We consider another non-classical of optimal control problem that is spurred by some current research on the nonlinear income issue in the field of financial matters. This class of issue can be set up as a maximizing issue in the area of Optimal Control. In any case, the state value at the final fixed time, y(T), is priori unknown and the integrand is an element of the unknown y(T). This is a non-classical optimal control problem. In this paper we apply the new costate value conditions p(T) in the definition of the optimal control problem. We solve some examples in this issue using the numerical shooting method to illuminate the subsequent Two Point Boundary Value Problem (TPBVP) and join the free y(T) as an additionally unknown. Basically similar outcomes are obtained through the nonlinear programming (NP) discrete-time results

Ali, Maselan

Rusiman, Mohd Saifullah

Sufahani, Suliadi

Wan Ahmad, Wan Noor Afifah

English

Journals of Universiti Tun Hussein Onn Malaysia (UTHM)

Journal of Science and Technology, Vol. 10 No. 1 (2018) p. 28-32 *Corresponding author: suliadi@uthm.edu.my2018 UTHM Publisher. All right reserved. penerbit.uthm.edu.my/ojs/index.php/jst A Non-Classical Optimal Control Problem Wan Noor Afifah Wan Ahmad, Suliadi Sufahani*, Mohd Saifullah Rusiman, Maselan AliDepartment of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Pagoh, Johor, Malaysia. Received 6 February 2018; accepted 30 May 2018; available online 27 June 2018 1. IntroductionCalculus of Variations (CoV) gives the mathematical theory to take care of extremizing functional issues for which a given functional has a stationary value either minimum or maximum [8]. Optimal control is an expansion of CoV and it is a mathematical optimization method for determining optimal control strategies. A couple of standard cases that mirror the utilization of optimal control are the medication bust technique, optimal generation, optimal control in discrete mechanics, strategy plan and the royalty installment issue [3,4,7]. Consider the framework in the time area displayed by the differential equation    y t u t  ,  0y  known  1with the unknown endpoint of state value  y T  at time .t T  We wish to decide thecontrol function  u t  for  0,t T  thatmaximizes         0, , ,TJ u f t y t u t y T dt      2Note that the integrand relies on the priori unknown final value,  y T . This paper issorted out as follows. In Section 2, we build up the necessary conditions for the extremizing solution. Then in Section 3, we consider a simple illustrative example. We finish the last section with conclusions. 2. The Non-Classical Optimal ControlProblemWe start by building up the necessary conditions for the extremizing solution. Suppose J  be a functional of the form         , , ,TaJ y f t y t y t y T dt      3where .T a  We consider the issue of deciding the functions   1y C   with the end goal that  J   has an extremum. Anunderlying condition  y a   is forced on  ,y   however  y T  is unknown.Assume that  J   has an extremum at .y %  We can continue as Lagrange did [2],by considering the estimation of J  at a close-Abstract:  We consider another non-classical of optimal control problem that is spurred by some current research on the nonlinear income issue in the field of financial matters. This class of issue can be set up as a maximizing issue in the area of Optimal Control. In any case, the state value at the final fixed time, y(T), is priori unknown and the integrand is an element of the unknown y(T). This is a non-classical optimal control problem. In this paper we apply the new costate value conditions p(T) in the definition of the optimal control problem. We solve some examples in this issue using the numerical shooting method to illuminate the subsequent Two Point Boundary Value Problem (TPBVP) and join the free y(T) as an additionally unknown. Basically similar outcomes are obtained through the nonlinear programming (NP) discrete-time results. Keyword: Mathematics; Calculus of Variation; Optimal Control Problem; Nonlinear Programming; Shooting Method DOI: https://10.30880/jst.2018.10.01.005Journal of Science and Technology, Vol. 10 No. 1 (2018) p. 28-32  29  by function y y h % , where   is a small parameter,   1h C   and   0.h a   Since  y T  is unknown, we do not expect  h   to vanish at .T  Let                , , ,TaJ y ht y t h t y t h tf dty T h T              %% %%          4  A fundamental condition for  y %  to be an extremizer is given by             000Ty ya zf h t f h tdtf h T        L LL 5  where         , , ,t y t y t y T % % %L .  Integration by parts gives             TyaTTy yaaf h t dtdf h t f h t dtdt  LL L 6  Since   0,h a   the necessary condition (4) can be then composed as               0 , , ,y yTyazdf f h tdtdtf T y T y T y Th TT af                   L L% % %L         7   for all   1h C   with the end goal that   0.h a   Specifically, condition (8) holds for the subclass of function   1h C   that do vanish at  .h T Hence, the classical arguments apply, and in this way      0y ydf fdt L L      8  Condition (7) must be fulfilled for all   1h C   with   0,h a   which incorporates functions  h   that do not vanish at .T  Thus, conditions (7) and (8) infer that                   , , ,0, , 0TyzaTy zaf T y T y T y Tf h T dtT ah T f T y T y T f dt          % % %L% % L 9   That is,          , , , 0Ty zaf T y T y T y T f dt   % % % L 10   Note that in the function f  does not depend on  y T  in the classical setting, which is, 0.zf   All things considered (10) lessens to the outstanding normal boundary condition     , , 0yf T y T y T  % %  (or, from a Hamiltonian optimal control point of view,   0).p T  We have quite recently demonstrated the accompanying outcome:  Theorem 2.1: Let a  and T  be given real numbers, .a T  If  y %  is a solution of the problem  Journal of Science and Technology, Vol. 10 No. 1 (2018) p. 28-32  30                1, , ,,,TaJ y f t y t y t y T dty a y T freey C     11   then              , , ,, , ,yydf t y t y t y Tdtf t y t y t y T% % %% % %               12   for all  , .t a T  Moreover,              , , ,, , ,yTzaf T y T y T y Tf t y t y t y T dt % % %% % %       13  From an optimal control point of view one has          , , ,yp T f T y T y T y T  % % %      14   where  p t  is the Hamiltonian multiplier. Theorem 2.1 states that the standard necessary optimality conditions (the Euler-Lagrange equation [2] or the Pontryagin maximum principle [5] hold for issue (11) by substituting the classical transversality condition   0p T   with        ( ) , , ,Tzap T f t y t y t y T dt  % % %  15 3. Numerical Example  Consider the Ordinary Differential Equation system below     y t u t  ,  0 0y  .                     16   We wish to maximize        0, , ,TJ u f t y t u t z dt       17  where   1, , , sin2 10tf t y u z a u z u            18  is a continuous function. The initial known state is  0 0y   and final state value  z y T  is unknown. In this paper, we set 50.T  The Hamiltonian is  , , ,H t y u p f p u    and                  , , ,, , , .pyy t H t y t u t p tp t H t y t u t p t          19  Function f  does not rely upon y  and for an ideal (maximum in this illustration) the costate fulfills  0.yp H p                                20  The stationary condition is  0uH                                                   21   and this yields     21sin4 10tu t z p t                  22 From (15)          500, , ,zp T f t y t y t y T dt  % % %  23 Holds    500sin10tp T u t dt                    24 4. Results  Let us consider the necessary condition that should be fulfilled. For the system of Ordinary Differential Equations (16) and (20) with control (22), the known zero initial condition Journal of Science and Technology, Vol. 10 No. 1 (2018) p. 28-32  31   0y  and a guessed initial value  0 ,p  we have to guarantee that the normal boundary condition (24) is fulfilled. The two point boundary value problem need to be solved. The value of z  that has been used in (22) also need to iterate so that the z  value obtained will be the same as  y t  at .t T  When one has gotten convergence in regards to the values  y T  utilized as a part of (22) and  p T  (24), at that point the necessary condition is fulfilled and we ought to have the optimal solution. Utilize the Newton shooting technique with two guessed value 1v  and 2v  [1]. We want  1 0v p  and  2v p T  as determined by condition (24). At the point when the program acquires the outcome with these two conditions holding to a high level of accuracy, the necessary conditions hold and we ought to have the optimal solution. We have tackled the shooting technique issue utilizing C++ and the profoundly precise Numerical Formulas library routines [6]. Integrate the system of Ordinary Differential Equations  (16) and (20), and   sin10ptg u t                                   25  J g                                                    26  The outcomes are ( ) 0.900000,y T   ( ) 0.015431p T  ,   3.275093pg T    and   6.25257J T                                  5. Conclusions  In this paper we have demonstrated to solve a nonstandard optimal control issue. We have introduced the fundamental conditions and computational techniques so as to acquire optimal solutions. A shooting method together with an expanding approach was utilized to acquire an exceedingly precise solution and compared with a discrete-time nonlinear programming solution. Our systems can be connected to the real problem rather more entangled financial matters issue where the Lagrangian integrand is piecewise constant in many stages and relies on the  y T  which is priori unknown.  Acknowledgements  The authors would like to thank the responsible community of Faculty of Applied Sciences and Technology, UTHM for giving a useful opportunity to publish paper. We are additionally grateful to anonymous referees and journal editors for their careful reading the paper and insightful comments and remarks that assist us in enhancement of this paper.  References  [1]  Betts, J. T. (2001). Practical Methods for Optimal Control Using Nonlinear Programming. Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. [2] Gelfand, I. M. & Fomin, S. V. (1963). Calculus of Variations. Revised English Edition Translated and Edited by Richard A. Silverman, Prentice Hall, Englewood Cliffs, N. J. [3] Leonard, D. & Long, N. V. (1992). Optimal Control Theory and Static Optimization in Economics. Cambridge University Press, Cambridge. [4] Pedro, A. F. C., Torres, D. F. M. & Zinober, A. S. I. (2010). “A Non Classical Class of Variational Problems with Application in Economics” in International Journal of Mathematical Modeling and Numerical Optimization, Vol. 1. No. 3. pp. 227-236. [5] Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V. & Mishchenko, E. F. (1986). Selected Works. The Mathematical Theory of Optimal Processes. Translated from the Russian by K. N. Trirogoff and Translation Edited by L. W. Neustadt, Reprint of the 1962 English Translation, Gordon & Breach, New York. [6] Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. (2007). Numerical Recipes - The Art of Scientific Computing. Cambridge University Press, Cambridge. [7] Sethi, S. P. & Thompson, G. L. (2000). Optimal Control Theory - Applications on Journal of Science and Technology, Vol. 10 No. 1 (2018) p. 28-32  32  Management Science and Economics. Kluwer Academic Publishers, Boston, MA. [8] Pinch, E. R. (1993). Optimal Control and the Calculus of Variations. Oxford University Press, Oxford. 

A Non-Classical Optimal Control Problem

Sufahani, Suliadi Firdaus

UTHM Institutional Repository

A non-classical optimal control problem

In this paper, triangular fuzzy numbers (TFN) are used to represent the uncertainty of data in the transportation problems (TP), which are referred to as fuzzy transportation problems (FTP). The main issues in the FTP are the lack of information, error of the basic trading rules, incomplete fuzzy data, and the limitations of the existing fuzzy-ranking functions which is failed to compare two TFN. Segregated advancement (SA) is a separation approach where the TFN represented by low, middle, and upper are solved part by part. The flaw of SA is that it uses the classical algorithms which are NWC, LCM, and VAM. Therefore, we proposed the improvement based on the combination of total ratio cost matrix and total difference method by column without using classical ranking functions. The first and second examples illustrate the existing methods without using the SA approach. The results show that the proposed method obtained the optimal solution of FTP, whereas the existing

methods produced infeasible solution. The third example presents the limitation of existing methods along with the SA approach. The results show that the proposed method is capable in solving the third example whereas the existing approach failed to solve the said example

Muhammad Sam’An, Muhammad Sam’An

Yosza Dasril, Yosza Dasril

Chandrasekar Ramasamy, Chandrasekar Ramasamy

Nazarudin Bujang, Nazarudin Bujang

Yahya Nur Ifriza, Yahya Nur Ifriza

Improved Segregated Advancement (I-SA): a new method for solving full triangular fuzzy transportation problems

Journal of Science and Technology, Vol. 10 No. 1 (2018) p. 28-32 
*Corresponding author: suliadi@uthm.edu.my
2018 UTHM Publisher. All right reserved. 
penerbit.uthm.edu.my/ojs/index.php/jst 
A Non-Classical Optimal Control Problem 
Wan Noor Afifah Wan Ahmad, Suliadi Sufahani*, Mohd Saifullah Rusiman, Maselan Ali
Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, 
Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Pagoh, Johor, Malaysia. 
Received 6 February 2018; accepted 30 May 2018; available online 27 June 2018 
1. Introduction
Calculus of Variations (CoV) gives the 
mathematical theory to take care of 
extremizing functional issues for which a 
given functional has a stationary value either 
minimum or maximum [8]. Optimal control is 
an expansion of CoV and it is a mathematical 
optimization method for determining optimal 
control strategies. A couple of standard cases 
that mirror the utilization of optimal control 
are the medication bust technique, optimal 
generation, optimal control in discrete 
mechanics, strategy plan and the royalty 
installment issue [3,4,7]. Consider the 
framework in the time area displayed by the 
differential equation 
   y t u t  ,  0y  known  1
with the unknown endpoint of state value 
 y T  at time .t T  We wish to decide the
control function  u t  for  0,t T  that
maximizes 
        
0
, , ,
T
J u f t y t u t y T dt      2
Note that the integrand relies on the priori 
unknown final value,  y T . This paper is
sorted out as follows. In Section 2, we build up 
the necessary conditions for the extremizing 
solution. Then in Section 3, we consider a 
simple illustrative example. We finish the last 
section with conclusions. 
2. The Non-Classical Optimal Control
Problem
We start by building up the necessary 
conditions for the extremizing solution. 
Suppose J  be a functional of the form 
        , , ,
T
a
J y f t y t y t y T dt      3
where .T a  We consider the issue of 
deciding the functions   1y C   with the end 
goal that  J   has an extremum. An
underlying condition  y a   is forced on
  ,y   however  y T  is unknown.
Assume that  J   has an extremum at
 .y %  We can continue as Lagrange did [2],
by considering the estimation of J  at a close-
Abstract:  We consider another non-classical of optimal control problem that is spurred by some current 
research on the nonlinear income issue in the field of financial matters. This class of issue can be set up as a 
maximizing issue in the area of Optimal Control. In any case, the state value at the final fixed time, y(T), is 
priori unknown and the integrand is an element of the unknown y(T). This is a non-classical optimal control 
problem. In this paper we apply the new costate value conditions p(T) in the definition of the optimal 
control problem. We solve some examples in this issue using the numerical shooting method to illuminate 
the subsequent Two Point Boundary Value Problem (TPBVP) and join the free y(T) as an additionally 
unknown. Basically similar outcomes are obtained through the nonlinear programming (NP) discrete-time 
results. 
Keyword: Mathematics; Calculus of Variation; Optimal Control Problem; Nonlinear Programming; 
Shooting Method 
DOI: https://10.30880/jst.2018.10.01.005
Journal of Science and Technology, Vol. 10 No. 1 (2018) p. 28-32 
 
29 
 
by function y y h % , where   is a small 
parameter,   1h C   and   0.h a   Since 
 y T  is unknown, we do not expect  h   to 
vanish at .T  Let 
 
    
       
   
, , ,T
a
J y h
t y t h t y t h t
f dt
y T h T
  
 

    
   
  
  

%
% %
%
         
 4
 
 
A fundamental condition for  y %  to be an 
extremizer is given by 
 
 
       
   
0
0
0
T
y y
a z
f h t f h t
dt
f h T

 


 
  
  
  

L L
L
 5
  
where         , , ,t y t y t y T % % %L .  
Integration by parts gives 
 
   
        
T
y
a
T
T
y ya
a
f h t dt
d
f h t f h t dt
dt

 

 


L
L L
 6
  
Since   0,h a   the necessary condition 
(4) can be then composed as 
 
     
      
 
 
0 , , ,
y y
T
y
a
z
d
f f h t
dt
dtf T y T y T y T
h TT a
f


  
   
  
 
   
  
  
    

L L
% % %
L
        
 7  
 
for all   1h C   with the end goal that 
  0.h a   Specifically, condition (8) holds for 
the subclass of function   1h C   that do 
vanish at  .h T Hence, the classical 
arguments apply, and in this way 
 
    0y y
d
f f
dt
 L L
     
 8
 
 
Condition (7) must be fulfilled for all 
  1h C   with   0,h a   which incorporates 
functions  h   that do not vanish at .T  Thus, 
conditions (7) and (8) infer that 
 
      
   
        
, , ,
0
, , 0
T
y
z
a
T
y z
a
f T y T y T y T
f h T dt
T a
h T f T y T y T f dt


 
  
  
 
   
 


% % %
L
% % L
 9  
 
That is, 
 
        , , , 0
T
y z
a
f T y T y T y T f dt   % % % L
 10  
 
Note that in the function f  does not 
depend on  y T  in the classical setting, 
which is, 0.zf   All things considered (10) 
lessens to the outstanding normal boundary 
condition     , , 0yf T y T y T  % %  (or, from 
a Hamiltonian optimal control point of view, 
  0).p T  We have quite recently 
demonstrated the accompanying outcome: 
 
Theorem 2.1: Let a  and T  be given real 
numbers, .a T  If  y %  is a solution of the 
problem 
 
Journal of Science and Technology, Vol. 10 No. 1 (2018) p. 28-32 
 
30 
 
        
    
  1
, , ,
,
,
T
a
J y f t y t y t y T dt
y a y T free
y C

   

 

 11  
 
then 
 
      
      
, , ,
, , ,
y
y
d
f t y t y t y T
dt
f t y t y t y T



% % %
% % %              
 12  
 
for all  , .t a T  Moreover, 
 
      
      
, , ,
, , ,
y
T
z
a
f T y T y T y T
f t y t y t y T dt


 
% % %
% % %
      
 13
 
 
From an optimal control point of view one has 
 
        , , ,yp T f T y T y T y T  % % %      14  
 
where  p t  is the Hamiltonian multiplier. 
Theorem 2.1 states that the standard necessary 
optimality conditions (the Euler-Lagrange 
equation [2] or the Pontryagin maximum 
principle [5] hold for issue (11) by substituting 
the classical transversality condition 
  0p T   with 
 
      ( ) , , ,
T
z
a
p T f t y t y t y T dt  % % %
 
 15
 
3. Numerical Example 
 
Consider the Ordinary Differential Equation 
system below 
 
   y t u t  ,  0 0y  .                     16  
 
We wish to maximize 
 
      
0
, , ,
T
J u f t y t u t z dt    
  
 17
 
 
where 
 
 
1
, , , sin
2 10
t
f t y u z a u z u
  
     
    
 18
  
is a continuous function. The initial known 
state is  0 0y   and final state value 
 z y T  is unknown. In this paper, we set 
50.T  The Hamiltonian is 
 , , ,H t y u p f p u    and 
 
        
        
, , ,
, , , .
p
y
y t H t y t u t p t
p t H t y t u t p t
  

       
 19
 
 
Function f  does not rely upon y  and for 
an ideal (maximum in this illustration) the 
costate fulfills 
 
0.yp H p                               
 20
 
 
The stationary condition is 
 
0uH                                                   21  
 
and this yields 
 
   
2
1
sin
4 10
t
u t z p t
  
   
            
 22
 
From (15) 
 
        
50
0
, , ,zp T f t y t y t y T dt  % % %
 
 23
 Holds 
   
50
0
sin
10
t
p T u t dt
 
  
 

               
 24
 
4. Results 
 
Let us consider the necessary condition that 
should be fulfilled. For the system of Ordinary 
Differential Equations (16) and (20) with 
control (22), the known zero initial condition 
Journal of Science and Technology, Vol. 10 No. 1 (2018) p. 28-32 
 
31 
 
 0y  and a guessed initial value  0 ,p  we 
have to guarantee that the normal boundary 
condition (24) is fulfilled. 
The two point boundary value problem 
need to be solved. The value of z  that has 
been used in (22) also need to iterate so that 
the z  value obtained will be the same as  y t  
at .t T  When one has gotten convergence in 
regards to the values  y T  utilized as a part 
of (22) and  p T  (24), at that point the 
necessary condition is fulfilled and we ought 
to have the optimal solution. 
Utilize the Newton shooting technique with 
two guessed value 1v  and 2v  
[1]. We want 
 1 0v p  and  2v p T  as determined by 
condition (24). At the point when the program 
acquires the outcome with these two 
conditions holding to a high level of accuracy, 
the necessary conditions hold and we ought to 
have the optimal solution. We have tackled the 
shooting technique issue utilizing C++ and the 
profoundly precise Numerical Formulas 
library routines [6]. 
Integrate the system of Ordinary 
Differential Equations  (16) and (20), and 
 
 sin
10
p
t
g u t
 
   
                              
 25
 
 
J g                                                    26  
The outcomes are ( ) 0.900000,y T   
( ) 0.015431p T  ,   3.275093pg T    and 
  6.25257J T 
                                
 
5. Conclusions 
 
In this paper we have demonstrated to 
solve a nonstandard optimal control issue. We 
have introduced the fundamental conditions 
and computational techniques so as to acquire 
optimal solutions. A shooting method together 
with an expanding approach was utilized to 
acquire an exceedingly precise solution and 
compared with a discrete-time nonlinear 
programming solution. Our systems can be 
connected to the real problem rather more 
entangled financial matters issue where the 
Lagrangian integrand is piecewise constant in 
many stages and relies on the  y T  which is 
priori unknown. 
 
Acknowledgements 
 
The authors would like to thank the 
responsible community of Faculty of Applied 
Sciences and Technology, UTHM for giving a 
useful opportunity to publish paper. We are 
additionally grateful to anonymous referees 
and journal editors for their careful reading the 
paper and insightful comments and remarks 
that assist us in enhancement of this paper. 
 
References 
 
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Optimal Control Using Nonlinear 
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[2] Gelfand, I. M. & Fomin, S. V. (1963). 
Calculus of Variations. Revised English 
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A. Silverman, Prentice Hall, Englewood 
Cliffs, N. J. 
[3] Leonard, D. & Long, N. V. (1992). 
Optimal Control Theory and Static 
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[4] Pedro, A. F. C., Torres, D. F. M. & 
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[5] Pontryagin, L. S., Boltyanskii, V. G., 
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[6] Press, W. H., Teukolsky, S. A., 
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