Computation of Optimal Controls for a Class of Distributed Parameter Systems

ISSN 2319-8133 (Online)

(An International Research Journal), www.compmath-journal.org

Computation of Optimal Controls for a Class of Distributed Parameter Systems

M. H. Farag^1,2 , T. A. Nofal^1,2, M. A. El-Saied^3,4 and N. M. Al-Baqmi⁵

1Mathematics Department,

Faculty of Science, Taif University, Hawia (888), Taif, KSA.

2Mathematics Department,

Faculty of Science, Minia University, Mina, EGYPT.

3Department of Inf. Tech.,

College of Computers and Inf. Tech., Hawia (888), Taif, KSA.

4Department Computer Sciences,

Faculty of Science, Fayoum University, Fayoum, EGYPT.

5College of Applied Medical Sciences, Taif University, Torba, KSA.

(Received on: July 24, 2015) ABSTRACT

Our work is devoted to a class of optimal control problems of parabolic partial differential equations. Because of the partial differential equations constraints, it is rather difficult to solve the optimization problem. The gradient of the cost function can be found by the adjoint problem approach. The partial quadratic interpolation technique is applied to solve this optimization problem and this algorithm is given.

The numerical results of two types of optimal control parabolic problems are also given.

Keywords: Optimal Control, Parabolic Equations, Partial Quadratic Interpolation Technique, Sufficient Differentiability Conditions, Numerical Methods.

1. INTRODUCTION

The optimal control problems governed by partial differential equations have developed very fast in the last 30 years, and it has brought a promising and vital researching domain to the subject of mathematics. The optimal control problems governed by partial differential equations concern many applications in physics, chemistry, biology, etc., such as materials design, crystal growth, temperature control, petroleum exploitation, and so on. The relative details can be seen in^1-5, and so on. The partial differential equations involved in these problems include elliptic equations, parabolic equations and hyperbolic equations. In the meanings of constrained conditions, these optimal control problems can be divided into control

constrained problems and state constrained problems. In each of the branches referred above, there are many excellent works and also many difficulties to be solved. The optimal control problems have had a rather complete theoretical framework in the last few decades’

development and the relative software have also improved rapidly. Many researchers considered a similar optimal control problem of partial differential equations (OCP)^6,7:

  ^     ^    ^     ^   



T l

V

v

f v y x T z x dx v t w t dt

0

2 0 2

,

2

min

_

 

(1)

Subject to



x t



x v x y

x t

y ₂ ( ) ,

2



  















 



 ,



x,t



T, (2)



x



v

 

x

y ,0  ₁ ,x

 

0,l ,(3)

 0 , t   0

y

_x ,

    l y

_x

 l , t   k  y  l , t   v

₂

  t 

, t



0,T



(4) where _T {



x,t



:x(0,l) , 0  t  T }, y



x,t;v



is the solution of the parabolic system (1)-(4), the constant k >0 is called the convection coefficient or heat transfer coefficient and,⁰ 

  

^x  ^L



^0,l



and z

 

x , w

 

t are given functions, respectively, from

 

^l

L₂ 0, and ^L2



^0,T



,

T

is a fixed time and

 , 

are given positive numbers.Vis the set of admissible controls, where

V  { v : v

_

  x , t  L

₂

    

_T

, v

₁

x  L

₂

    0 , l , v

₂

t  L

₂

 0 , T  }

⊂

 

^L

 

^{l L}

 

^T

L₂_T  ₂ 0,  ₂0, .

In this paper, a class of optimal control problems of parabolic partial differential equations is studied. The gradient of the cost function can be found by the adjoint problem approach. The partial quadratic interpolation technique is applied to solve this optimization problemand this algorithm is given. The numerical results of two types of optimal control parabolic problems are also given.

2. CORRECTNESS OF STATEMENT OPTIMAL CONTROL PROBLEM

The correctness of the boundary value problem (1)-(4) for given vV is as follows:

We define a solution of the optimal control problem (1)–(4), according to⁸, as a solution of the minimization problem for the cost functional

^f

_

  ^v

, given by (1):

 

^{v f}

 

^v

f

V v





inf



  (5) Evidently, if

f

_

  v

_ = 0, then the solution v_∈V is also a strict solution of the optimal control problem (1)-(4), since v_∈V satisfies the functional equation

y  x t v  z   x

T

t



;



,

,

 

l

x 0, . Further, in the view of the weak solution theory for parabolic problems, one can prove that if the sequence

  ^v

^{ n}

^{ V}

weakly converges to the function

v V

, then the sequence of traces

 ^u  ^{x , T ; v}

^{ n}

 

of corresponding solutions of problem (1)-(4) converges in

norm

L₂( )_T  to the solution {u(x, T; v)}, which means f_

 

v^{ n}  f_

 

v , as n→∞. This means the functional

f

_

  v

is weakly continuous on V, hence due to the Weierstrass existence theorem⁹ the set of solutions

V

_

  v  V : f

_

  v

_

 ( f

_

)

_

 inf f

_

  v 

of the minimization problem (1)-(4) is not an empty set.

3. THE VARIATION OF THE COST FUNCTIONAL

The sufficient differentiability condition of the functional (1)-(4) and its gradient formulae is as follows¹⁰:

Lemma 3.1.

Let

v  { v

₀

      x , t , v

₁

x , v

₂

t }

,vv



v₀

 

x,t v₀

   

x,t,v₁ x v₁

   

x,v₂ t v₂

 

t



V be given elements. If y y



x,t;v



L₂



_T



is the corresponding solution of the direct problem (1)-(4) and

 

x,t;v



L₂



_T



is the solution of the adjoint parabolic problem

 



_x



_x

t

 x 

  

,

  x , t  

_t



x,T



2

 

y



x,T;v



z

 

x





, x

 

0,l (6)

  0 , t  0



x ,

    l 

_x

  l , t  k    l , t

, t



0,T



then for all v ∈ V the following integral identity holds:

   



y x T v z x



u



xT v



dx

l , ; , ;

2

^ 

0   ⁼



0^l

 ^

^x,0;^v

^

^v1

^{ }

^{x dx}

+

    





T

dt dx t x v v t

x , ;

₀

,



+k



0^T

 

l,t;v



^v2

 

t dt ,

 v  V

(7)

Lemma3.2.

Let yy



x,t;v



L₂()be the solution of the parabolic problem corresponding to a given v∈ V.

 

 x y  v   x t y

_t

 

_{x x}

 

₀

,

 

,

  x , t  

_t



x



v

 

x y ,0  ₁

 , x

 

0,l (8)

  0 ,  0

 y

_x

t

,

    l  y

_x

  l , t  k   y   l , t   v

₂

  t 

, t



0,T



Then the following estimate holds:

 

 



0^{l y xT v dx}

; 2

, ^{0 2}

v

W

c 

 

^,

^{  v  V}

(9)

where

     

12

0

2 0 2

2 1 2

0

,





























   

_T

T l

W

v x t dx dt v x dx v t dt

v

is the norm L₂( )_T normof the function

 v  V

, and the constants

c

₀, ε >0 are defined as follows:

  k

c

₀

 max 1 ,

,

min   0

0





 



x

l

x





,









 

 k l

k l

2 , 2 min

2

 

(10)

Thus by the definition

  v f

_

  

v V

v

f 

 _^/ , ^



0^T



^v

 

^t



^dt

2



2 +



0^l



^y



^{x T v}

 

^dx

; 2



, , (11) and from the above lemma implies that the last integral in (11) is bounded by the term



vV²



O  . Then the functional

 f

_

  v

is Fréchet-differential at v∈V we obtain the following theorem:

Theorem 3.1.

Let conditions in the considered problem hold. Then the cost functional is Fréchet- differentiable,

f

_

  v

∈

C

¹

( V )

. Moreover, Fréchet derivative at

v  V

of the cost functional

  v

f

_ can be defined by the solution

  W

^1,0

 

_T



of the adjoint problem (6) as follows:

 

^v

 

^{x t v}

 

^{x v}



^k



^{l t v}

 

f_^/ 



, ; ,



,0; ;



, ; (12)

4. NUMERICAL SOLUTION OF THE OPTIMAL CONTROL PROBLEM 4.1 The partial quadratic interpolation technique (PQI )

The essential steps to apply the partial quadratic interpolation technique¹¹ can be summarized as follows:

1) Choose some starting point

x

^(r)

 

_m and

r  1

.

2) Approximate the function

F ( x )

about

x

^r in the quadratic form

] )][

( [ ] 2 [

] 1 [

)]

( [ )

( x a B

_m

x

^{r Tr}

x x

^r

x x

^{r Tr}

A

_m

x

^r

x x

^r

F      

(13)

where

[ A

_m

]

and

[ B

_m

]

represent the gradient vector and the Hessian matrix of the function F(x) respectively. To compute particular values for

a , A

_m

, B

_mwe choose a set of interpolation points as follows:

i)mpoints[x_i^r_],i 1(1)m [x_i^r_](x₁^r, x₂^r,, x_i^r_1,x_i^r l_i,x_i^r_1,,x_m^r ) ii)mpoints[x_i^r_],i 1(1)m [x_i^r_](x₁^r, x₂^r,, x_i^r_1,x_i^r l_i,x_i^r_1,,x_m^r) iii)

2

) 1 ( m 

m

points

[ x

_ij^r

] , i  1 ( 1 ) m  1 , , j  i  1 ( 1 ) m

where

[ x

_ij^r

]  ( x

₁^r

, x

₂^r

,  , x

_i^r_1

, x

_i^r

 l

_i

, x

_i^r_1

,  , x

^r_j

 l

_j

,  , x

_m^r

)

Using these interpolation points it can be shown that

a  F ( x

^r

)

and the elements

ij i a

b , of

B ,

_m

A

_mrespectively are given by

2

) ( ) ( 2 ) , (

2

) ( ) (

i

r i r

r i ii

i r i r

i

i l

x F x F x

a F l

x F x

b F ^{  }   ^

 

 (14)

j i

r r

j r

i r

ij

ij

l l

x F x

F x F x

a F ( )  ( )  ( )  ( )



^{ } (15)

The

l

_i are a set of constants which determine the accuracy of the interpolation.

3) Extract the symmetric positive definite matrix [A_q]from the symmetric matrix

[ A

_m

]

using Choliski’s method,

q  m

, by cancelling certain rows and columns. Essentially we write

Tr m m

m S S

A [ ][ ] . From this we have

s

₁₁²

 a

₁₁ . If

a

₁₁

 0

then we eliminate the first row and column in each of [A_m ],[S_m] and [S_m]^Tr and perform the calculation on the

Tr m m

m S and S

A ],[ ] [ ]

[ _{1 1 1} .If

a

₁₁

 0

then we have and

j m

s

s

_j

a

^j

, 2 ( 1 )

11 1

1

  s 

₁₁²

a

₁₁

.Let us now suppose that we have operated on the first j1 columns of [S], i.e. We have either calculated the elements or eliminated them. The operation on the

j

^thcolumn gives

.

, 1

2

2











j i k i

ij jj

jj

a s

s

(16)

where k1 is the set of indices of rows and columns not eliminated. If s_jj  0 we eliminate the

j

^th columns and rows [A_q] and [S_q] where [A_q] and [S_q] are the current reduced matrices derived to date from

[ A

_m

]

and

[ S

_m

]

,

q  m

. Otherwise we take

jj j

i k i

ij ij

ij j

i k i

ij jj

jj

a s s a s s

s  

















, 1

2 ,

1

2

,

(17)

This process is repeated for each column until we finally obtain the reduced matrix [A_q]given by

Tr q q

q

S S

A ] [ ][ ]

[ 

(18) 4) Solve the system of the linear equations

[ A

_q

] [  x

_i

]  [ B

_q

]

where B_q is the reduced form of gradient vector corresponding to A_q. 5) Compute a new point

x

^v1 where

















 





q i v

i

q i i

v v i

i x for x

x for x x ₁ x ¹



and



is a parameter which takes values

,  4 , 1 2 , 1

1

and we use the first value of



which satisfies

F ( x

^v1

)  F ( x

^v

)

. If



becomes too small without satisfying this condition, the calculation can be restarted with a finer approximation of the matrices

[ A

_m

]

and

[ B

_m

]

, i.e. smaller values

l

_i.

4.2 The numerical procedure

The outlined of the numerical procedure suggested with PQI for solving the optimal control problem is as follows:

1) Given

k  0 , 

^*

 0 ,   0

and

v

^(k)

 V

. 2) Solve state system (1)-(4), then find

y (., v

^k

)

.

3) Minimize f_(v^(k)) to find optimal control

v

_*^(k1) using PQI technique.

4) If f_(v^(k1)) f_(v^(k)) 



, then Stop, else, go to Step 2.

5) Set

v

^(k1)

 v

^(k)

, k  k  1

and go to Step 2 4.3 Numerical Results

In this section we illustrate the numerical procedure suggested with PQI to solve two cases numerically as follows:

OCP1: In the first problem the control takes the form

v (t )

which appears in the second boundary condition.

OCP2: In the second problem the control takes the form

v (x )

which appears in the initial condition of the parabolic equation.

Example 1(OCP1): Let



be a number in

( 0 ,  ]

. Then

y ( x , t )  e

^2t

cos (  x )

is the solution of the problem

] , 0 [ ] 1 , 0 [ ) , ( , )

( x t T

x x y x

t

y   















 



  (19)

l x x

x

y ( , 0 )  cos (  ) , 0  

(20)

T

t x e

t x x y x

t x

y

_t

x x







 

 

 



_





0 , ) ( ) sin

, ) ( ( , ) 0

,

(

²

1 0







^ (21)

Let

v ( t )  e

^42t

cos

⁴

(  )   e

^2t

sin (  )

. We see that y(1,t)_x y⁴(1,t)v(t) . Thus, in this example we set

G ( y )   y

⁴. To test our method we set

z ( x )  y ( x , T )  e

^2T

cos (  x )

. In this example,

T

is set to be 1 and the space and time gird sizes are taken to be 0.05 and 0.05, respectively. To analyze the effects of the initial guess on the reconstruction of the control function, the algorithm was run with three initial guesses functions in figures 1. We tested our algorithm for

  

, then

v ( t )  e

^42t

, z ( x )  e

^2T

cos (  x )

.

Figure 1: Values of control function

v (t )

with 3 initial guesses

The numerical results are given in figures 2-5. In figure 2 shows the computing control function compared with the exact one for OCP. But in figures 3 and 4 show the values of the functional and gradient functional

f

_

(v )

various iterations numbers very close to one another and close to zero.

Figure 2: The values of exact and approximate control function

v (t )

for OCP.

Figure 3: The values of function

f

_

( v )  L

₂

[ 0 , T ]

various iteration numbers

Figure 4: The values of f_^/

 

v various iteration numbers

But in Figure 7 shows the exact state results and the present numerical results for y_i^j in

x  i h

and

t  j 

when

h    1 20

20 at N  M  8 for OCP.

Figure 7: The values of state y_i^j at

i  j  8

for OCP

Example 1(OCP2): Let  {(x,t):xD,t(0,T)}and D be a bounded domain in E_N. Consider the following process

 

T t t

g t

x y t x t

g t

x y t x

l x D

x x v x

y

t x t x x f

t y x x

t y

l x

x    















 















 







 ( ), ( , ) ( , ) ( ) , 0

) , ( ) , (

) 0

( ,

) ( ) 0 , (

) , ( , , )

, (

1

0 0 





(22)

On the set { : ( ) [0, ], , 0}

] , 0 2 [

2









 v v v x L l v R R

V L l , we are required to minimize

the functional

   

²_[0,]

0

2 1 1

0

2 0 0

0 ( , ) ( ) ( , ) ( ) ₂

)

( _{L l}

T

l x T

x z t dt y x t z t dt v

t x y v

f_  



^   



^    (23)

where

z

_i

( t )  L

₂

( 0 , T ) , i  0 , 1

are given functions.

The numerical results were carried out for the following example:

2 1

2 0

2 2

) 3 . 1 sin(

) ( , ) sin(

) ( , ) cos(

) sin(

) ( , 1

0004 . 0 0

, 3 . 1 0

, ) cos(

) sin(

x t

z t t

z x x

x v

t x

t x t

x y



























(24)

2 1

2 0

2 2

2 2

) 3 . 1 sin(

) ( , ) sin(

) (

) cos(

) ( 2 ) cos(

2 ) sin(

) ( 4 ) , (

t t

g t t

g

t x t

x t

x t

x t

x t

x f





















 (25)

The numerical results are given in figures 8-11. In figure 8 shows the estimated control function compared with the exact control

v (x )

various iteration (NR).

Figure 8: Approximate and exact controls values

v (x )

various iteration numbers

But in figures 9 and 10 show the values of the functional and gradient functional

)

(v

f

_ various iterations numbers very close to one another and close to zero. Figure 11 shows by increasing the values of



approximate values of v(x)L₂[0,l] very close to the exact vales of control.

Figure 9: The values of function

f

_

( v )  L

₂

[ 0 , l ]

various iteration numbers

Figure 10: The values of f_^/

 

v various iteration numbers

Figure 11: The values of v(x)L₂[0,l] various different values of



5. CONCLUSION

Our work is devoted to a class of optimal control problems of parabolic partial differential equations. Because of the partial differential equations constraints, it is rather difficult to solve the optimization problem. The gradient of the cost function can be found by the adjoint problem approach. The partial quadratic interpolation technique is applied to solve this optimization problem and this algorithm is given. The numerical results of two types of optimal control parabolic problems are also given.

REFERENCES

1. R. Gabasov, F. M. Kirillova, D. S. Kuzmenkov, Real -Time optimal control of a special distributed parameter systems, Computational Mathematics and Mathematical Physics, 4(12), 1765-1775, (2014).

2. Chunyan Du and Guansheng Xing, Control of nonlinear distributed parameter systems based on global approximation, Journal of Applied Mathematics, V. 2014, 1-6, (2014).

3. M. H. Farag, On an optimal control constrained problem governed by parabolic type equations, Palestine Journal of Mathematics, 4(1),136-143, (2015).

4. K.D. Do, Inverse Optimal control of linear distributed parameter systems, Applied Mathematical Sciences, Vol. 8, No. 7, 293 – 313, (2014).

5. A.Ya. Akhundov, A.I. Gasanova, On an inverse problem for a semilinear parabolic equation in the case of boundary value problem with nonlinear boundary condition, Azerbaijan Journal of Mathematics, 4(2), (2014).

6. R. K. Tagiyev, Optimal control for the coefficients of a quasilinear parabolic equation, Automation and Remote Control, 70(11), 1814-1826, (2009).

7. N. U. Ahmed and X. Xiang, Nonlinear boundary control of semilinear parabolic systems, SIAM Control and Optim. 34(2), 473-490, (1996).

8. A. Hasanov, Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach, J. Math. Anal. Appl. ,330, 766–

779, (2007).

9. K. Gelashvili, The existence of optimal control on the basis of Weierstrass’s theorem, Journal of Mathematical Sciences, 177(3), 373-382, (2011).

10. M. H. Farag, T. A. Nofal, Mohamed A. El-Sayed and T. S. M. AL-Qarni, An optimization problem with controls in the coefficients of parabolic equations, British Journal of Mathematics & Computer Science, 6(6), 532-543, (2015).

11. M. H. Farag, F. H. Riad and W. A. Hashem, A penalty/MPQI Method for constrained parabolic optimal control problems, IPASJ International Journal of Computer Science (IIJCS), 2(4), pp. 14-22, (2014).