International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org ǁ Volume 2 ǁ Issue 3 ǁ March 2014 ǁ PP-64-72

Duality between initial and terminal function problems Ukwu Chukwunenye Department of Mathematics, University of Jos, P.M.B 2084 Jos, Nigeria.

ABSTRACT: This paper successfully investigated the topological duality between initial and terminal function problems with respect to single-delay autonomous linear control systems using the expressions for the solution matrices and indices of control systems developed and proved in Ukwu and Garba [2014a, c] for single – delay autonomous linear systems, as well as the general variation of constants formula for the relevant systems. The paper obtained independent proofs that the solution matrices and the indices of control systems are analytic almost everywhere and differentiable except at a single point. The proofs were achieved using an algorithm involving key transformations from one problem to the other, effectively establishing that the results from one problem can be realized from the other.

KEYWORDS: Duality, Initial, Problems, Terminal, Transformations. I.

INTRODUCTION

The qualitative approach to the controllability of functional differential control systems have been areas of active research for the past fifty years among control theorists and applied mathematicians in general. This circumvents the severe difficulties associated with the search for and computations of solutions of such systems. Unfortunately computations of solutions cannot be wished away in the tracking of trajectories and many practical applications. Literature on state space approach to control studies is replete with variation of constants formulas, which incorporate the solution matrices of the free part of the systems. See Chukwu (1992), Gabsov and Kirillova (1976), Hale (1977), Manitius (1978), Tadmore (1984), Ukwu (1987, 1992, 1996) and Ukwu and Garba [2014a]. Furthermore the importance of indices of control systems matrices stems from the fact that they not only pave the way for the derivation of determining matrices for the determination of Euclidean controllability and compactness of cores of Euclidean targets but can be used independently for such determination, Ukwu and Garba [2014c]. Regrettably no author had made any noteworthy attempt to obtain general expressions for such solution matrices and indices of control systems matrices involving the delay h until Ukwu and Garba [2014a,c]. This article makes further contribution to knowledge in controllability studies by establishing that initial and terminal function problems are duals of each other, using Ukwu and Garba [2014a,c] as key components in the ensuing analyses and much more.

II.

PRELIMINARIES

Consider the initial function problem:

x (t )  A0 x(t )  A1 x(t  h)

(1)

x(t )   (t ), t [h,0]

(2)

and the terminal function problem:

y ( )   y( ) A0  y(  h) A1

(3)

y( )   ( ),  [t1  h, t 1 ]

(4)

where A0 and A1 are n  n constant matrices; x (.) and y (.) are column and row n-vector functions respect ively. Let Y (t , ) and X ( , t ) be solution matrices of systems (1) and (3) respectively. Then X ( , t ) and Y (t ,  ) satisfy respectively the following matrix differential equations:

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Duality between initial and terminal function problem…

I , t    Y  t ,   A0Y  t ,   A1Y  t ,  h  , where Y (t , )   n t 0, t  

(5)

 X   , t    X  , t  A0  X   h, t  A1 , 

(6)

for 0    t ,   t  k h, k  0,1,... where

In ;   t X  , t    0;  t

 

(7)

See Chukwu (1992), Hale (1977) and Tadmore (1984) for properties of importance

is

the

fact

that

    , t 

is

X t,   . of particular analytic

on

the

intervals  t1   j  1 h, t1  j h  , j  0,1,...; t1   j  1 h  0 . Any such    t1   j  1 h, t1  j h  is called a regular point of     t ,   . The first result we will establish is that Y (t ,  ) and X ( , t ) are topological duals of each other. This result will rely on and exploit the following Ukwu-Garba’s theorems on global expressions for the matrices Y (t ,  ) and X ( , t ). See Ukwu and Garba [2014 a,c].

Let J k i  [(k  i)h, (k  1  i)h], k  {0,1, }, i  {0,1}; let K j  [t 1  ( j  1)h, t 1  jh], j  0 : t 1  ( j  1)h  0. THEOREM: UKWU-GARBA’S SOLUTION MATRIX FORMULA FOR AUTONOMOUS, SINGLE – DELAY LINEAR SYSTEMS e A0t , t  J 0 ; (8)  t e A0t  e A0 (t  s ) A e A0 ( s h ) ds, t  J ; (9) 1 1 h    t Y (t )   A0t A (t  s ) A ( s h ) e  h e 0 A1e 0 ds   si  h j   k t A (t  s )  A0 ( si  h  si 1 ) A0 ( s1  h ) A e A e ds , t  J k , k  2 (10)     e 0 j    1 1)h 1 j  2  i  j , j  1,  ,2   1     jh ( i     III.

IV.

THEOREM: UKWU-GARBA’S CONTROL INDEX THEOREM FOR AUTONOMOUS LINEAR DELAY CONTROL SYSTEMS:

e A0 t 1   ,   K ; (11) 0   A0 t 1    A0 (t 1  h  s 1 ) A0 ( s 1  )   e A1e ds1 ,   K1 ; (12) e t 1 h   X ( , t1 )   A0 t 1    A0 (t 1  h  s 1 ) A0 ( s 1  )   e A1e ds1 (13) e t 1 h    j si  h   k  k 1 A0 ( t 1  h  s 1 ) A0 ( sq  h  sq1 )  A0 ( sk  )    1 j   ds ,  K j , j  2 e A e A e  1  1        1 k 2 i  k , k  1,  ,2 q  1   t 1  kh ( i 1) h     www.ijmsi.org

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Duality between initial and terminal function problem… Our current result is captured in the following theorem.

THEOREM: UKWU’S TOPOLOGICAL DUAL ALGORITHM FOR AUTONOMOUS LINEAR DELAY SYSTEMS Y (t ,  ) and X ( , t ) are topological duals of each other subject to the transformations below. To obtain X ( , t ) from Y (t ,  ) ) perform the following operations; V.

(i) (ii)

t  t  in the formula for Y (t ). (.)  t  (.) in the lower integral limits, as well as in the leading integral limits in (10) with the preceding

sign

( 1) j , for 1  j  k , followed

convention

j  k, Jk   K j ;

k

j

j 2

k 2

by

the

notational

changes

 (.)  (.)

(iii) s i  h  s i  h in the upper integral limits in Y (t ,  ).

(iv) The leading exponential function e

A 0 ( t   s j )

in Y (t , ) and the trailing exponential function e

 e

A0 ( s 1 h )

A 0 (t hs 1 )

 e

, j  {2, 3,  k }, in the integrals

A 0 ( s j  )

.

To obtain Y (t ,  ) from X ( , t ) perform the following operations: (v)

(.)  t  (.) in the leading integral limits, as well as in the lower integral limits. Omit the sign convention

( 1) j , for 1  j  k , and effect the notational changes K j  J k   ; (vi) Replace the leading exponential function e and the trailing exponential function e

A0 (t h s 1 )

by e

A 0 ( s j  )

A 0 ( t   s j )

j

k

k 2

j 2

 (.)   (.) , j  {1, 2,  k},in X ( , t )

A0 ( s 1 h )

by e (vii) Replace the upper integral limits s i  h in X ( , t ) by s i  h. (viii) Given the initial function problem (1) and (2) the equivalent terminal function problem is given by:

y ( )   y( ) A0  y(  h) A1

(14)

y( )   ( ),   [t1  h, t 1 ]

(15)

where

 ( )   T (  t1 ), [t1  h, t1 ];(.)T denotes the transpose of (.).

(16)

Proof We need to prove that the continuity and analytic dispositions of X ( , t1 ),  K j can be inferred from those of Y (t , ), t  J k   and vice-versa. In particular, we prove that the continuity and non-analytic behavior

of

X ( , t 1 ) , for   t 1  jh : t 1  ( j  1) h  0

correspond

to

those

of

Y (t , ), for t    kh : t  t 1. We start from the expression for Y (t ,  ). Note that Y (t ,  )  Y (t   ) since the systems (1) and (5) are autonomous. So we may replace t by t   in Ukwu and Garba’s solution matrix theorem to secure Y (t , ). This justifies (i). With t  t 1 , in Ukwu and Garba’s Control Index theorem, the transformation (.)  t  (.) , in (ii) ensures that t  t 1   , jh  t1  jh, t     , (i  1) h  t 1  (i  1) h. These, together with the sign convention and the notational changes in (ii) guarantee that (8) and (9) with t replaced by t  transform to (11) and (12) respectively. We need to prove that the multiple integral in (10) transforms to the multiple integral in (13). To achieve the proof, combine the foregoing with j  k and the transformations in (iii) and (iv) to deduce immediately that the multiple integral in (10) transforms to the

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Duality between initial and terminal function problem… multiple integral in (13), proving that X ( , t 1 ) has been obtained from Y (t , ). The proof that X ( , t 1 ) transforms to Y (t , ) is similar. Finally we investigate the continuity and analytic dispositions of Y (t , ) and X ( , t 1 ). Being sums of integrals of products of constant matrices and exponential functions we deduce that

Y (t , ) and X ( , t 1 ) are analytic in the interiors of J k   and K j respectively, since e

Y (t , ) and X ( , t 1 ) on the boundaries of

Finally we must examine

A 0 (.)

is C  .

J k   and K j respectively.

Y (t , )  0 t    lim Y (t , )  0, lim Y (t , )  0; t 

t 

Y (t , )  e  t  [ ,  h]  lim Y (t , )  I , lim Y (t , )  A0 . A0 t

t 

Therefore Y (t ) is not continuous at analytic at t   .

t 

t   and certainly not differentiable there. It follows that Y (t ) is not

Ah Ah lim Y (t , )  A0 e ; lim  Y (t ,  )  A0 lim  Y (t ,  )  A1 lim  Y (t ,   h)  A0Y ( h)  AY (0)  A0e  A1 , 1 0

0

t (  h )

t (  h )

by the continuity of Y (t ) for

t (  h )

t (  h )

t  0 and the appropriate evaluations using the interval J 0   for the left limit

and J1   for the right limit. Alternatively, we apply Leibniz’s rule for differentiating an integral to

lim  Y (t , ) to obtain A0e

A0 (2 h  h )

t (  h )

 A1e

A0 ( h  h )

 h

0

 

A0e A0 (t s ) A1e A0 (t s ) ds  A0e A0h  A1.

h

Therefore Y (t , ) is not differentiable at

t    h and hence not analytic there. Equivalently using the fact

that (1) and (5) are autonomous, Y (t ) is not differentiable at t  h and hence not analytic there. t

Y (t )  e A0t   e A0 (t  s ) A1e A0 ( s  h ) ds h

si  h j  k t A (t  s )  A0 ( si  h  si 1 ) A0 ( s1  h )    e 0 j A e A e ds , t  J k , k  2  1   1  i j , j 1,,2 ( i 1) h  1  j  2 jh 

kh

 lim  Y (t )  e A0 kh  t ( kh )

e

A0 ( kh  s )

A1e A0 ( s  h ) ds

h

si  h j  k kh A0 ( kh  s j )  A0 ( si  h  si 1 ) A0 ( s1  h )    e A e A e ds , k  2    1  1 i j , j 1,,2 ( i 1) h  1  j  2 jh 

kh

lim  Y (t )  e A0kh 

t ( kh )

e

A0 ( kh  s )

A1e A0 ( s h ) ds

h si  h j  k 1 kh A0 ( kh  s j )  A0 ( si  h  si1 ) A0 ( s1  h )    e A e A e ds , k  1  2    1  1 i j , j 1,,2 ( i 1) h  1  j  2 jh 

since the integral contribution at j  kh is zero. t

On

J k 1 , Y (t )  e

A 0t

  e A0 (t  s ) A1e A0 ( s h ) ds h

si  h j   A (t  s ) A0 ( si  h  si 1 ) A0 ( s1  h )    e 0 j A e A e ds , k  1  2  1   1  i j , j 1,,2 ( i 1) h  1  j  2 jh  k 1 t

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Duality between initial and terminal function problem… kh

 lim  Y (t )  e

A 0 kh

t ( kh )

  e A0 ( kh  s ) A1e A0 ( s  h ) ds h

si  h j   A0 ( s 1  h ) A ( kh  s j ) A0 ( si  h  si 1 )    e 0 A e A e ds , k  1  2  1   1  i j , j 1,,2 ( i 1) h  1  j  2 jh  This completes the proof of the continuity of Y (t ), for t  0, and hence that of Y (t ,  ), t   k 1 kh

t

Y (t )  A0e A0t  A1e A0 (t  h )   A0e A0 (t  s ) A1e A0 ( s  h ) ds h k



si  h

j

j  2 i j , j 1,,2 ( i 1) h

A1e A0 ( si  h  si1 ) A1e A0 ( s1  h )  ds  1

j  k t  A (t  s ) A0 ( si  h  si 1 ) A0 ( s1  h )     A0 e 0 j A e A e ds , t  J k , k  2    1  1 i j , j 1,,2 ( i 1) h  1  j  2 jh  si  h

kh

 lim  Y (t )  A0e A0 kh  A1e A0 [ k 1]h   A0e A0 ( kh  s ) A1e A0 ( s h ) ds t ( kh )

h

k



si  h

j

j  2 i j , j 1,,2 ( i 1) h

A1e A0 ( si  h  si1 ) A1e A0 ( s1  h )  ds  1

si  h j  k 1 kh  A (t  s ) A0 ( si  h  si 1 ) A0 ( s1  h )     A0 e 0 j A e A e ds , k  3  1   1  i j , j 1,,2 ( i 1) h  1  j  2 jh 

since the integral contribution at t  kh is zero. kh

t  J k 1  lim  Y (t )  A0e A0 kh  A1e A0 [ k 1]h   A0e A0 ( kh  s ) A1e A0 ( s  h ) ds t  ( kh )

h k 1



si  h

j  2 i j , j 1,,2 ( i 1) h

j

A1e A0 ( si  h  si1 ) A1e A0 ( s1  h )  ds  1

si  h j  k 1 t  A ( kh  s j ) A0 ( si  h  si 1 ) A0 ( s1  h )     A0 e 0 A e A e ds , k  1  2  1   1  j  2 i  j , j  1,  ,2   1   ( i 1) h  jh 

We proceed to take the second derivative of Y( t ). t

Y(t )  A02e A0t  A1 A0e A0 (t h )  A0 A1e A0 (t h )   A02e A0 (t  s ) A1e A0 ( s  h ) ds h

k

  A0 j 2

si  h

i j , j 1,,2 ( i 1) h

j

A1e A0 ( si  h  si1 ) A1e A0 ( s1 h )  ds  1

si  h j  k t  A (t  s ) A0 ( si  h  si 1 ) A0 ( s1  h )     A02 e 0 j A e A e ds , t  J k , k  2  1   1  i j , j 1,,2 ( i 1) h  1  j  2 jh 

kh

 lim  Y(t )  A02e A0kh  A1 A0e A0 [ k 1]h  A0 A1e A0 [ k 1]h   A02e A0 ( kh  s ) A1e A0 ( s h ) ds t ( kh )

h

k

  A0 j 2

si  h

i j , j 1,,2 ( i 1) h

j

A1e A0 ( si h  si1 ) A1e A0 ( s1 h )  ds  1

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Duality between initial and terminal function problem…

si  h j  k kh 2 A0 ( kh  s j )  A0 ( si  h  si 1 ) A0 ( s1  h )     A0 e A e A e ds , k  2    1  1 i j , j 1,,2 ( i 1) h  1  j  2 jh  kh

lim  Y(t )  A02 e A0 kh  A1 A0e A0 [ k 1]h  A0 A1e A0 [ k 1]h   A02e A0 ( kh  s ) A1e A0 ( s h ) ds

t ( kh )

h

k 1

  A0 j 2

si  h

i j , j 1,,2 ( i 1) h

j

A1e A0 ( si  h  si1 ) A1e A0 ( s1  h )  ds  1

si  h j  k 1 kh  A ( kh  s j ) A0 ( si  h  si 1 ) A0 ( s1  h )     A02 e 0 A e A e ds , k  2    1  1 i j , j 1,,2 ( i 1) h  1  j  2 jh 

Clearly, lim  Y(t )  lim  Y(t )  A0 t ( kh )

Clearly,

t ( kh )

si  h

ik , k 1,,2 ( i 1) h

k

A1e A0 ( si h  si1 ) A1e A0 ( s1 h )  ds  1

lim Y(t )  A e , 2 A0 h 0

t h 

t

Y(t )  A02e A0t  A1 A0e A0 (t h )  A0 A1e A0 (t h )   A02e A0 (t s ) A1e A0 ( s h ) ds h

 lim Y(t )  A02e A0h  A1 A0  A0 A1  lim Y(t )  lim Y(t )  A1 A0  A0 A1 t h

t h

t h

Therefore, Y (t ) is differentiable except at 0, but not twice differentiable at kh, for any nonnegative integer k. This completes the proof that

Y (t ) is non-analytic at kh , for k nonnegative.

Additional information revealed is the differentiability of Y (t ), for t  0. We conclude the following: (i) Y (t , ) is analytic except at t    kh, k {0,1, }  Y (t , ) is analytic for t  R \ {  kh : k  0,1, }

(ii) Y (t , ) is differentiable at t    kh, k  0.

X ( , t1 )  e

  t1

A0 (t 1  )

, for   K0  lim X ( , t1 )  I n ; lim X ( , t1 )  0  X ( , t1 ) is not continuous at  t 1

 t 1

and hence not analytic there.

lim

  K1  X ( , t1 )  e

X ( , t 1 )  e

  ( t 1  h )  A0 (t 1  ) A0 (t 1  h  s1 )

e t h

A1e

A0h

,

A0 ( s1  )

ds1  lim  X ( , t1)  e A0h .  ( t 1  h )

1

Therefore X ( , t 1 ) is continuous at

  K1  X ( , t1 )  A0e

  t 1  h. By Leibniz’s rule of integral differentiation,

A0 (t 1  )

e

A0 ( t 1  h  )

A1  0 

e t h

A0 (t 1  h  s1 )

A1 A0e

A0 ( s1  )

ds1

1

lim

 ( t 1  h )

X ( , t1)  A0e

Therefore X ( , t 1 ) is not differentiable at

A0 h

 A1 ;

A h lim  X ( , t 1 )   A0 e 0 .

 ( t 1  h )

  t 1  h, and hence not analytic there.

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Duality between initial and terminal function problem… 

 A  t   A (t h s ) A (t h s ) A ( s  ) X ( , t1 )   A0e 0 1  e 0 1 1 A1   e 0 1 1 A1 A0e 0 1 ds1  t 1 h si  h k  j   k 1 j A (t h s ) A0 ( sq  h  sq1 )      1 e 0 1 1 A e A  1  1  ds    ik , k 1,,2 ( i 1) h  q 1  k  2    1 si  h   j  k  k 1 j 1 A0 ( t 1  h  s 1 ) A0 ( sq  h  sq1 )  A0 ( sk  )   ds ,  K j , j  2    1 e A e A A e  1 0    1   k  2   1 ik , k 1,,2 ( i 1) h  q 1 t 1  kh 

  lim   X ( , t1 )    A0e A0 jh  e A0 [ j 1]h A1    t 1  jh    

t1 j h

e

A0 ( t 1  h  s 1 )

A1 A0e

A0 ( s 1  jh t 1 )

ds 1

t 1 h

si  h k  j   k 1 j A0 ( t 1  h  s 1 ) A0 ( sq  h  sq1 )      1 e A e A ds       1 1  ik , k 1,,2 ( i 1) h  q 1   1  k  2  t 1  jh si  h  j 1  k  k 1 j 1 A0 ( t 1  h  s 1 ) A0 ( sk  jh t 1 ) A0 ( sq  h  sq1 )    ds , j  1  2     1 e A e A A e  1 0    1   k  2   1 ik , k 1,,2 ( i 1) h  q 1 t 1  kh  since the integral contribution from k  j is zero.

    K j 1  lim   X ( , t1 )    A0e A0 jh  e A0 [ j 1]h A1    t 1  jh    

t1 j h

e

A0 ( t 1  h  s 1 )

A1 A0e

A0 ( s 1  jh t 1 )

ds 1

t 1 h

si  h k  j   k 1 j A0 ( t 1  h  s 1 ) A0 ( sq  h  sq1 )      1 e A e A ds       1 1   ik , k 1,,2 ( i 1) h  q 1   1  k  2  t 1  jh si  h  j 1  k  k 1 j 1 A0 ( t 1  h  s 1 ) A0 ( sk  jh t 1 ) A0 ( sq  h  sq1 )    ds , j  1  2     1 e A e A A e     1 1 0    k  2   1 ik , k 1,,2 ( i 1) h  q 1 t 1  kh  Therefore X ( , t 1 ) is differentiable except at   t 1 . We proceed to take the second derivative:

2 A  t   A ( t  h  ) A (t h  s ) A ( s  ) X ( , t1 )  A02e 0 1  e 0 1 A1 A0   e 0 1 1 A1 A02e 0 1 ds 1 2  t 1 h si  h k  j  k  k 1 j 1 A0 ( t 1  h  s 1 ) A0 ( sq  h  sq1 )      1 e A e A A ds   ds      1 1 0    1 ik , k 1,,2 ( i 1) h  q 1   k  2   1 si  h   j  k  k 1 j 2 A0 ( t 1  h  s 1 ) A0 ( sq  h  sq1 )  2 A0 ( sk  )   ds ,  K j , j  2     1 e A e A A e      1 1 0   k  2   1 ik , k 1,,2 ( i 1) h  q 1 t 1  kh  t  jh

1  2  A (t h  s ) A ( s  jh t 1 ) A0 [ j 1] h 2 A0 jh  lim   2 X ( , t1 )   A0 e  e A1 A0   e 0 1 1 A1 A02e 0 1 ds 1   t 1  jh     t 1 h

si  h k  j  k  k 1 j 1 A ( t  h  s ) A0 ( sq  h  sq 1 )      1 e 0 1 1 A e A A ds  1  1 0     ds    1 k  2 i  k , k  1,  ,2 q  1   ( i 1) h      1 t 1  jh si  h  j 1  k  k 1 j2 A0 ( t 1  h  s 1 ) A0 ( sq  h  sq1 )  2 A0 ( sk  jh t 1 )   ds , j  1  2     1 e A e A A e  1  1 0     k  2   1 i  k , k  1,  ,2 q  1   t 1  kh ( i 1) h  

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Duality between initial and terminal function problem… because the integral contribution from

k  j is zero. t  jh

1  2  A (t h s ) A ( s  jh t )   K j 1  lim   2 X ( , t1 )   A02e A0 jh  e A0 [ j 1]h A1 A0   e 0 1 1 A1 A02e 0 1 1 ds 1   t 1  jh     t 1 h

si  h k  j 1  k  k 1 j 1 A0 ( t 1  h  s 1 ) A0 ( sq  h  sq 1 )      1 e A e A A ds  1 0     ds    1  1 ik , k 1,,2 ( i 1) h  q 1  k  2   1  t 1  jh si  h  j 1  k  k 1 j 2 A0 ( t 1  h  s 1 ) A0 ( sq  h  sq1 )  2 A0 ( sk  jh t 1 )   ds , j  1  2     1 e A e A A e  1 0    1   k  2   1 q  1 i  k , k  1,  ,2   t 1  kh ( i 1) h   Hence lim  X ( , t 1 )  lim  X ( , t 1 ), j  2 : t1  [ j  1]h  0.   t 1  jh   t 1  jh  Therefore, X ( , t 1 ) is differentiable except at   t 1 , but not twice differentiable at   t 1  jh , for

any nonnegative integer j. This completes the proof that X ( , t 1 ) is not analytic at   t 1  j h , for any nonnegative integer j. Additional information revealed is the differentiability of

X ( , t 1 ), for   t 1.

We conclude the following: (i) X ( , t 1 ) is analytic except at   t 1  jh, j  {0,1, } : t 1  [ j  1]h  0

 X ( , t 1 ) is analytic for   R \ {t 1  jh : j  0,1, }

(ii) X ( , t 1 ) is differentiable at   t 1  jh, j  0. Proof of (viii) Given (14), (15) and (16), observe that

 (t 1 )   (0) and  (t 1  h)   (  h); t 1 is the terminal point for  , while 0 is the initial point for  ; t 1  h is T

T

the initial point for  , while  h is the initial point for  . The variation of constants formula for the initial function problem (1) and (2) is: 0

x(t )  Y (t ) (0)   Y (t  h  s) A1 (s)ds, t  0

(17)

h

Cf. Chukwu ( 1992, pp. 125-126, 345) and Manitius (1978, pp. 6, 12) , while the variation of constants formula for the terminal function problem (14) and (15) is: 

y ( )   (0) X (t 1   )    (t 1  h  s ) A1 X ( s  h   ) d ,   0

(18)

t1

This completes the proof that the stated initial and terminal function problems are duals of each other.

VI.

CONCLUSION

This article has established the duality between the solution matrices in Ukwu and Garba [2014a] and the corresponding indices of control systems in Ukwu and Garba [2014c], as well as the duality between their variation of constants formulas, culminating in the conclusion that the associated initial and terminal function problems are duals of each other. The proofs were achieved using an algorithm involving deft application of rules of integral differentiation and key transformations from one problem to the other, leading to the assertion that the results from one problem can be realized from the other. The ideas exposed in this paper can be exploited to extend the results to double-delay and neutral systems if the structures of the associated solution and control index matrices can be determined.

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Ukwu, C. and E. J. D. Garba (2014a). A derivation of an optimal expression for the index of control systems matrices for single – delay differential systems. Journal of Mathematical Science, Vol. 25 Number-1, January, 2014. Ukwu, C. and E. J. D. Garba (2014c). Optimal expressions for solution matrices of single – delay differential systems. International Journal of Mathematics and Statistics Invention (IJMSI), Volume 2, Issue 2, February 2014. Chukwu, E. N. Stability and time-optimal control of hereditary systems (Academic Press, New York, 1992). Gabasov, R. and Kirillova, F. The qualitative theory of optimal processes ( Marcel Dekker Inc., New York, 1976). Hale, J. K. Theory of functional differential equations. Applied Mathematical Science, Vol. 3, 1977, Springer-Verlag, New York. Manitius, A. (1978). Control Theory and Topics in Functional Analysis. Vol. 3, Int. Atomic Energy Agency, Vienna. Tadmore, G. Functional differential equations of retarded and neutral types: Analytical solutions and piecewise continuous controls, Journal of Differential Equations, Vol. 51, No. 2, 1984, Pp. 151-181. Ukwu, C. (1987). Compactness of cores of targets for linear delay systems., J. Math. Analy. and Appl., Vol. 125, No. 2, August 1. pp. 323-330. Ukwu, C. Euclidean controllability and cores of euclidean targets for differential difference systems Master of Science Thesis in Applied Math. with O.R. (Unpublished), North Carolina State University, Raleigh, N. C. U.S.A., 1992. Ukwu, C. An exposition on Cores and Controllability of differential difference systems, ABACUS, Vol. 24, No. 2, 1996, pp. 276285.

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