International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
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Duality between initial and terminal function problem…
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     0 1
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, , , , where ( , ) (5)...
Duality between initial and terminal function problem…
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Our current result is captured in the fo...
Duality between initial and terminal function problem…
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multiple integral in (13), proving that ...
Duality between initial and terminal function problem…
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 
0 0 0
0 10 0 1
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1
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Duality between initial and terminal function problem…
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Duality between initial and terminal function problem…
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 
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because the integral contribution from k...
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REFERENCES
[1] Ukwu, C. and E. J. D. Gar...
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International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.

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F023064072

  1. 1. 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 www.ijmsi.org 64 | P a g e 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 huntil 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: 0 1( ) ( ) ( ) (1) ( ) ( ), [ ,0] (2) x t A x t A x t h x t t t h        and the terminal function problem: 0 1 1 1 ( ) ( ) ( ) (3) ( ) ( ), [ , ] (4) y y A y h A y t h t                0 1 where and are constant matrices; (.) and (.) are column and row n-vector functionsA A n n x y respect ively. Let ( , ) and ( , )Y t X t  be solution matrices of systems (1) and (3) respectively. Then ( , ) and ( , )X t Y t  satisfy respectively the following matrix differential equations:
  2. 2. Duality between initial and terminal function problem… www.ijmsi.org 65 | P a g e      0 1 , , , , , where ( , ) (5) 0, nI t Y t A Y t A Y t h Y t t t                    0 1, , , , (6)X t X t A X h t A          for 0 , , 0,1,...t t k h k      where   ; 0; (7), nI t t X t         See Chukwu (1992), Hale (1977) and Tadmore (1984) for properties of  , .X t  of particular importance is the fact that  ,t   is analytic on the intervals     1 1 1 1 , , 0,1,...; 1 0t j h t j h j t j h       . Any such   1 1 1 ,t j h t j h     is called a regular point of  ,t   . The first result we will establish is that ( , ) and ( , )Y t 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 ( , ) and ( , ).Y t X t  See Ukwu and Garba [2014 a,c]. 1 1 1 et [ ( 1) , ],Let [( ) ,( 1 ) ], {0,1, }, {0,1};l 0: ( 1) 0. k i jK t j h t jhJ k i h k i h k i j t j h                III. THEOREM: UKWU-GARBA’S SOLUTION MATRIX FORMULA FOR AUTONOMOUS, SINGLE – DELAY LINEAR SYSTEMS   0 0 0 0 0 0 0 0 0 1 0 1 0 ( ) ( ) 1 1 ( ) ( ) 1 ( ) ( ) ( ) 1 1 2 , 1, ,2 1( 1) , ; (8) , ; (9) ( ) , , 2 (10) i j i i A t t A t A t s A s h h t A t A t s A s h h s ht jk A t s A s h s A s h k j i j jjh i h e t J e e Ae ds t J Y t e e Ae ds e Ae Ae ds t J k                                              IV. THEOREM: UKWU-GARBA’S CONTROL INDEX THEOREM FOR AUTONOMOUS LINEAR DELAY CONTROL SYSTEMS:           0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 1 0 ( ) ( ) 1 1 1 ( ) ( )1 1 1 1 ( ) ( ) ( ) 1 1 , 1, ,2 1( 1) , ; (11) , ; (12) ( , ) (13) 1 i q q k A t A t A t h s A s t h A t A t h s A s t h s h k j A t h s A s h s A s i k k qt i h e K e e Ae ds K X t e e Ae ds e Ae Ae                                                   12 , , 2 k j j k kh ds K j                            
  3. 3. Duality between initial and terminal function problem… www.ijmsi.org 66 | P a g e Our current result is captured in the following theorem. V. THEOREM: UKWU’S TOPOLOGICAL DUAL ALGORITHM FOR AUTONOMOUS LINEAR DELAY SYSTEMS ( , ) and ( , )Y t X t  are topological duals of each other subject to the transformations below. To obtain ( , ) from ( , ) )X t Y t  perform the following operations; (i) t t   in the formula for ( ).Y t (ii) (.) (.)t  in the lower integral limits, as well as in the leading integral limits in (10) with the preceding sign convention ( 1) , for 1 ,j j k   followed by the notational changes 2 2 , ; (.) (.) jk k j j k j k J K        (iii) in the upper integral limits in ( , ).i i Y ts h s h    0 0 1( ) ( ) (iv) The leading exponential function , {2,3, },in the integralsjA t s A t h s e e j k         in ( , )Y t  and the trailing exponential function 0 1 0( ) ( ) .jA s h A s e e    To obtain ( , ) from ( , )Y t 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) , for 1 ,j j k   and effect the notational changes 2 2 ; (.) (.) j k j k k j K J        (vi) Replace the leading exponential function 0 1 0( ) ( ) by , {1,2, },in ( , )jA t h s A t s e e j k X t          and the trailing exponential function 0 0 1( ) ( ) byjA s A s h e e   (vii) Replace the upper integral limits in ( , ) by .i is h X t s h  (viii) Given the initial function problem (1) and (2) the equivalent terminal function problem is given by: 0 1 1 1 ( ) ( ) ( ) (14) ( ) ( ), [ , ] (15) y y A y h A y t h t                T T 1 1 1 denotes the transpose of (.). where ( ) ( ), [ , ];(.) (16)t t h t        Proof We need to prove that the continuity and analytic dispositions of 1( , ), jX t K   can be inferred from those of ( , ), kY t t J   and vice-versa. In particular, we prove that the continuity and non-analytic behavior of 1 1 1 ( , ),for : ( 1) 0X t t jh t j h       correspond to those of 1( , ),for : .Y t t kh t t    We start from the expression for ( , ). Note that ( , ) ( )Y t Y t Y t    since the systems (1) and (5) are autonomous. So we may replace byt t  in Ukwu and Garba’s solution matrix theorem to secure ( , ).Y t  This justifies (i). With 1t t , in Ukwu and Garba’s Control Index theorem, the transformation (.) (.)t  , in (ii) ensures that 1 1 1, , ,( 1) ( 1) .t t jh t jh t i h t i 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
  4. 4. Duality between initial and terminal function problem… www.ijmsi.org 67 | P a g e multiple integral in (13), proving that 1 ( , )X t has been obtained from ( , ).Y t  The proof that 1 ( , )X t transforms to ( , )Y t  is similar. Finally we investigate the continuity and analytic dispositions of ( , )Y t  and 1 ( , ).X t Being sums of integrals of products of constant matrices and exponential functions we deduce that ( , )Y t  and 1 ( , )X t are analytic in the interiors of andk j J K respectively, since 0 (.) is . A e C Finally we must examine ( , )Y t  and 1( , )X t on the boundaries of andk jJ K respectively. ( , ) 0 lim ( , ) 0, lim ( , ) 0; t t Y t t Y t Y t                0 0 ( , ) [ , ] lim ( , ) , lim ( , ) . A t t t Y t e t h Y t I Y t A                  Therefore ( )Y t is not continuous at t  and certainly not differentiable there. It follows that ( )Y t is not analytic at .t  0 0 0 0 1 0 1 0 1 ( ) ( ) ( ) ( ) lim ( , ) ; lim ( , ) lim ( , ) lim ( , ) ( ) (0) , A h A h t h t h t h t h Y t A e Y t A Y t A Y t h A Y h AY A e A                             by the continuity of ( )Y t for 0t  and the appropriate evaluations using the interval 0J  for the left limit and 1J  for the right limit. Alternatively, we apply Leibniz’s rule for differentiating an integral to ( ) lim ( , ) t h Y t     to obtain 0 0 0 0 0(2 ) ( ) ( ) ( ) 0 1 0 1 0 10 . h A h h A h h A t s A t s A h h A e Ae A e Ae ds A e A              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.   0 0 0 0 0 1 0 1 ( ) ( ) 1 ( ) ( ) ( ) 1 1 2 , 1, ,2 1( 1) ( ) , , 2 i j i i t A t A t s A s h h s ht jk A t s A s h s A s h k j i j jjh i h Y t e e Ae ds e Ae Ae ds t J k                                0 0 0 0 0 1 0 1 ( ) ( ) 1 ( ) ( ) ( ) ( ) 1 1 2 , 1, ,2 1( 1) lim ( ) , 2 i j i i kh A kh A kh s A s h t kh h s hkh jk A kh s A s h s A s h j i j jjh i h Y t e e Ae ds e Ae Ae ds k                                  0 0 0 0 0 1 0 1 ( ) ( ) 1 ( ) 1 ( ) ( ) ( ) 1 1 2 , 1, ,2 1( 1) since the integral contribution at is zero. lim ( ) , 1 2 i j i i kh A kh A kh s A s h t kh h s hkh jk A kh s A s h s A s h j i j jjh i h j kh Y t e e Ae ds e Ae Ae ds k                                   0 0 0 0 0 1 0 1 ( ) ( ) 1 1 1 ( ) ( ) ( ) 1 1 2 , 1, ,2 1( 1) On , ( ) , 1 2 i j i i t A t A t s A s h k h s ht jk A t s A s h s A s h j i j jjh i h J Y t e e Ae ds e Ae Ae ds k                              
  5. 5. Duality between initial and terminal function problem… www.ijmsi.org 68 | P a g e   0 0 0 0 10 0 1 ( ) ( ) 1 ( ) 1 ( )( ) ( ) 1 1 2 , 1, ,2 1( 1) This completes the proof of the continuity of ( ), for 0, lim ( ) , 1 2 i j i i kh A kh A kh s A s h t kh h s hkh jk A s hA kh s A s h s j i j jjh i h Y t t Y t e e Ae ds e Ae Ae ds k                                 and hence that of ( , ),Y t t      0 0 0 0 0 1 0 1 0 0 1 0 1 ( ) ( ) ( ) 0 1 0 1 ( ) ( ) 1 1 2 , 1, ,2 1( 1) ( ) ( ) ( ) 0 1 1 2 , 1, ,2 1( 1) ( ) i i i i j i i t A t A t h A t s A s h h s h jk A s h s A s h j i j j i h s ht jk A t s A s h s A s h j i j jjh i h Y t A e Ae A e Ae ds Ae Ae ds A e Ae Ae ds                                                  , , 2kt J k      0 0 0 0 0 1 0 1 0 0 1 [ 1] ( ) ( ) 0 1 0 1 ( ) ( ) ( ) 1 1 2 , 1, ,2 1( 1) 1 ( ) ( ) 0 1 2 , 1, ,2 ( 1) lim ( ) i i i i j i i kh A kh A k h A kh s A s h t kh h s h jk A s h s A s h j i j j i h s hkhk A t s A s h s j i j jjh i h Y t A e Ae A e Ae ds Ae Ae ds A e Ae A                                                 0 1( ) 1 1 since the integral contribution at is zero. , 3 j A s h t kh e ds k          0 0 0 0 0 1 0 1 0 0 1 [ 1] ( ) ( ) 1 0 1 0 1 ( ) 1 ( ) ( ) 1 1 2 , 1, ,2 1( 1) 1 ( ) ( ) 0 1 2 , 1, ,2 ( 1) lim ( ) i i i i j i i kh A kh A k h A kh s A s h k t kh h s h jk A s h s A s h j i j j i h s htk A kh s A s h s j i j jjh i h t J Y t A e Ae A e Ae ds Ae Ae ds A e Ae                                             0 1( ) 1 1 , 1 2 j A s h Ae ds k            We proceed to take the second derivative of Y( t ).   0 0 0 0 0 0 1 0 1 ( ) ( ) ( ) ( )2 2 0 1 0 0 1 0 1 ( ) ( ) 0 1 1 2 , 1, ,2 1( 1) ( ) i i i t A t A t h A t h A t s A s h h s h jk A s h s A s h j i j j i h Y t A e A A e A Ae A e Ae ds A Ae Ae ds                           0 0 1 0 1 ( ) ( ) ( )2 0 1 1 2 , 1, ,2 1( 1) , , 2 i j i i s ht jk A t s A s h s A s h k j i j jjh i h A e Ae Ae ds t J k                           0 0 0 0 0 0 1 0 1 [ 1] [ 1] ( ) ( )2 2 0 1 0 0 1 0 1 ( ) ( ) ( ) 0 1 1 2 , 1, ,2 1( 1) lim ( ) i i i kh A kh A k h A k h A kh s A s h t kh h s h jk A s h s A s h j i j j i h Y t A e A A e A Ae A e Ae ds A Ae Ae ds                           
  6. 6. Duality between initial and terminal function problem… www.ijmsi.org 69 | P a g e   0 0 1 0 1 ( ) ( ) ( )2 0 1 1 2 , 1, ,2 1( 1) , 2 i j i i s hkh jk A kh s A s h s A s h j i j jjh i h A e Ae Ae ds k                          0 0 0 0 0 0 1 0 1 [ 1] [ 1] ( ) ( )2 2 0 1 0 0 1 0 1 ( ) 1 ( ) ( ) 0 1 1 2 , 1, ,2 1( 1) lim ( ) i i i kh A kh A k h A k h A kh s A s h t kh h s h jk A s h s A s h j i j j i h Y t A e A A e A Ae A e Ae ds A Ae Ae ds                             0 0 1 0 1 1 ( ) ( ) ( )2 0 1 1 2 , 1, ,2 1( 1) , 2 i j i i s hkh jk A kh s A s h s A s h j i j jjh i h A e Ae Ae ds k                          0 1 0 1( ) ( ) 0 1 1 ( ) ( ) , 1, ,2 1( 1) Clearly, lim ( ) lim ( ) i i i s h k A s h s A s h t kh t kh i k k i h Y t Y t A Ae Ae ds                    02 0Clearly, lim ( ) ,A h t h Y t A e   0 0 0 0 0( ) ( ) ( ) ( )2 2 0 1 0 0 1 0 1( ) t A t A t h A t h A t s A s h h Y t A e A A e A Ae A e Ae ds         02 0 1 0 0 1 1 0 0 1lim ( ) lim ( ) lim ( )A h t h t h t h Y t A e A A A A Y t Y t A A A A               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 ( ),for 0.Y t t  We conclude the following: ( , )(i)Y t  is analytic except at , {0,1, } is analytic for { : 0,1, }( , )t kh k kh kY t t       R (ii) ( , )Y t  is differentiable at , 0.t kh k   0 1 1 1 1 1 0 1 1 ( ) lim limfor ;, ( , ) ( , ) 0 ( , )( , ) n t t A t K X t I X t X tX t e                 is not continuous at 1t  and hence not analytic there. 0 1 1 ( ) lim ( , ) , A h t h X t e      1 01 1 1 1 1 0 1 0 0 1 ( ) ( ) ( ) ( ) 1 1 1lim( , ) ( , )h A h t h A t A t s A s t h AK X t e e e ds X t e                  . Therefore 1( , )X t is continuous at 1 .t h   By Leibniz’s rule of integral differentiation, 0 1 1 0 1 1 1 1 ( ) 1 0 1 1 0 0 1 1 0 ( ) 0 1 0 0 1 0 ( ) ( ) ( ) ( ) 1 1 1 0 ; lim ( , ) .lim ( , ) ( , ) A t h h A hh t ht h A t A t s A s t h A e A A A A X t A e A A K X t e e e ds eX t                                    Therefore 1( , )X t is not differentiable at 1 ,t h   and hence not analytic there.
  7. 7. Duality between initial and terminal function problem… www.ijmsi.org 70 | P a g e         0 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 0 1 1 0 1 1 ( ) ( ) ( ) 1 0 1 1 0 1 1 ( ) ( ) 1 1 2 , 1, ,2 1( 1) 1 ( ) ( ) 1 1 ( , ) 1 1 i q q q q k A t A t h s A t h s A s t h s hj k j A t h s A s h s k i k k qi h k j A t h s A s h s q ds X t A e e A e A A e ds e Ae A e Ae                                                           0 1 1 1 ( ) 1 0 2 , 1, ,2 ( 1) , , 2 i k k j s hj A s k i k kt kh i h ds K jA A e                                      1 0 1 1 0 1 10 0 1 1 0 1 1 0 1 0 1 1 1 ( ) ( )[ 1] 1 0 1 1 0 1 1 ( ) ( ) 1 1 2 , 1, ,2 1( 1) 1 ( ) lim ( , ) 1 1 i q q k t jh A t h s A s jh tA jh A j h t jh t h s hj k j A t h s A s h s k i k k qi h j A t h s ds X t A e e A e A A e ds e Ae A e                                                        1 0 10 1 1 1 1 1 ( )( ) 1 1 0 2 , 1, ,2 1( 1) , 1 2 i kq q k t jh s hj k A s jh tA s h s k i k k qt kh i h ds jAe A A e                                since the integral contribution from k j is zero.         1 0 1 1 0 1 10 0 1 1 0 1 1 0 1 0 1 ( ) ( )[ 1] 1 1 0 1 1 0 1 1 ( ) ( ) 1 1 2 , 1, ,2 1( 1) 1 ( lim ( , ) 1 1 i q q k t jh A t h s A s jh tA jh A j h j t jh t h s hj k j A t h s A s h s k i k k qi h j A t ds K X t A e e A e A A e ds e Ae A e                                                          1 1 1 0 10 1 1 1 1 1 ) ( )( ) 1 1 0 2 , 1, ,2 1( 1) , 1 2 i kq q k t jh s hj k h s A s jh tA s h s k i k k qt kh i h ds jAe A A e                                  Therefore 1( , )X t is differentiable except at 1.t  We proceed to take the second derivative:         0 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 2 ( ) ( ) ( )2 2 1 0 1 0 1 0 12 1 1 ( ) ( ) 1 1 0 2 1 2 ( ) 1 , 1, ,2 ( 1) ( , ) 1 1 i q q k k A t A t h A t h s A s t h s hj k j A t h s A s h s k q j A t h s A i k k i h ds ds X t A e e A A e A A e ds e Ae A A e Ae                                                              0 1 0 1 1 1 ( ) ( )2 1 0 2 1, 1, ,2 ( 1) , , 2 i q q k k j s hj k s h s A s k qi k kt kh i h ds K jA A e                                      1 0 1 1 0 1 10 0 1 1 0 1 1 0 1 1 1 2 ( ) ( )[ 1]2 2 1 0 1 0 1 0 12 1 1 ( ) ( ) 1 1 0 2 , 1, ,2 1( 1) lim ( , ) 1 i q q k k t jh A t h s A s jh tA jh A j h t jh t h s hj k j A t h s A s h s k i k k qi h ds ds X t A e e A A e A A e ds e Ae A A                                                           1 0 1 1 0 10 1 1 1 1 1 2 ( ) ( )( ) 2 1 1 0 2 , 1, ,2 1( 1) , 1 21 i kq q k t jh s hj k j A t h s A s jh tA s h s k i k k qt kh i h ds je Ae A A e                                    
  8. 8. Duality between initial and terminal function problem… www.ijmsi.org 71 | P a g e because the integral contribution from k j is zero.       1 0 1 1 0 1 10 0 0 1 1 0 1 1 1 1 2 ( ) ( )[ 1]2 2 1 1 0 1 0 1 0 12 1 1 1 ( ) ( ) 1 1 0 2 1, 1, ,2 ( 1) lim ( , ) 1 i q q k t jh A t h s A s jh tA jh A j h j s hj k j A t h s A s h s k q t jh t h i k k i h ds K X t A e e A A e A A e ds e Ae A A                                                            1 0 1 1 0 10 1 1 1 1 1 1 2 ( ) ( )( ) 2 1 1 0 2 1, 1, ,2 ( 1) , 1 21 i kq q k k t jh s hj k j A t h s A s jh tA s h s k qi k kt kh i h ds ds je Ae A A e                                           1 1 1 1 1 Hence lim ( , ) lim ( , ), 2: [ 1] 0. t jh t jh X t X t j t j h               1 Therefore, ( , )X t is differentiable except at 1t  , but not twice differentiable at 1t jh   , for any nonnegative integer j. This completes the proof that 1 ( , )X t is not analytic at 1 t j h   , for any nonnegative integer j. Additional information revealed is the differentiability of 1 1( , ),for .X t t   We conclude the following: 1 ( , )(i) tX  is analytic except at 1 1 , {0,1, }: [ 1] 0t j tjh j h      1 1 is analytic for { : 0,1, }( , )X t jt jh   R 1 (ii) ( , )X t is differentiable at 1 , 0.t jh j    Proof of (viii) Given (14), (15) and (16), observe that T T 1 1 1 1 ( ) (0) and ( ) ( ); is the terminal point for , while 0 is the initial point for ; is the initial point for , while is the initial point for . t t h h t t h h               The variation of constants formula for the initial function problem (1) and (2) is: 0 1( ) ( ) (0) ( ) ( ) , 0 (17) h x t Y t Y t h s A s ds t        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: 1 1 1 1( ) (0) ( ) ( ) ( ) , 0 (18) t y X t t h s A X s h d                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.
  9. 9. Duality between initial and terminal function problem… www.ijmsi.org 72 | P a g e REFERENCES [1] 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. [2] 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. [3] Chukwu, E. N. Stability and time-optimal control of hereditary systems (Academic Press, New York, 1992). [4] Gabasov, R. and Kirillova, F. The qualitative theory of optimal processes ( Marcel Dekker Inc., New York, 1976). [5] Hale, J. K. Theory of functional differential equations. Applied Mathematical Science, Vol. 3, 1977, Springer-Verlag, New York. [6] Manitius, A. (1978). Control Theory and Topics in Functional Analysis. Vol. 3, Int. Atomic Energy Agency, Vienna. [7] 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. [8] 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. [9] 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. [10] Ukwu, C. An exposition on Cores and Controllability of differential difference systems, ABACUS, Vol. 24, No. 2, 1996, pp. 276- 285.

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