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International Journal of Mathematics and Statistics Invention (IJMSI)

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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 …

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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  • 1. International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 || P-ISSN: 2321 - 4759 www.ijmsi.org || Volume 2 || Issue 1 || February - 2014 || PP-11-20 Formulations and proofs of optimal expressions for control index matrices for a class of double – delay differential equations. C. Ukwu and E.J.D. Garba Department of Mathematics, University of Jos, P.M.B 2084, Jos, Nigeria ABSTRACT: This paper derived the structure of the indices of control systems for a class of double – delay autonomous linear differential equations on any given interval of length equal to the delay h for non –negative time periods. The formulations and the development of the theorem relied on an earlier work by Ukwu (2013i) on the interval [ t  4 h , t ] . The derivation of the associated solution matrices exploited the continuity of these 1 1 matrices for positive time periods, the method of steps and backward continuation recursions to obtain these matrices on successive intervals of length equal to the delay h. The proofs were achieved using ingenious combinations of summation notations, greatest integer functions and multiple integrals. The indices were derived using the stage – wise algorithmic format, starting from the right – most interval of length h. Our results globally extend the time scope of applications of these matrices to the solutions of terminal function problems and rank conditions for controllability and cores of targets. KEYWORDS: Index, Matrices, Recursions, Scalar, Structure. I. INTRODUCTION 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. In sharp contrasts to determining matrices the use of indices of control systems for the investigation of the Euclidean controllability of systems can be quite computationally challenging; however this difficulty can be mitigated if the coefficient matrix associated with the state variable at time t is diagonal. This paper pioneers the development of the structure of these indices. Literature on state space approach to control studies is replete with indices of control systems as key components for the investigation of controllability. See Chukwu (1992), Gabsov and Kirillova (1976), Manitius (1978), Tadmore (1984), and Ukwu (1987, 1992, 1996). Regrettably no author has made any attempt to obtain general expressions for the associated matrices or special cases of such matrices involving the double delay h an d 2 h . Effort is usually focused on the single – delay mode with the usual approach being to start from the interval [ t 1  h , t 1 ] and compute the index matrices for given problem instances; then the method of steps and backward continuation recursive procedure are deployed to extend these to the intervals [ t  ( k  1) h , t  k h ] , for positive integral k , not exceeding 2, for the most part. Such approach is rather 1 1 restrictive and doomed to failure in terms of structure for arbitrary k . In other words such approach fails to address the issue of the structure of control index matrices. The need to address such short-comings has become imperative; this is the major contribution of this paper, in the case the scalar counterparts, with its wide-ranging implications for extensions to systems and holistic approach to controllability studies. II. PRELIMINARIES Consider the system:   X   ,t  X  , t  A0 for 0    t ,   t  k h , k  0 ,1, ...  X   h, t  A1  X   2h, t  A 2 (1) where www.ijmsi.org 11 | P a g e
  • 2. Derivation of an optimal expression for control index matrices…  I ;  t X  , t    n 0;   t  (2) A 0 , A 1 , A 2 a r e n  n c o n s t a n t m a t r i c e s a n d   X (  , t ) ,   X (  , t  h ) a r e n  n m a t r i x f u n c t i o n s . See Chukwu (1992), Hale (1977) and Tadmore (1984) for properties of X  t ,   . Of particular importance is the fact that   X   , t  is analytic on the intervals  t such    t1   j  1  h , t1  j h  1   j  1  h , t 1  j h  , j  0 , 1, ...; t 1   j  1 h  0 . Any is called a regular point of   X  t ,   . 2.1 Definition T h e e x p r e s s i o n c X ( , t 1 ) B * is called the index of a given control system, where c is an n - dimensional constant column vector, X ( , t 1 ) is defined in (1 ), B is an with the control system: n m constant matrix associated  x (t )  A 0 x (t )  A 1 x (t  h )  A 2 x (t  2 h )  B u (t ) and u ( .) is an m - vector admissible control function. Thus the control index matrix, X ( , t 1 ) determines the structure of the index of a given control system. We proceed to determine the structure of the above matrix. This will be achieved using the method of steps and a Backward Continuation Recursive procedure. Let K j   t 1  ( j  1) h , t 1  jh  ,  j : t 1  ( j  1) h  0 , a n d fix e d t1  0 .   Ukwu (2013i) obtained the following expressions for the control index matrices, X ( , t 1 ) o n K j , f o r j  {0 ,  , 3} : A ( t  ) e 0 1 ,  K 0 ;    A0 ( t 1   ) A0 ( t 1  h  s 1 ) A0 ( s 1   ) e   e A1 e d s1 ,   K 1 ;  t1h   s h   X ( , t 1 )   A ( t   ) A0 ( t 1  h  s 1 ) A0 ( s 1   ) 0 1 e   e A1 e d s1     t 2h t h t1h    A (t  2 h  s ) A ( s  )    e 0 1 A2 e d s3 , fo r   K 2 ;  t 2h  (2) (3) 2 1 3 0 e A0 ( t 1 hs 1 ) A1 e A0 ( s 1 hs 2 ) A1 e A0 ( s 2  ) d s 1d s 2 1 (4) 3 1    A0 ( t 1   ) A0 ( t 1  h  s 1 ) A0 ( s 1   ) A (t  h s ) A (s hs ) A ( s  ) e   e A1 e d s1   A1 e A1 e d s 1d s 2   e t 2h t h t1h   s3  h s h  A (t  h s ) A (s hs ) A ( s  h  s3 ) A ( s3  )   A1 e A1 e A1 e d s 1d s 2 d s3    e  t13h t 2h t h  X ( , t 1 )   t1  h   A0 ( t 1  2 h  s 3 ) A 0 ( t 1  h  s1 )  A 0 ( s1  2 h  s 3 ) A ( s  ) A0 ( s 3   )   e A2 e d s3  A1 e A2e 0 3 d s1 d s 3   e  t12h t13h s3  2 h  t12h   A (t  2 h s ) A (s s h) A ( s  )   A2 e A1 e d s2 d s3 ,  K 3 (5 )   e  t13h s h  s2  h 0 1 1 1 0 1 2 0 2 1 2 0 1 1 1 0 1 2 0 2 0 1 0 2 3 0 1 2 0 3 3 He also interrogated some topological dispositions of the solution matrices and deduced that the solution matrices are continuous on the interval [t1 – 4h, t1 ] but not analytic due to the break-down of analyticity for www.ijmsi.org 12 | P a g e
  • 3. Derivation of an optimal expression for control index matrices…   { t 1 , t1  h , t 1  2 h , t 1  3 h } . These results are consistent with the existing qualitative theory on X ( , t ). See Chukwu (1992), Hale (1977), Tadmore (1984) and Ukwu (1987, 1996). See also analytic function (2010) and Chidume (2007) for discussions on analytical functions and topology. The objective of this paper is to formulate and prove a theorem on the general expression for  0 , 1, 2 , 3  , by appropriating the above expression for X ( , t 1 ) o n K j , f o r j  X ( , t 1 ) , f o r t h e c a s e n  1, where n is the dimension of the state space. Let r0 , r1 , r2 be nonnegative integers and let P0  r 0  , 1  r1  , 2 ( r2 ) denote the set of all permutations of 0 , 0 , ...0 1, 1, ...1 2 , 2 , ...2 : th e p e rm u ta tio n s o f th e o b je c ts 0 , 1, a n d 2 in w h ic h i a p p e a rs ri tim e s ,    r1 tim e s r0 tim e s r2 tim e s i   0 , 1, 2  . 3. Theorem: Ukwu-Garba’s Control Index Formula for Autonomous, Double – Delay Linear Systems (1), with state space dimension n  1 . In ( 1 ) , s e t n  1, A 0  a , A 1  a 1  b , A 2  a 2  c . T h e n :   a ( t  ) 1 ,   K 0; e  i j  t 1  [  ih ]  a  t 1  [  ih ]   a ( t 1  ) i X ( , t 1 )   e   b e i! i 1       j     2  j 2 k   k 1 i t k b c 1  [  ( i  2 k ) h ]  (6 ) ik e  a t 1  [  ( i  2 k ) h ]  i!k ! i0 ,  K j, j 1 (7 ) The third component:  j     2  j 2k   k 1 i b c k  [  ( i  2 k ) h ]  i0 ik e  a t 1  [  ( i  2 k ) h ]  i!k !   k 1 1 i0  j     2  j 2 k  t  ( v 1 ,v 2 , , v av av  av 1 2 t  [  ( i  2 k ) h ]  1 ik (i  k ) ! i k i  k )P 1( i ) , 2 ( k ) e  a t 1  [  ( i  2 k ) h ]  (8 ) The equivalent form in (8) will be exploited to achieve the proof of the theorem. Note that the formula can be rewritten in the form: X ( , t 1 )  I n e a ( t 1  ) j   b i t 1  [  ih ]     k 1 i0  ( v 1 ,v 2 , , v i  k )  P 1( i ) , 2 ( k ) e  a t 1  [   ih ]  s g n (m a x {0 , j } ) i! i 1  j   2  j2k   i av av  av 1 2 t i k 1  [  ( i  2 k ) h ] (i  k ) !  ik e  a t 1  [  ( i  2 k ) h ]  s g n (m a x { 0 , j  1} ) (9 ) Proof First, we prove that the theorem is true for   K j , j  { 0 , 1, 2 , 3} by comparing the results with expressions (2) through (5) above. Then we use induction to complete the proof: www.ijmsi.org 13 | P a g e
  • 4. Derivation of an optimal expression for control index matrices… a ( t 1  ) t  K 0  X ( , t 1 )  e a ( t 1  ) t  K 1  X ( , t 1 )  e ; ( 2 )  X ( , t 1 )  e a ( t 1  )  a (  h  t 1 )  b (  h  t 1 ) e ;  e a ( t 1  )  b ( t 1  [  h ] ) e  (3 )  X ( , t 1 )  e a ( t 1  )   be a ( t 1  [  h ]) ds  e a ( t 1  )  b ( t 1  [  h ] ) e a ( t 1  [  h ] ) a ( t 1  [  h ] ) t1h   K 2  X ( , t 1 )  e t 2 a ( t 1  )   b i 1  [  ih ]    v1  X ( , t 1 )  e t 2   i b 1 a ( t 1  )   av t  [  ih ]  e   a t 1  [   ih ]  a t 1    h     b  t 1    h   e  e be 0 1 e  c  t 1    2 h   e a ( t 1  ) A0 ( s 1  h  s 2 ) b  a t 1    2 h   be A0 ( s 2  )  d s 1d s2   e a ( t 1  2 h  s3 ) ce a ( s3  ) d s3 t1  2 h  a t 1    h     b  t 1    h   e   t 2   t12h t1h  X ( , t 1 )  e  a t 1  [  ( 0  2 ) h ] i i! A0 ( t 1  h  s 1 )  ( 0  1) ! 1 s2  h    [  ( 0  2 ) h ] 1  P 1 ( 0 ) , 2 (1 ) i 1 ( 4 )  X ( , t 1 )  e  a t 1  [   ih ] e i! i 1 a ( t 1  ) i    h     1   c  t 1    2 h   e    t 1    2 h   2 e  a t 1    2 h     2  X ( , t 1 )  e a ( t 1  ) 2   b i t     ih   1  a t 1    ih    e  i! i 1   K 3  X ( , t 1 )  e i a ( t 1  ) j   b i t 1  [  ih ]   c  t 1    2 h   e i  a t 1  [   ih ] e t 1   i  0 ( v 1 ,v  X ( , t 1 )  e a ( t 1  ) j   b i 2 t , , v 1  i! i 1   t 1    2 h   av av  av 1 2 1  [  ( i  2 ) h ] i 1 ( i  1) ! i 1 )  P 1 ( i ) , 2 ( 1 )  [  ih ]    bc  cb  t 1 i 1 e  a t 1  [  ( i  2 ) h ]  i e a t 1  [   ih ]     i! i 1   [  3 h ]  c  t 1  [  2 h ]  e  a t 1  [  2 h ]  2 e a t 1  [  3 h ]  2! (5 )  X ( , t 1 )  e a ( t 1  ) 2   b i t 1     ih   t 1    2 h     e  a t 1    ih   i! i 1  c  t 1    2 h   e i s3  h s2  h    t13h t1 2h e A0 ( t 1  h  s 1 ) be a(s1hs 2 ) be a(s 2  h  s3 ) be a ( s3  ) d s 1d s 2 d s3 t1h www.ijmsi.org 14 | P a g e
  • 5. Derivation of an optimal expression for control index matrices…    e a ( t 1  2 h  s3 )  a ( s3  ) ce t1  2 h  t13h    a ( s1  2 h  s 3 ) e be a ( t 1  h  s1 ) ce a ( s3  ) d s1 d s 3 s3  2 h t1  2 h t13h   d s3  t1  h s3  h e a ( s2  s3  h ) a ( t 1  2 h  s2 ) 3 a ( t 1  )  X ( , t 1 )  e ce   b i t 1 A0 ( s 3   ) be d s2 d s3     ih   i e  a t 1    ih    c  t 1    2 h   e i! i 1   bc  cb  t  [  3 h ] 1  t 1    2 h   2  a t 1  [  3 h ] e  , 2! n e e d le s s to s a y :  bc  cb  t 1  [  3 h ]  2 e  a t 1  [  3 h ]    b c t 1  [  3 h ] 2!  2 e  a t 1  [  3 h ]  Therefore the theorem has been be verified for j  {0 ,1, 2 , 3} . Assume that the theorem is valid for   K p , 4  p  j,  , s j 1  K for some integers  t 1  [ j  1] h  K j , j 1 p an d j. Then s j 1  h  K a n d s j 1  2 h  K j j 1 . Hence:   X ( , t 1 )  X ( t 1  [ j  1] h , t 1 ) e a ( t 1 [ j 1] h  )   X ( s j  1  h , t1 ) b e a(s j 1  ) ds j t 1  [ j 1]h    X ( s j  1  2 h , t1 ) c e a(s j 1  ) d s j 1 (1 0 ) t 1 [ j 1]h  e  a t 1   j   b i  [ j  1  i ]h  i0  a t 1    ih   (1 1) j2k k 1  e i! i 1  j     2  i    ( v 1 , v 2 , , v i k av av  av 1 )  P 1 ( i ),  b  t 1     ( j  1) h   e  j   b i 1 t 1   t 1  ( j 1) h  j   2  j2k     k 1 i0  ( v 1 ,v 2 av av  av 1 2 ik (1 2 ) (i  k ) ! i k  [ s j  1  ( i  1) h ]  i e  a t 1  [  ( i  1 ) h ] i! t 1  ( j 1) h i 1  j  1  i  2 k ]h  2(k ) t a ( t 1  ) 2 [ t 1  [ s j 1  (i  1  2 k ) h ] i k , , v i  k )  P 1( i ) , 2 ( k ) www.ijmsi.org (i  k ) !   d s j 1 (1 3 ) ik be  a t 1  [  ( i  1  2 k ) h ]  d s j  1 (1 4 ) 15 | P a g e
  • 6. Derivation of an optimal expression for control index matrices…   c  t 1     ( j  1) h   e a ( t 1  )    t j i b c 1  [ s j 1  ( i  2 ) h ]    j 1    2  j2 k       k 1 t 1  ( j 1) h  i0 ( v 1 ,v 2 av av  av 1  a t 1  [  ( i  2 ) h ] e  i! t 1  ( j 1) h i 1  i 2 t  [ s j 1  (i  2  2 k ) h ] 1  , , v i  k )  P 1(i ) , 2 ( k ) (1 5 ) ik ce (i  k ) ! i k d s j 1  a t 1  [  ( i  2  2 k ) h ]  d s j  1 (1 6 ) The expression (13) yields: b  t 1     ( j  1) h   e j 1 a ( t 1  )   i b t 1  [  ih ]  i e  a t 1  [   ih ] j1    i! i2 [ i b j  1  i ]h  i e  a t 1  [   ih ]  (1 7 ) i! i2 The expression (14) yields:  j     2  j 2 k   k 1 i0    k 1 1 ( v 1 , v 2 , , v i  k )  P  j   2  j2k    av av  av i0 2 1  [  ( i  2 k ) h ]  ik be (i  k ) ! i k  a t 1  [  ( i  2 k ) h ]  (1 8 ) 1 ( i 1 ) , 2 ( k )  ( v 1 ,v 2 t av av  av 1 2 [ j  1  i  2 k ]h ]  ik be (i  k ) ! i k , , v i  k )  P 1 ( i 1 ) , 2 ( k )  a t 1  [  ( i  2 k ) h ]  (1 9 ) since the summations with i = 0 are infeasible and so may be equated to zero, yielding:  j     2  j 2 k   k 1 i0  Av Av  Av 1 ( v 1 , v 2 , , v i  k )  P 2 t 1  [  ( i  2 k ) h ]  ik e (i  k ) ! i k  a t 1  [  ( i  2 k ) h ]  , (20) 1( i ) , 2 ( k ) ( w ith a tr a ilin g a 2  c )  j     2  j 2 k    k 1 i0  ( v 1 , v 2 , , v i  k )  P av av  av 1 2 [ j  1  i  2 k ]h  ik e (i  k ) ! i k  a t 1  [  ( i  2 k ) h ]  , ( 2 1) 1( i ) , 2 ( k ) w ith a tr a ilin g a 2  c The expression (15) yields: c  t 1     ( j  1) h   e a ( t 1  ) j 1   i b c t 1  [  ( i  2 ) h ]    e ( i  1) ! i 1 j 1 i 1 b c  [ j  1  i ]h  i i 1 ( i  1) ! i 1 e  a t 1  [  ( i  2 ) h ]  a t 1  [  ( i  2 ) h ]   (22) The expression (16) yields:   j 1     2   j 1 2 k     k2 i0  ( v 1 ,v 2 , , v i  k )  P 1( i ) , 2 ( k ) av av  av 1 2 t i k 1  [  ( i  2 k ) h ] (i  k ) !  ik e  a t 1  [  ( i  2 k ) h ]  (2 3 ) (w ith a tra ilin g a 2  c ) www.ijmsi.org 16 | P a g e
  • 7. Derivation of an optimal expression for control index matrices…   j 1       2   j 1 2 k   k2  i0  ( v 1 ,v 2 [ av av  av 1 j  1  i  2 k ]h  , , v i  k )  P 1(i ) , 2 ( k )  a t 1  [  ( i  2 k ) h ] e (i  k ) ! i k 2 ik  (2 4 ) ( w ith a tr a ilin g a 2  c )   j 1       2   j 1 2 k   k 1   i0 av av  av 1 ( v 1 , v 2 , , v i  k )  P 2 t 1  [  ( i  2 k ) h ]  ik e (i  k ) ! i k  a t 1  [  ( i  2 k ) h ]  (2 5 ) 1( i ) , 2 ( k ) (w ith a tra ilin g a 2  c ) j 1   i b c t  [  ( i  2 ) h ]  1  a t 1  [  ( i  2 ) h ] [ j 1 2 k   k 1  e ( i  1) ! i0   j 1      2  i 1  i0 ( v 1 , v 2 , , v i k av av  av 1 )  P 1 ( i ), 2  (26) ik j  1  i  2 k ]h  e (i  k ) ! i k 2(k )  a t 1  [  ( i  2 k ) h ]  (27) ( w ith a tr a ilin g a 2  c ) j 1   i b c  [ j  1  i ]h  i 1 e ( i  1) ! i0  a t 1  [  ( i  2 k ) h ]  (2 8 ) Therefore: Y ( t )  (1 1)  (1 2 )  (1 7 )  ( 2 0 )  ( 2 1)  ( 2 2 )  ( 2 5 )  ( 2 6 )  ( 2 7 )  ( 2 8 )  ( 2 9 )  ( 3 0 )    ( 3 8 )  e  a t 1   j   i b [ j  1  i ]h    e  i0  ( v 1 , v 2 , , v b  t 1     ( j  1) h   e i k av av  av 1 )  P 1 ( i ), a ( t 1  )  b i t 1  k 1 i0  ( v 1 , v 2 , , v i k  i e  a t 1  [   ih ]  j1  i! av av  av 1 )  P 1 ( i ), 2 k ]h  2 ik e (i  k ) ! i j  [  ih ]  b i  a t 1  [  ( i  2 k ) h ]  [ j  1  i ]h  1  [  ( i  2 k ) h ]  i k 2(k ) (i  k ) ! i e  (3 0 )  a t 1  [   ih ]  (3 1) i! i 2 t j 1 2 k  (29) 2(k) j 1  2 i 2   [ j  1  i  j 1 2 k j 1   j 1      2   a t 1    ih    i! i 1   j 1      2  i ik e  a t 1  [  ( i  2 k ) h ]  (3 2 ) ( w ith a le a d in g a 1  b )    j   j   j   j  1  1 a n d th e f a c t th a t   n o tin g th a t k e v e n           ; j o d d           2   2   2   2    www.ijmsi.org 17 | P a g e
  • 8. Derivation of an optimal expression for control index matrices…    j  1  j  1  2k  0   j odd, k      2      j 1      2    k 1 i0  ( v 1 , v 2 , , v  w ith a le a d in g i k av av  av 1 )  P 1 ( i ), a1  b  c  t 1     ( j  1) h   e 2 t j 1 a ( t 1  )   i b c 1  [  ( i  2 ) h ]   b c  [ j  1  i ]h  i  i0 ( v 1 , v 2 , , v ( w ith a le a d in g a j 1   i b c i0   j 1      2   2 i k av av  av 1 )  P 1 ( i ), 2 e  a t 1  [  ( i  2 ) h ]  a t 1  [  ( i  2 ) h ] 2 j ]h  (i  k ) ! i k 2(k )  t  [i  2 ]h  i 1 ( i  1) !  i0  a t 1  [  ( i  2 k ) h ]  (3 3 ) ik e   (3 4 )  a t 1  [  ( i  2 k ) h ]  (3 5 )  c)  j 1 e  t  [i  e  a t 1  [  ( i  2 ) h ] ( v 1 , v 2 , , v i k av av  av 1 )  P 1 ( i ), 2  (3 6 ) [ j 1 2 k  i 1 ( i  1) ! j 1 2 k k 1 e i 1 ( i  1) ! i 1  ik   (.) is a p p ro p ria te .       2 k ]h  (i  k ) ! i k 2(k) j 1  k 1 [ j  1  i  i 1   j 1      2   (.)  0 , b e in g in fe a s ib le. S o i0 j 1 2 k     j 1      2  j 1 2 k j  1  i  2 k ]h  i k 2(k ) (i  k ) ! ik e  a t 1  [  ( i  2 k ) h ]  (3 7 ) ( w ith a le a d in g a 2  c ) j 1   i b c [ j  1  i ]h  i0 ( i  1) ! i 1 e  a t 1  [  ( i  2 ) h ]  (3 8 ) (29)+(31) yields: www.ijmsi.org 18 | P a g e
  • 9. Derivation of an optimal expression for control index matrices… e a ( t 1  ) j 1   i b t 1  [  ih ]  i e  a t 1  [   ih ]  (3 9 ) i! i 1 (32) + (35)   j 1      2    k 1 yields: t j 1 2 k i0  av av  av 1 ( v 1 , v 2 , , v i  k )  P 1 ( i ), 2 1  [  ( i  2 k ) h ]  (i  k ) ! i k 2(k ) Expressions (30) + (33) + (37) yield zero; the expressions cancel out. Expressions (34) + (34) + (38) yield zero; the expressions cancel out. Therefore, on K j  1 , X ( , t 1 ) reduces to X ( , t )  e x p re s s io n (3 9 )  1  X ( , t 1 )  e a ( t 1  ) j 1  b i t 1 for   K   i0 j 1  a t 1  [  ( i  2 k ) h ]  (40) e x p re s s io n (4 0 ) i e  a t 1  [   ih ]  t j 1 2 k k 1   e i! i 1   j 1      2   [  ih ] ik  ( v 1 , v 2 , , v i  k )  P 1 ( i ), av av  av 1 2 1  [  ( i  2 k ) h ]  i k 2(k ) (i  k ) ! ik e  a t 1  [  ( i  2 k ) h ]  ( 4 1) . This completes the proof of the theorem. 3.1 Corollary If A 0  a  0 , t h e n    1,   K 0 ;  i j  t 1  [  ih ]   i X ( , t 1 )   1   b i! i 1    j  ik     2  j 2 k  t 1  [  ( i  2 k ) h ]  i k ,   Kj, j1    b c i! k ! k 1 i  0  (42) (4 3) Proof T h e p ro o f is im m ed iate, n o tin g th at A 0  0  e III. a (.)  1. CONCLUSION This paper relied greatly on the optimal deployment of combinatorial analysis and change of variables technique, without which its development would be impossible. The paper has provided a sound basis for its extension to more general systems. Such extension must of necessity rely on multinomial distributions, the method of steps and backward continuation recursive procedure. www.ijmsi.org 19 | P a g e
  • 10. Derivation of an optimal expression for control index matrices… REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] Analytic function. (2010). Retrieved September 11, 2010 from http://en.wikipedia.org/wiki/Analytic_function. Chidume, C. (2007). Applicable Functional Analysis. The Abdus Salam, International Centre for Theoretical Physics, Trieste, Italy. Chukwu, E. N. (1992). Stability and Time-optimal control of hereditary systems. Academic Press, New York. Gabasov, R. and Kirillova, F. (1976). The qualitative theory of optimal processes. Marcel Dekker Inc., New York. Manitius, A. (1978). Control Theory and Topics in Functional Analysis. Vol. 3, Int. Atomic Energy Agency, Vienna. Tadmore, G. (1984). Functional differential equations of retarded and neutral types: Analytical solutions and piecewise continuous controls. J. Differential equations, Vol. 51, No. 2, 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. (1992). 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. Ukwu, C. (1996). An exposition on Cores and Controllability of differential difference systems, ABACUS Vol. 24, No. 2, pp. 276-285. Ukwu, C. (2013a). On Determining Matrices, Controllability and Cores of Targets of Certain Classes of Autonomous Functional Differential Systems with Software Development and Implementation. Doctor of Philosophy Thesis, UNIJOS (In progress) Ukwu, C. (2013i). The structure of indices of control systems for a class of double – delay systems on the interval [t 1  4h, t ]. 1 (Manuscript). www.ijmsi.org 20 | P a g e