Exact Solution of a Linear-Quadratic Inverse Eigenvalue Problem on a Certain Hamiltonian Symmetric Matrices

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 10, October 2015)

Problem on a Certain Hamiltonian Symmetric Matrices

.

1

2

, F.T Oduro

3

1

University for Development Studies, Navrongo

2,3Kwame Nkrumah University of Science and Technology, Kumasi

Abstract-- This paper investigate the exact solution of an inverse eigenvalue problem (IEP) on a certain Hamiltonian symmetric matrices namely singular symmetric matrices of rank 1 and non-singular symmetric matrices in the neighborhood of the first type of matrices via the Newton’s iterative method.

Keywords-- Inverse, Hermiltian, eigenvalue, exact, symmetric, iterative

I. INTRODUCTION

Various theoretical results on the solvability of the inverse eigenvalue problem for Hamilton matrices together with numerical examples are systematically reviewed and discussed in respect of the inverse eigenvalue problems for certain singular and non-singular Hamilton matrices in Oduro et al (2012),Oduro (2012a, b), Baah Gyamfi (2012) as well as Oladejo et.al (2014) and (2015).This paper investigate and established the exact solution of an inverse eigenvalue problem (IEP) on a certain Hamilton matrices consist of both singular and non-singular symmetric matrices of rank 1 via Newton’s iterative method.

Linear-quadratic optimal control system

Throughout this paper we define our notation as follows: We let

nn nm nn T

and

mm T

Where

Q

is a symmetric positive semi definite matrix and

R

is a symmetric positive definite matrix

We find the linear- quadratic optimal control for the functional:

1

0

t

t

T T

i

Ix

Subject to differential equation

0

1

i

i



Constructing the Hamiltonian equation from equation (1) and (2) yields:

T

T

T



Given any optimal input

u

and the corresponding state

 

T

T

Thus:

1

T

5

and the adjoint equation yields:

T

 

1

1 0

  

T T T

Thus;

0 1 1

 

T

Consequently;

,

, ( ) , ( ) 0

 

8

, ) (

) ( )

( ) (

1 0

1 0

1

 

         

 

 

 

     

 

  

 

t p x t x t t t

t p

t x

A Q

B BR A

t p

t x dt

d

i T

T

Equation (8) is then a linear, time variant differential equation in



Inverse Eigenvalue-Problem for singular

n

symmetric matrix of rank 1

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 10, October 2015)

44 43 42 41

34 33 32 31

24 23 22 21

14 13 12 11

A

We assume that the singularity is due to the row dependence relations specified below:

1 1

i



i

14 1 24 13 1 23 12 1 22 11 1

21



14 2 34 13 2 33 12 2 32 11 2

31

a

14 3 44 13 3 43 12 3 42 11 3

41

11 2 1 21 1 12 1

22

11 2 1 31 1 13 1

23

11 3 1 41 1 14 1

24

11 2 2 31 2 13 2

33

11 3 2 41 2 14 2

34

11 2 3 41 3 14 3

44

Thus:

2 3 3 2 3 1 3

3 2 2 2 2 1 2

3 1 2 1 2 1 1

3 2 1

44 43 42 41

34 33 32 31

24 23 22 21

14 13 12

11

A

To solve the inverse eigenvalue problem (IEP) we use the given nonzero eigenvalue as follows

11

12

22

32

2 3 2 2 2 1 11

2 3 3 2 3 1 3

3 2 2 2 2 1 2

3 1 2 1 2 1 1

3 2 1

Illustration

Given that

1

2

3

b

Hence, matrix

is a

4

singular symmetric matrix which has been reconstructed from the given nonzero eigenvalue and the prescribed dependence relation parameters.

Inverse Eigenvalue Problem for a non-singular

4

symmetric matrix- Newton’s method

We construct a characteristic (Polynomial) function of the diagonal elements of the matrix

44 43 42

41

34 33

32 31

24 23

22 21

14 13

12 11

44 43 42 41

34 33 32 31

24 23 22 21

14 13 12 11

A

In other words, consider the function with independent variables defined on 4 selected elements of matrix

A

, precisely, the diagonal elements:

11

22

33

44

2

det

Thus, given four distinct eigenvalues

1

2

3



4 we have the following four (4) functions with 4 independent variables being the diagonal element of

A

       

 

       

 

 

 

 

 

 

 

 

 

 

 

 

 

  

33 42 21 14 32 43 21 14 42 34 21 13

44 32 21 13 43 34 21 12 44 33 21 12 43 34 22 11

44 33 22 11 1 42 21 14 32 21 13 44 21 12 33 21 12

43 34 11 44 33 22 43 33 11 44 33 11 44 21 11

33 22 11 1 2 21 12 43 34 44 33 44 22 33 22

44 11 33 11 22 11 1 3 44 33 22 11 1 4

44 33 22 11 1

) ( ) (

) (

) , , , (

a a a a a a a a a a a a

a a a a a a a a a a a a a a a a

a a a a a

a a a a a a a a a a a

a a a a a a a a a a a a a a a

a a a a

a a a a a a a a a

a a a a a a a

a a a

a a a a f

  

(3)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 10, October 2015)

295

                                                   33 42 21 14 32 43 21 14 42 34 21 13 44 32 21 13 43 34 21 12 44 33 21 12 43 34 22 11 44 33 22 11 2 42 21 14 32 21 13 44 21 12 33 21 12 43 34 11 44 33 22 43 33 11 44 33 11 44 21 11 33 22 11 2 2 21 12 43 34 44 33 44 22 33 22 44 11 33 11 22 11 2 3 44 33 22 11 2 4 44 33 22 11 2 ) ( ) ( ) ( ) , , , ( a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a f



                                                   33 42 21 14 32 43 21 14 42 34 21 13 44 32 21 13 43 34 21 12 44 33 21 12 43 34 22 11 44 33 22 11 3 42 21 14 32 21 13 44 21 12 33 21 12 43 34 11 44 33 22 43 33 11 44 33 11 44 21 11 33 22 11 3 2 21 12 43 34 44 33 44 22 33 22 44 11 33 11 22 11 3 3 44 33 22 11 3 4 44 33 22 11 3 ) ( ) ( ) ( ) , , , ( a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a f



                                                   33 42 21 14 32 43 21 14 42 34 21 13 44 32 21 13 43 34 21 12 44 33 21 12 43 34 22 11 44 33 22 11 4 42 21 14 32 21 13 44 21 12 33 21 12 43 34 11 44 33 22 43 33 11 44 33 11 44 21 11 33 22 11 4 2 21 12 43 34 44 33 44 22 33 22 44 11 33 11 22 11 4 3 44 33 22 11 4 4 44 33 22 11 4 ) ( ) ( ) ( ) , , , ( a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a f



Derivation of an Explicit Formula for the Jacobian in the 4

4 case

22 33 44

 

1 22 33 33 44 22 44

1 2 1 3 11 1

11 33 44

 

1 11 33 33 44 11 44

1 2 1 3 22 1

11 22 44

 

1 11 22 22 44 11 44

1 2 1 3 33 1

11 22 33

1



11 22 22 33 11 33

1 2 1 3 44 1

22 33 44

2



22 33 33 44 22 44

2 2 2 3 11 2

11 33 44

2



11 33 33 44 11 44

2 2 2 3 22 2

11 22 44

2



11 22 22 44 11 44

2 2 2 3 33 2

:

11 22 33

2



11 22 22 33 11 33

2 2 2 3 44 2

22 33 44

3



22 33 33 44 22 44

3 2 3 3 11 3

:

11 33 44

3



11 33 33 44 11 44

3 2 3 3 22 3

11 22 44

3



11 22 22 44 11 44

3 2 3 3 33 3

11 22 33

3



11 22 22 33 11 33

3 2 3 3 44 3

22 33 44

4



22 33 33 44 22 44

4 2 4 3 11 4

11 33 44

4



11 33 33 44 11 44

4 2 4 3 22 4

11 22 44

4



11 22 22 44 11 44

(4)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 10, October 2015)

296

       

 

        

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

44 4 33 4 22 4 11 4

44 3 33 3 22 3 11 3

44 2 33 2 22 2 11 2

44 1 33 1 22 1 11 1

a f a

f a

f a

f

a f a

f a

f a

f

a f a

f a

f a

f

a f a

f a

f a

f

J





       

 

         

 

 

     

     

     

    

 

     

     

     

    

 

     

     

     

    

 

     

     

     

    

33 11 33 22 22 11 4

33 22 11 4 2 4 3

44 11 44 22 22 11 4

44 22 11 2 2 4 3

44 11 44 33 33 11 4

44 33 11 4 2 4 3

44 22 44 33 33 22 4

44 33 22 4 2 4 3

33 11 22 11 33 22 3

33 22 11 3 2 3 3

44 22 44 11 22 11 3

44 22 11 3 2 3 3

44 11 44 33 33 11 3

44 33 11 3 2 3 3

44 22 44 33 33 22 3

44 33 22 3 2 3 3

33 11 33 22 22 11 2

33 22 11 2 2 2 3

44 11 44 22 22 11 2

44 22 11 2 2 2 3

44 11 44 33 33 11 2

44 33 11 2 2 2 3

44 22 44 33 33 22 2

44 33 22 3 2 3 3

33 11 33 22 22 11 1

33 22 11 1 2 1 3

44 11 44 22 22 11 1

44 22 11 1 2 1 3

44 11 44 33 33 11 1

44 33 11 1 2 1 3

44 11 44 33 33 11 1

44 33 22 1 2 1 3

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

a a a a a a

a a a

   

  

  

 

  

  

  

 

  

  

  

 

   

  

  

 

II. NUMERICAL EXAMPLE

Consider the nonsingular symmetric Coefficient matrix of the system:

2 22 1 2

2 1 11 1



Given the eigenvalues

1

2

3

i.e., given a target solution of the form

t t

1



2 3

Assuming the initializing matrix

) 0 (

Moreover,

A

(0) is singular with

11

Also

(0)

5

We first compute the values of the functions at the initial point:

11 22

1

11 22

1

11 22 122

2

1

11 22

2

11 22

2

11 22 122



2

2

(0)

f

Using the formula for the inverse of the Jacobian matrix, we have:

2 11 2 22

1 11 1 22

11 22 2 1

1

11 22 2 1 1



Substitute the values into the Newton’s iterative equation. i.e.

( ) ( )

1 ) ( ) 1

(n n n n

Then:

(1)

(0)

1

(0)

(0)

) 0 (

X

                     

1 1

3 0

4 1

     

1 1 )

1 (

(5)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 10, October 2015)

1

A

Hence, we obtain the (exact) solution.

III. CONCLUSION

Theoretical results on the solvability of the inverse eigenvalue problem for Hamilton matrices was systematically reviewed and discussed in respect of the inverse eigenvalue problems (IEP) for certain singular and non-singular Hermitian matrices. Base on this we have successfully investigate and established the exact solution of an inverse eigenvalue problem (IEP) on a certain Hermiltian matrices consist of both singular and non-singular symmetric matrices of rank 1 via Newton’s iterative method.

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[3] Cai, Y.F. Kuo, Y.C Lin W.W and Xu, S.F (2009) Solutions to a quadratic inverse eigenvalue problem, Linear Algebra and its Applications. Vol. 430. Pp.1590-1606

[4] Datta B.N and Sarkissian, D.R (2004) Theory and computations of some Inverse Eigenvalue Problems for the quadratic pencil. Journal of Contemporary Mathematics,Vol.280 221-240.

[5] Miranda.J.O.A.O. Optimal Linear Quadratic Control. Control system, Robotics and Automation.Vol.VIII INESC.JD/IST,R.Alves,Redol 9.1000-029.Lisboa, Portugal [6] Oduro F.T., A.Y. Aidoo, K.B, Gyamfi, J. and Ackora-Prah, (2012)

Solvability of the Inverse Eigenvalue Problem for Dense Symmetric Matrices‟ Advances in Pure Mathematics, Vol. 3, pp 14- 19, [7] Oladejo N.K, Amponsah S.K and Oduro F.T (2014).Linear–

Quadratics optimal Control problem(LQOCP) and the Definiteness of an inverse eigenvalue Problem (IEP) on a certain Hermitian Matrices. International Journal of Emerging Technology and Advance Engineering Vol.5,Issue 5

[8] Oladejo N.K, Amponsah S.K and Oduro F.T (2014): An inverse eigenvalue problem for linear-quadratic optimal control Problem. International Journal of mathematical Archive 5(4) 306-314. [9] Ram Y.M and El-Golhary (1996). ‘An inverse eigenvalue problem

for the symmetric tridiagonal quadratic pencil with application of damped oscillatory systems’ SIAM J. Applied Math., Vol. 56 pg. 232–244

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