fixed)
(.,.)f
[0, ]TU
m
[0, ]
s
T the set
 
,:

f
suu U is a closed interval in .

Proof: Assume that [0, ]
s
T be given. Let


u 


,

f
su . Obviously, (.)

is a continuous function on U.
Since, continuous function preserve compactness and
connectedness, the set is compact and
connected. Therefore, is a closed inter-
val in.□
 
:uu 
 
:uu  


U
U

For any [0, ]
s
T, we may suppose the lower and
upper bounds of interval
 
,:
F
suu U are


g
s
and respectively. Thus we have:

ws
 

,,0,
g
sfsuwss T  (5)
In other words
   

min, :,0,
u
g
sfsuuUsT (6)
   

max, :,0,
u
wsf suuUsT (7)
Theorem 2.2: Let functions (.)
g
and be de-
fined by relations (6) and (7). Then they are uniformly
continuous on .
(.)w
[0,]T
Proof: We will show that (.)
g
is a uniformly con-
tinuous function. It is sufficient that we show that for any
given 0

, there exists 0

 such that if
12

s
Ns

 then



12
gsgs

 where


Nz

is a


(.,.f
neighborhood of . Since, any continuous
function on compact set is a uniformly continuous. The
function on compact set is a uni-
formly continuous, i.e. for any
z
)[0, ]TU
0

 there exists
0

, such that if


12

,,
s
uNs

u then

12
,u f,sufs

. Thus

fsu

12

,,suf

.
In addition, by (5), 11
 
,
g
sfsu and so


1
gs



2,fsu

. Now, by taking infimum on the right hand
side of the last inequality

gs

1
gs

2



gs


21
. By a simi-
lar procedure, we havegs

. Thus
 
12
gs gs

. The proof of uniformly continuity of
is similar. □ (.)w
By linear combination property of intervals and rela-
tion (5), we have for any [0, ]
s
T:


 



,fsuswsgss gss   ,[0,1]
,
(8)
Thus, we transform COCV problem (3)-(4) by relation
(6) as the following continuous-time problem:
 
0d
T
minimizectytt
 (9)
  
 
0,d
01,0,
t
s
ubject toytqthssdtsyss
ttT


 






where and
 
0d
t
qtptgs s






htwtgt
for any [0, ].tT

Note that in the new problem (9),
which is a optimal control of linear Volterra integral
equation, (.)

is the new control function. Next section,
converts the problem (9) to the corresponding linear pro-
gramming problem.
3. Discrete-Time Problem
Now, discretization method enables us transforming con-
tinuous problem (9) to the corresponding discrete form.
Consider equidistance points 012
0N
s
sssT 
of interval which defined as
[0, ]T
jT
s
j
N
, 0,1,,jN
where N is a given big number. Also, we set
j
j
ts

for 0,1,,jN

. By trapezoidal approximation in nu-
merical integration, problem (9) is converted to the fol-
lowing discrete-time problem:
1
00
1
minimize 22
N
jj NN
j
TT T
cycyc y
NNN






 (10)

1
00 0
1
00
00
subject to
122 2
,
2
01,,0,1, 2,,
j
j
jji jiijj
i
jj
jN
TTTT
dyhh dyh
jjj j
T
qdy
j
yqy jN










 


where


j
j
yyt,


j
j
hhs, ,

,
ijij
ddts


j
j
ccs,


j
j
t   and


j
j
qqt for all .
In problem (10), final state is
,0,1,2,,ij N

that is a known number.
By solving problem (10), which is a linear programming
problem, we are able to obtain the optimal solution
j


and
j
y

for all 0,1,2,,jN

. Note that, for evaluat-
ing optimal control variable , we must use the fol-
lowing equation:
(.u)


 
,.
f
suhsgs 


(11)
4. Numerical Examples
Here, we use our approach to obtain approximate optimal

M. H. N. SKANDARI ET AL.123
solution of the following two COCV problems by solv-
ing linear programming (LP) problem (10) via simplex
method [26] in MATLAB software.
Example 4.1: Consider the following COCV problem:

1
0
minimizecos 3πdtyt t
 (12)

  
 
0
subject to
π
sin 3πsin 4
cos3πd
00.5,0,1
(1) 1.
t
yttus s
tsys s
ut t
y

 




 

 


Here,

π
,sin
4
f
suu s




,



,cos3 πdtst s




cos 3πct t andfor all
 
sin 3πptt[0,1]t

,
and u. Thus by (6), (7) [0,1]s[0,0.5]
  
[0,0.5]
π
min sinsin,0,1
4
u
gsu sss









 
[0,0.5]
π
max sinsin,0,1
48
u
wsu sss








 
.
hence
  
0dsin3πcos1,0,1
t
qtptgs sttt 

  
π
sinsin ,0,1
8
hswsgsss s

 


Assume that and 100N100
jj
s for all 0,1,j

2, ,100. The optimal solutions
j
y and
j

, 0,1,j

of problem (12) is obtained by solving prob-
lem (10) which is illustrated in Figures 1 and 2 respec-
tively. Here, the value of optimal solution of objective
function is –0.470. The corresponding Equation (11) for
this example is
2, ,100
Figure 1. Optimal trajectory of Ex.12. (.)y
Figure 2. Corresponding optimal control of Ex.12. (.)


 
π
sin,0,1, 2,,100
4jjjj j
ushsgs j 


  
 



therefore
  


1
4sin,0,1, 2,,100
π
jjjjj
uhsgssj 
 

The optimal control *
j
u, of this
example is showed in Figure 3.
0,1, 2,,100j
Example 4.2: Consider the following COCV problem:

1
0
minimize 0.5dtyt t
 (13)
 




 
0
3
subject to3ln2d
01,0,1
(1) 1.
tts
yttu ssteyss
ut t
y
 



Here
 


,3lnfsuu ss
32

,

,)dtste
ts

,


ct 0.5 t

 and


pt t

for all , [0,1][0,1]st

and [0,1]u

. In this example for all [0,1]s
 





[0,1]
3
min3ln23ln3 ,
u
gsu sss

 
 




[0,1]
3
max 3ln23ln(2),
u
wsu sss








 
3ln3ln2 .hsws gsss 

Moreover for all [0,1]t

 
  
0d
33ln3 33log33
t
qtptgs s
ttt t



,
Let 100N

and 100
jj
t for all .
0,1, 2,,100j
We obtain the optimal solution
j
y and
j


, 0,1,j

of problem (13) by solving problem (10)
which is illustrated in Figures 4 and 5 respectively. In
2, ,100 

M. H. N. SKANDARI ET AL. 124
Figure 3. Optimal control of Ex.12. (.)u
Figure 4. Optimal trajectory of Ex.13. (.)y
Figure 5. Corresponding optimal control of Ex.13. (.)


addition, by (11) the corresponding for this ex-
ample is
(.)u
 

1/3
32,0,1,2,,100
jj j
hs gs
jj
ues j








The optimal control *
j
u, of prob-
lem (10) is showed in Figure 6. Here, the value of opti-
mal solution of objective function is 0.071.
0,1, 2,,100j
Figure 6. Optimal control of Ex.13. (.)u
5. Conclusions
In this paper, we posed a different approach for a class of
nonlinear optimal control problem including Volterra
integral equations. In our approach, the linear combina-
tion property of intervals is used to obtain the new cor-
responding problem which is a linear problem. The new
problem can be converted to a LP problem by discreteza-
tion method. Finally, we obtain an approximate solution
for the main problem. In next works, we are going to use
our approach for subclasses of problem (1)-(2) which
Volterra integral equation is similar to Equation (4), but
objective functional is quadratic or nonlinear with re-
spect to state variables.
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