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Abstract 

      This paper deals with the numerical solution of the discrete classical optimal control 

problem (DCOCP) governing by linear hyperbolic boundary value problem (LHBVP). The 

method which is used here consists of: the GFEIM " the Galerkin finite element method in 

space variable with the implicit finite difference method in time variable" to find the solution 

of the discrete state equation (DSE) and the solution of its corresponding discrete adjoint 

equation, where a discrete classical control (DCC) is given.  The gradient projection method 

with either the Armijo method (GPARM) or with the optimal method (GPOSM) is used to 

solve the minimization problem which is obtained from the necessary condition for optimality 

of the DCOCP to find the DCC.An algorithm is given and a computer program is coded using 

the above methods to find the numerical solution of the DCOCP with step length of space 

variable        , and step length of time variable        . Illustration examples are given 

to explain the efficiency of these methods. The results show the methods which are used here 

are better than those obtained when we used the Gradient method (GM) or Frank Wolfe 

method (FWM) with Armijo step search method to solve the minimization problem.  
 
 

Keywords: Numerical classical optimal control, hyperbolic boundary value problem, finite 

element method, Gradient Projection method, Armijo step search method, Optimal step 

method. 
 
 
 

1. Introduction  

    Optimal control problems of partial differential equations PDEs have wide applications in 

many real live problems such as in economic, electromagnetic waves, biology, robotics, 

dynamical elasticity, medicine, air traffic and many others. The numerical solution of the 

DCOCP is studied by many researches.  The GPARM or GPOSM are used to find the 

numerical solution of the DCOCP governing either by systems of nonlinear elliptic PDEs as 

in [1, 2], or by systems of semi linear parabolic PDEs as in [3, 4], or by systems of nonlinear 

ordinary differential equations (ODEs) as in [5, 6], or by systems of LHBVP so as our 

previous work [7]. Since the GFEM is one of the most an efficient and fast methods for 

Numerical Solution for Classical Optimal Control Problem Governing 

by Hyperbolic Partial Differential Equation via Galerkin Finite Element-

Implicit method with Gradient Projection Method 
 

Jamil A. Ali Al-Hawasy 

 

Eman H. Al-Rawdanee  

 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

emanalquraishi91@gmail.com 
 

Article history: Received 25 February 2019, Accepted 4 March 2019, Publish May 2019     

10.30526/32.2.2141 Doi: 

Department of Mathematics, College of Science, University of Mustansiriyah 

 jhawassy17@uomustanriyah.edu.iq 
 

mailto:tibataiee92@gmail.com
mailto:tibataiee92@gmail.com
jhawassy17@uomustanriyah.edu.iq


  

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solving different types of differential equations in general [8, 9], and DCOCP in particular. In 

fact, the GFEIM is used in [7]. To find the numerical solution for the state of the LHBVP and 

its adjoint equations while the Gradient method (GM)and Frank Wolfe method (FM) with the 

Armijo method is used there to solve the minimization problem which is obtained from the 

necessary condition for optimality of the DCOCP to find the DCC. The our previous work 

push us to continue in studying the numerical solution for the DCOCP governed by LHBVP 

via GFEIM but instead of GM or FWM with the Armijo method to find the numerical DCOC, 

the GPM with both the ARM (GPARM) and the optimal step method (GPOSM) is used to 

find the numerical DCOP to solve the minimization problem which is obtained from the 

necessary condition for optimality of the DCOCP to find the DCC An algorithm is given and 

a computer program is coded in Mat lab software to solve the DCOCP, the results are drawing 

by figures and show that the GPARM is better than those methods which are used in [7], to 

solve the minimization problem. 

    

2. Description of the CCOC and the DCOC Problems [7]  

2.1. Description of the CCOC Problem 

     Consider the bounded and open region      with Lipschitz boundary   , let     ,T), 

0<T< ,       and    =         . The CCOC of LHBVP is given by: 

             ⃗          , in  ,  ⃗                                                              (1) 

with the BC: 

   ⃗     , in                                                                                                                      (2) 

and the ICs: 

   ⃗        ⃗ , in                                                                                                              (3) 

    ⃗     
   ⃗ , in                                                                                                             (4) 

    Where       ⃗     
    ̅  is the state which corresponds to the continuous classical 

control (CCC)      ⃗          ,       ⃗     
      is a given function and      is the 

second-order operator         
     

 
   

      
. 

     The set of the CCCs is    ,           

where             |    ⃗     , a.e. in   },      is a convex. 

 

     The cost functional is given by 

     =∫  
 

 

 

 
      

  
 

 
      

    ⃗                                                                         (5) 

where        ⃗   and         ⃗    are the desired state and the desired control 

respectively. 

    The CCOC problem is to minimize the cost functional (5) subject to    . 

In this work, the inner product and the norm in        it will be indicated by        and      

respectively, while the norm in Sobolev space          by     , and the norm in  
     by 

    . 

    Now, the weak form (WF) of the problem (1- 4) is given by: 

                                                                           (6) 

with the (ICs) 

       , in                                                                                                                         (7) 

       
 , in                                                                                                                        (8) 



  

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     Where                   is a bilinear form which is symmetric, and satisfies the 

following assumptions,       ,   ̅ and for some positive scalars    and   . 

 

(i)                      

(ii)                  
  

 

Suppose     , and then equations (6-8) can be written by: 

                                                                           (6a) 

with the (ICs) 

       , in                                                                                                                       (7a) 

       , in                                                                                                                        (8a) 

 

2.2. Description of the DCCOCP 

      The discretization of the CCOC is obtained by using the GFEM. Assume that   is 

polyhedron domain. For every integer (s), let   
     

    
 be an admissible regular triangulation 

of  ̅ ,    
     

      
 be a subdivision of the interval  ̅ and        

      be the space of 

continuous piecewise affine mapping in  .  

 

      For each     , the discrete state equations (DSEs) of (1-4) is written by (for   

           

(    
    

   )
 

     (    
   )    ( (  

 )  )
 

   (    
   )

 
   (  

   )
 

   

                                                                  
   

                                                               (9)        

    
    

        
 

                                                                                                              (10) 

(  
         

                                                                                                                   (11) 

   
         

                                                                                                                    (12) 

Where   
      

   ,   
      

        for          . 

     The discrete cost functional (DCF)    
      is defined by  

  
         

   

   

∫
 

 

 

 
      

     
     

     
    ⃗                                                      (13) 

     The DCOC problem here is to find      , such that 

  
          ̅       

   ̅    

 

3. The solution of the DCOCP 

     This section deals with some theorems and lemmas which are important in the next section 

they can be proved by using the same techniques which are used in [7].  

 

Theorem 1: For any fixed j (        , and       , the DSEs (9-12) has a unique 

solution    
        

    
      

   for sufficiently small   .  

                                

Theorem 2: The operator          
 

 is continuous.                                                   



  

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Lemma 3: If the DCCs     and  ̃  are bounded in      ,   
 
 and    

    
      

 
 (with 

   is a small positive number) are corresponding discrete states solutions to the DCCs   
 
 and 

   
    

       
 
  respectively (             then:   

     
   

          
   

  and      
   

          
   

  

Or      
   

   , and      
   

         

                                                                                   

Theorem 4 (Existence of DCOCP): Consider the DCF in equation (13). Assume that    is 

convex and closed. If   
      is coercive, then there exists a discrete classical optimal 

control.         

Theorem 5 (The Necessary conditions for DCOCP):  The discrete classical adjoint state 

    
 

 =       
    

        
   is given by (for              ) 

(    
    

   )      (  
   )    (  

   )     (    
    

   )                                       (14) 

    
    

       
 
                                                                                                                  (15) 

  
    

                                                                                                                            (16) 

where   
    

    ,(              

 

    The discrete directional derivative of G in equation (13) is given by: 

D  
    

    
     

      
   

   

(  
 (  

      
    

    
 )    

 )
 

 

                                        
    

    
     

         
  ,        

                            (17)                 

   Where   
     

    ,    
    

     
 
 for (         ), and   

  is called the discrete 

Hamiltonian functional.                                                                                                          

Remark: To prove equation (17)     
      is equivalent to the minimum principle block 

wise. 

 (  
    

    
    

 )
  

    
      

(  
    

    
    

  )
  

,                                 (18) 

Let          
     

                     and   
     

 
, for all j except once(say 

 ) i.e.   
     

 , then    
    

    
    

      
      

   
    

    
    

    , since   is arbitrary 

then we have equation (18), and converse is very clear. 

4. The GPM Method [10].  

    The GPM is an iterative method used to find the point that minimizes the problem. The 

following algorithms describe the GP, GPARM and GPOSM; we will use the norm     with 

respect to vector space Q. 

 

4.1. The basic algorithm GPM [10] 

    Let Q be a Hilbert space,   is a convex subset of an open set     ,        , and 

let         ,      be a sequence with          , for each  , and let      be an initial 

classical control, 

(1) Find a direction point      and set    =    −    

(2) Set    := max 
  subject to                

        
                

       
     

(3) Set                 

 



  

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4.2. Algorithm [10] 

    Let Q be a Hilbert space,   is a convex subset of an open set     ,        , and 

let  ,        ,      be a sequence with          , for each  .    , and let      be an 

initial control. 

Step1: Set     , solve the system of the WF (9-12) (the system of  the adjoint WF (14-16)) 

by GFEM to get     (    , and then Calculate  
      from equation (17) and       from 

equation (13). 

Step 2: Find a direction point     , (i.e. a direction      ) by applying the GPM as 

follows:  Find the unique      , such that 

         
             

 

 
       

  

             
   

              
 

 
      

  

Step 3: Solve the system of the WF (9-12) to find the new state    corresponding to the new 

control   . 

Step 4: Choose    using one of the following methods:  

ARM: Assume an initial value            . Find the  (           ) from equation 

(13). If   satisfies the inequality  

       (           )             

we set     ⁄ , and choose for   , the last         ) that satisfies the above inequality. If 

unsatisfied, set     , and choose for    the first           that satisfies this inequality. 

OPSM: Find an          , such that 

(             )              
              

Step 5: Set                 ,        and we go to step 2. 

    Practically the ARM is faster and is a finite procedure than OPSM. The following examples 

are solved by using the above algorithm; a computer program in mat lab software version 

8.1.0.604 is coded. The solution    and     in step (1) are found using the GFEM with 

        , (        ), the parameters in ARM are taken the value        , and 

the parameter     0.5 in the GPM.  

 

5. Numerical Examples  

Example 1: Consider the following classical optimal control problem (COCP) associated 

with the linear hyperbolic equation: 

          ⃗          , in      ,  ⃗                 

   ⃗     , in    =         .   

and the (ICs) 

   ⃗                    , in                       

     ⃗                      , in                                         

Where   [0,1],              , and 

    ⃗           
      

     

    The control constraint is            and the cost functional is given by: 

      =∫  
 

 

 

 
      

  
 

 
      

    ⃗  ,    

  Where        ⃗   and         ⃗    are the desired state and control and are given by 

    ⃗                     
   ,    ⃗     , and 



  

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    ⃗    {
                        
                         

  

with the initial control, 

    ⃗           ,    ⃗      

Algorithm (5.1) is used here to solve this problem. The following figures are represented the 

initial control and its corresponding state. 

 
Figure 1. Initial control at t=0.5. 

 

 
Figure 2. Corresponding state (of initial control).at t= 0.5. 

 

    Depending on the above initial control and its corresponding state, we get: 

(I) the GPARM, after 6 iterations is used to get the optimal control and its corresponding 

state, the results show with 

    
  =5.9085e-08,    6.6150e-04, and   =7.6800e-05 

where    and    are the discrete maximum error of the state and the control respectively. 

    The following figures represent the optimal control and its corresponding state which are 

obtained by using GPARM. 

 
Figure 3. Optimal control at t=0.5. 

 



  

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Figure 4. Corresponding state (of optimal control) at t= 0.5. 

 

(II) The GPOSM, after 2 iterations is used to obtain the optimal control and its corresponding 

state, with: 

    
  =5.8424e-08,    6.6150e-04, and   =7.1644e-05 

    The following figures are posted by using GPOSM and are represented the optimal control 

and its corresponding state. 

 
Figure 5. Optimal control at t=0.5. 

 

 
Figure 6. Corresponding state (of initial control) at t=0.5.  

                                  

Example 2: Consider the COCP, which was considered in example 1, but the control 

constraint is         , and the desired control is given by 

    ⃗             

with the initial control,  

    ⃗    {
                   
                   
                   

 

    The following figures represent the initial control and its corresponding state. 



  

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.  32 (2) 2019 

 

 
Figure 7. Initial control at t=0.5. 

 

 
Figure 8. Corresponding state (of initial control).at t=0.5. 

  

(I) In the GPARM, after 9 iterations is used to get the optimal control and corresponding state, 

the results show with: 

    
  =5.9048e-08,    6.6150e-04, and   =4.0813e-05 

    The following figures are represented the optimal control and its corresponding state. 

 

 
Figure 9. Optimal control at t=0.5. 

 
Figure 10. Corresponding state (of optimal control) at t=0.5. 

 

(II) The GPOSM, after 2 iterations is used to get the optimal control and corresponding state, 

with 

    
  =5.8570e-08,    6.6150e-04, and   =7.4298e-05 

    The following figures are represented the optimal control and its corresponding state. 



  

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Figure 11. Optimal control at t=0.5. 

 

 
Figure 12. Corresponding state (of optimal control) at t=0.5. 

6. Conclusion 

       The following conclusions can be derived from the obtained results: 

(1) the GFEIM which is used here to find the DSE of LHBVP as well as the discrete adjoint 

equation for the state equation is fast and efficient than other method e.g., the finite difference 

methods, methods of lines, variational methods which clearly are needed more time and hence 

have more computations which are Grow up the error.  

(2) By the results of the above examples, with (step length of space variable      , and step 

length of time          we conclude that: 

(I) the GPARM and the GPOSM which are used to solve the minimization problem (the cost 

functional), are suitable and efficient methods to find the DCOC governed by hyperbolic 

boundary value problem, with parameters        ,       and      .    

(II) Although the GPOSM is needed less iterations than GPARM, but the GPARM remain 

better since it is used for general minimization problems, on the contrary of the GPOSM 

which is used only for quadratic functional. 
 
 

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