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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-31-47

The structure of determining matrices for single-delay autonomous linear neutral control systems Ukwu Chukwunenye Department of Mathematics,University of Jos, P.M.B 2084 Jos, Plateau State, Nigeria. ABSTRACT: This paper derived and established the structure of determining matrices for single – delay autonomous linear neutral differential systems through a sequence of lemmas, theorems and corollaries and the exploitation of key facts about permutations. The proofs were achieved using ingenious combinations of summation notations, the multinomial distribution,change of variables technique and compositions of signum and max functions. The paper also pointed out some induction pitfalls in the derivation of the main results. The paper has extended the results on single–delay models, with more complexity in the structure of the determining matrices. KEYWORDS: Delay, Determining, Neutral, Structure, Systems.

I.

INTRODUCTION

The importance of determining matrices stems from the fact that they constitute the optimal instrumentality for the determination of Euclidean controllability and compactness of cores of Euclidean targets. See Gabasov and Kirillova (1976) and Ukwu (1992, 1996 and 2013a). In sharp contrast to determining matrices, the use of indices of control systems on the one hand and the application of controllability Grammians on the other, for the investigation of the Euclidean controllability of systems can at the very best be quite computationally challenging and at the worst, mathematically intractable. Thus, determining matrices are beautiful brides for the interrogation of the controllability disposition of single-delay neutral control systems. See (Ukwu 2013a). However up-to-date review of literature on this subject reveals that there is currently no correct result on the structure of determining matrices for single-delay neutral systems. This could be attributed to the severe difficulty in identifying recognizable mathematical patterns needed for inductive proof of any claimed result and induction pitfalls in the derivation of determining matrices. This paper establishes valid expressions for the determining matrices in this area of acute research need.

II.

ON DETERMINING MATRICES AND CONTROLLABILITY OF SINGLE-DELAY AUTONOMOUS NEUTRAL CONTROL SYSTEMS

We consider the class of neutral systems:

d (1) where  x(t )  A1 x(t  h)  A 0 x(t )  A1 x  t  h  Bu  t  , t  0 dt A1 , A0 , A1 are n  n constant matrices with real entries and B is an n  m constant matrix with the real entries. The initial function  is in C

h, 0 , R  equipped with sup norm. The control u is in n

L 0, t1  , Rn  . Such controls will be called admissible controls. x  t  , x  t  h   R n for t   0, t1  . If

x  C  h, t1  , R n , then for t  0, t1 we define xt  C  h, 0 , Rn by

xt  s  x t  s, s   h, 0 . 2.1 Existence, Uniqueness and Representation of Solutions If A1  0 and  is continuously differentiable on   h, 0 , then there exists a unique function x :   h,  

which coincides with  on   h, 0 , is continuously differentiable and satisfies (1) except possibly at the points

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The structure of determining matrices for single-delay… jh; j  0, 1, 2,... . This solution x can have no more derivatives than  and continuously differentiable if and only if the relation:

  0  A1   h   A 0 (0)  A1   h   Bu  0

(2)

is satisfied. See Bellman and Cooke (1963) and theorem 7.1 in Dauer and Gahl (1977) for complete discussion on existence, uniqueness and representations of solutions of (1). 2.2 Let

Preliminaries on the Partial Derivatives

 k X (t , ) , k  0,1,   k

t ,   0, t1 for fixed t, let    t,   be the unique function satisfying the matrix differential

equation:

   t,      t ,   h  A 1    t ,   A 0    t ,   h  A 1  

(3)

0    t ,   t  jh; j  0,1,..., subject to: In ;   t 0,   t

(4)

 t,   = 

By Tadmore (1984 , p. 80),     t ,   is analytic on  t   j  1 h, t  jh  , j  0,1,.... and hence

    t ,   is C  on these intervals. The left and right-hand limits of

   t,  

exist at

  t  jh , so that   t ,   is of bounded

variation on each compact interval. Cf. Banks and Jacobs (1973), Banks and Kent (1972) and Hale (1977). We maintain the notation

  k  t ,  ,    k  t ,   as in the double-delay systems.

2.3 Investigation of the Expressions and Structure of Determining Matrices for System (1) In this section we establish the expressions and the structure of the determining matrices for system (1), as well as their relationships with X (t , ) through a sequence of lemmas, theorems and corollaries and the exploitation of key facts about permutations. The process of obtaining necessary and sufficient conditions for the Euclidean controllability of (1) in chapter four will be initiated in the rest of chapter three as follows: i. Obtaining a workable expression for the determining matrices of (k )

(1): Qk

 jh 

for

j : t1  jh  0, k  1, 2,..

(5)

ii.Showing that:

Qk  jh  for j : t1  jh  0, k  0, 1, 2,....,



k

t1  jh, t1    1

iii.Showing that

k

(6)

Q t1  is a linear combination of:

Q0  s  , Q1  s  ,..., Qn 1  s  , s  0, h,....,  n  1 h

(7)

We now define the determining equation of the n  n matrices, Qk  s . For every integer k and real number s, define Qk  s by:

Qk  s   A 1Qk  s  h   A 0 Qk 1  s   A1 Qk 1  s  h 

(8)

for k  0, 1,...; s  0, h, 2h,..... subject to Q0  0  I n , the n  n identity matrix and Qk  s   0 for k  0 or s  0 . 2.4

Preliminary Lemma on Determining matrices Qk ( s ), s  R

In (1), assume that A1  0 , then for j , k  1 , i. Qk  0

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The structure of determining matrices for single-delay… ii. Q0

 jh 

j

A1

iii. Qk  h  A 0 A1  k

iv. Q1

 jh 

k 

 1

 1

k

vii. 

r0

j 1

j 1

A

A0 

t , t    1   vi.   t , t   0 v. 

k 1

r 0

k

1

A0 k  1 r  A 1  A1 A 0 A r0

Ar1 A 0 A1  A 1 Aj1 1 r 

A0k

1

t1  jh, t1   Aj1

Remarks: Notice that

r j  1 r  A0r can switch places with A0k  1 r  in (iii); similarly for A1 and A 1 in

(iv), since r and k  1  r  have the same range for fixed k; similarly in (iv), r and j  1  r  have the same range for fixed j. Proof i.From (8) we have

Qk  0  A1Qk   h  A0 Qk 1  0  A1Qk 1   h  0  A0 Qk 1  0  0

Now, Q1  0  A 0 Q0  0  A 0 I n  A 0 . Assume that

Qk  0  A k0

for some k

 1. Then,

Qk 1  0   A1 Qk 1  h   A 0 Qk  0   A1 Qk  h   0  A1 Qk  0   0  A 0 Ak0  by the induction hypotheses   Ak01 Hence,

ii

Qk  0  A0k for all k  1,2,..... ; this proves (i).

Q0  jh   A1 Q0   j  1 h   A0 Q1  jh   A1 Q1   j  1 h   A1 Q0   j  1 h   0  0

Hence, Q0  h  A1 Q0  0  A1 The rest of the proof is by the principle of mathematical induction: Assume that Q0  jh   A1 for some j  1 . j

Then

,

Q0  j  1h  A1 Q0  jh  A0 Q1  j  1h  A1 Q1  j  1h  A1 Q0  jh  0  0  A1 Aj1  Aj11

by induction hypothesis. So Q0  jh   A1 for j  1, 2, ...... j

iii

From (8),

Q1 (h)  A1 Q1  0   A0 Q0  h   A1 Q0  0   A1 A0  A0 A1  A1.

.

Plugging in the right-hand side of iii yields 11

Q1 (h)  A0 A1  A011 r   A1  A1 A0  A0r  A0 A1  A1  A1 A0 . r 0

Therefore the formula (iii) is valid for k  1 . Assume the validity of (iii) for some k  1. Now

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The structure of determining matrices for single-delay…

Qk 1  h  A1Qk 1  0  A 0 Qk  k   A 1Qk  0

k 1

 A1 A k01  A 0 A k0 A1  A 0  A k0 1 r  A 1  A1 A 0 A r0  A 1 A k0 r0

 A  A  A

k 1 0

k 1 0

k 1 0

k 1

A1  A 0  A k0 1 r  A 1  A1 A 0 A r0  A 1  A1 A 0 A k0 . r0

k 1

A1 

A r0 k

A1 

A r0

k r 0

A

1

 A1 A 0 A r0  A k0 k A 1  A1 A 0 A k0

k r 0

A

1

 A1 A 0 A r0  Qk 1  h

k  1 and so true for k  1, 2,..... Q1  h  A0 A1  A1 A0  A1 .

Therefore (iii) is true for

iv Plug in the right-hand side of (iv) to obtain

A1 A0  A01  A0 A1  A1  A01  A1 A0  A0 A1  A1 It follows that (iv) is true for j  1. Assume that (iv) is valid for some j  1 . Now Q1   j  1 h   A1Q1  jh   A 0Q0   j  1 h   A1Q0  jh  j 1  j  1 r   A1  Aj1 A 0  Ar1  A 0 A1  A1  A1    r 0   j 1 j  A 0 A1  A1 A1  by the induction hypothesis  j 1 1

A

j 1

A 0  Ar11  A 0 A1  A1  Aj11 r    A 0 A1  A1  Aj1 r 0

j 1 1

A

j

A 0  Ar1  A 0 A1  A1  Aj1 r  Ar1  A 0 A1  A1  Aj1 r r 1

r 0

j

 Aj11 A 0  Ar1  A 0 A1  A1  Aj1 r  Q1   j  1 h  r 0

So, the formula (iv) is valid for v. For

j  1. This completes the proof that (iv) holds for all j  1, 2,...

k  1,

   , t1   t        t1 , t1  A0    t1  h  , t1 A1     h, t1  A1  t   A0  0. A1   0   A0    t 

k

t

 1

, t1   1  t1 , t1  

 1

by (3), noting that for Assume that

 1

sufficiently close to t1 ,   h  t1 ; so

 k   t1 , t1    1 A0k for some k  1. Then

 h  t1 . 

1

k

X

k 1

t

 1

  k , t1    X    , t1      t 

  X   (t1 , t1 ) A0  X  k

k

t  h  , t  A  X t  h  , t  A

k

 k 1

1

   1 A0k A0  0  0   1

Therefore,

t

 1

1

k 1

1

1

1

1

A0k 1

X  k   t1 , t1     A0  for all k  1, 2,... . k

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The structure of determining matrices for single-delay…

vi. X 

t , t   tlimtt h X    , t   0 , since   t

k

 1

k

1

1

1

1

1

1

vii Let j be a nonnegative integer such that t1  jh  0 . We integrate (3) and apply (4). Thus

 t1  jh 

0

Hence,

 t  jh    X  , t1   X   h, t1  A1  d     X  , t1  A0  X   h, t1  A1  d 0  

1

 

  t1  jh  , t1    t1   j  1 h  , t1 A1    0, t1     h, t1  A1 

 

 t1  jh 

0

Similarly

   , t1  A0     h, t1  A1  d

t  jh  , t   X t , t   j  1 h  , t  A 

X

1

 

1

( t1  jh )

1

   , t  A 1

0

0

1

 t  jh  , t   X t   j  1 h  , t  A 

1



1

 t1  jh 

 t1  jh 

1

 X  , t  A

1

a 

0

 X  0, t1     h, t1  A1

    h, t1  A1  d

Therefore

X 

1

1

1

1

  X  

t  jh  , t   X t   j  1 h  , t  A 

1

1

X

1

1

 

 X   h, t1  A1  d  0,



f t dt  0 , for any bounded integrable function f, and since  0 a  is bounded and integrable. Therefore we have deduced that lim

1

    , t1  A0     h, t1  A1

 t  jh  , t   X  t  jh  , t    X t   j  1 h  , t   X t   j  1 h  , t  A 

1

1

1

1

1

1

1

1

1

j  1 , j integer. For j  1 , we have for any

X

t  h  , t   X t  h  , t    X t , t   X t , t  A 

1

1

1

 1

1

by (vi). The rest of the proof is by induction on j. Assume the validity of (vii) for j  1,... , p for some

Then

X  t1   p  1 h, t1   X  X 

1

 I n A1  A1

p  1.

1

1

1

1

 Ap1 A1

1

1

by the induction hypothesis. So 2.5

1

t   p  1 h  , t   X t  p  1 h  , t 

t  ph  , t   X t  ph  , t  A 1

 1

1

1

1

 t1   p  1 h, t1   Ap11 and hence (vii) is valid.

Lemma on Qk ( h ) and Q1 ( jh ) as sums of products of permutations

For any nonnegative integers j and k , (i) (ii)

Qk (h) 

( v1 ,,vk 1 )P1(1),0( k )

Q1 ( jh) 

Av1 Av k 1 

( v1 ,,v j1 )P1( j ),0(1)

( v1 ,,vk )P0( k 1),1(1)

Av1 Av j1 

Av1 Av k

( v1 ,,v j )P1( j1), 1(1)

Av1 Av j

Proof of (i)

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The structure of determining matrices for single-delay… k 1

A

By lemma 2.4, Q0  jh   A1 and Qk  h   A 0 A1  j

k

k  0, j  1,  Q0  h   A1 ; rhs 

k 1 r 

0

r 0

v1P1(1), 0(0)

k  1  Q1  h   A0 A1  A1 A0  A1 ; rhs 

A

 A1 A 0  Ar0

1

Av1  0  A1

( v1 ,v2 )P1(1), 0(1)

Av1 Av2 

v1P0 (11),1(1)

Av1 A0 A1  A1 A0  A1

So, the lemma is valid for k {0,1}. Assume the validity of the lemma for 2  k  p, for some integer p 1

Q p  h   A p0 A1   A0

A

r 0

p  1 r 

Now Q p 1  h   A 0p 1 A1 

p 11

 r 0

p 1

 A0

1

 A1 A 0 Ar0

Av1 Av p1 

( v1 ,,v p1 )P1(1),0( p )

p . Then,

( v1 ,,v p )P0( p1),1(1)

Av1 Av p

A0 p 11 r   A1  A1 A 0  Ar0

p

A1   A0 p  r  A1  A1 A 0  Ar0 r 0

p 1

 A0

p 1

A1   A 1  A1 A 0  A 0p   A0 p  r  A1  A1 A 0  Ar0 r 0

p 1

 A0

p 1

A1   A 1  A1 A 0  A 0p  A 0  A0 p (1 r )  A1  A1 A 0  Ar0 r 0

p 1

  A1 A 0  Ar0    A 1  A1 A 0  A 0p   r 0    A0  Av1  Av p1  Av1  Av p   A1 A 0p 1  A 1 A 0p    ( v ,,v ) P  ( v1 ,,v p ) P0( p1) ,1(1)  1 p1 1(1) , 0( p )  p

 A 0  A 0 A1 

 A0

 

( v1 ,,v p1 ) P1(1) , 0( p )

(v1 ,,v p2 )P1(1),0( p1)

( v1 ,,v p1 ) P0( p11),1(1)

A

0

p  (1 r )

A

1

Av1  Av p1  A1 A 0p 1  A 0

Av1  Av p2 (with a leading A0 ) 

Av1  Av p1 (with a leading A0 ) 

( v1 ,,v p2 )P1(1),0( p1)

( v1 ,,v p ) P0( p1) ,1(1)

( v1 ,,v p2 )P1(1),0( p1)

( v1 ,,v p1 ) P0( p11),1(1)

Av1  Av p2 

( v1 ,,v p1 )P0( p ),1(1)

Av1  Av p  A 1 A 0p

Av1  Av p2 (with a leading A1 )

Av1  Av p1 (with a leading A1 )

Av1  Av p1

Therefore, part (i) of the lemma is valid for k  p  1, and hence valid for every nonnegative integer k . Proof of (ii) Using similar reasoning (ii) can be established by induction; but let us do it differently by using the expression for Q1 ( jh ) from lemma 2.4.

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36 | P a g e


The structure of determining matrices for single-delay… j 1

j  1 r 

Q1  jh   Aj1 A 0   Ar1 A 0 A1  A 1 A1 r 0

j

j 1

r 0

r 0

  Ar1 A 0 Aj1 r   Ar1 A1 Aj1 1r

( v1 ,,v j1 )P1( j ),0(1)

Av1  Av j1 

( v1 ,,v j )P1( j1),1(1)

Av1  Av j ,

noting that in the first summation over r A0 occupies positions 1, 2, , j  1 for r  0,1, , j respectively. A0 leads only once and trails only once.; in the second summation A1 occupies positions

1, 2, , j for r  0,1, , j  1 respectively. A1 leads only once and trails only once. Remarks: Observe that Q1 ( jh) may be deduced from Qk ( h ) by (a) replacing k by j in the expression for

Qk ( h ) (b) interchanging -1 and 0 in the expression for Qk ( h ) (c) replacing 0 by -1 in the summations or permutations involving only the indices 0 and 1.

X ( , t1 ) of uncontrolled part of (1)  j  1  (1)  (i) X ((t1  jh) , t1 )     X ((t1  ( j  (r  i ))h)  , t1 ) Ai  Ar1 r 0  i 0  j 1   (ii) X (1) ((t1  jh)  , t1 )     X ((t1  ( j  (r  i ))h)  , t1 ) Ai  Ar1 r 0  i 0  j 1   (iii) X (1) (t1  jh, t1 )     X (t1  ( j  (r  i ))h), t1 ) Ai  Ar1 r 0  i 0 

2.6

Lemma on

Proof of (i)

X (1) (t1 , (t1  jh)  )   X ((t1  jh)  , t1 ) A0  X ((t1  ( j  1)h)  , t1 ) A1  X (1) ((t1  ( j  1)h)  , t1 ) A1 X (1) ((t1  ( j  1)h)  , t1 ) A1  [ X ((t1  ( j  1)h)  , t1 ) A0  X ((t1  ( j  2)h)  , t1 ) A1 ] A1  X (1) ((t1  ( j  2)h)  , t1 ) A21

j  0  lhs  X (1) (t1 , t1 )   A0  rhs; j  1  lhs  X (1) ((t  h)  , t1 )   X ((t  h) , t1 ) A0  A1  A0 A1 (By (v) of lemma 2.4)

  (1  (r  i ))h)  , t1 ) Ai  Ar1 ; r  0, i  0 yield  ( X (t1 , (t1  h)  ) A0 r 0 i 0   r  0, i  1 yield  ( X (t1 , t1 ) A 1   A1 ; r  1, i  0 yield  A0 A 1 ; r  1, i  1 yield 0. Let us examine 

1

1

  X ((t

1

Sum the results for these contingencies to get 1  1     X ((t1  (1  (r  i ))h)  , t1 ) Ai  Ar1  ( X ((t1  h)  , t1 ) A0  A1  A0 A 1 r 0  i 0  So, the lemma is valid for j  1 . Let us explore j  2 .

j  2  lhs  X (1) ((t  2h) , t1 )   X ((t  2h)  , t1 ) A0  X ((t  h)  , t1 ) A1  X (1) ((t  h)  , t1 ) A1   X ((t  2h) , t1 ) A0  X ((t  h) , t1 ) A1  X ((t  h) , t1 ) A0 A1  A 1 A1  A 0 A21 www.ijmsi.org

37 | P a g e


The structure of determining matrices for single-delay… 2  1      X ((t1  (2  (r  i ))h)  , t1 ) Ai  Ar1 ; r 0  i 0   r  0, i  0 yield  ( X ((t1  2h) , t1 ) A0

Let us examine

r  0, i  1 yield  ( X ((t1  h)  , t1 ) A 1 ; r  1, i  0 yield  ( X ((t1  h)  , t1 ) A0 A 1; r  1, i  1 yield  A 1 A 1 ; r  2, i  0 yield  A0 A21 ; r  2, i  0 yield 0. Sum the results for these contingencies to get 2  1      X ((t1  (2  (r  i ))h)  , t1 ) Ai  Ar1 r 0  i 0     ( X ((t1  2h) , t1 ) A0  ( X ((t1  h) , t1 ) A 1  ( X ((t1  h)  , t1 ) A0 A 1  A 1 A 1  A0 A21.

So, (i) of the lemma is valid for j {0,1, 2}. Assume that (i) of the lemma is valid for 3  j  n , for some integer n. Then n 1  X (1) ((t1  nh) , t1 )    X ((t1  (n  (r  i))h)  , t1 ) Ai Ar1, by the induction hypothesis. r 0  i 0  (1)   Now, X ((t1  ( n  1) h) , t1 )    X ((t1  ( n  1) h) , t1 ) A0  X ((t1  nh)  , t1 ) A1 

 X (1) ((t1  nh)  , t1 ) A1 1

 X (1) ((t1  nh)  , t1 ) A1   X ((t1  (n  1  i )h)  , t1 ) Ai i 0

1       X ((t1  (n  (r  i ))h)  , t1 ) Ai  Ar11   X ((t1  (n  1  i )h)  , t1 ) Ai r 0  i 0 i 0  n 1 1      X ((t1  (n  1  (r  i ))h)  , t1 ) Ai  Ar1 , as desired. So (i) is true for j  n, and hence valid r 0  i 0  n

1

for all j. Proof of (ii)

X (1) ((t1  jh)  , t1 )   X ((t1  jh)  , t1 ) A0  X ((t1  ( j  1)h)  , t1 ) A1  X (1) ((t1  ( j  1)h)  , t1 ) A1 X (1) ((t1  ( j  1)h)  , t1 ) A1  [ X ((t1  ( j  1)h)  , t1 ) A0  X ((t1  ( j  2)h)  , t1 ) A1 ] A1  X (1) ((t1  ( j  2)h)  , t1 ) A21 j  0  lhs  X (1) (t1 , t1 )  0  rhs; j  1  lhs  X (1) ((t1  h)  , t1 )   X ((t1  h)  , t1 ) A0  X (t1 , t1 ) A1  X (1) (t1 , t1 ) A 1   X ((t1  h)  , t1 ) A0  0  0

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38 | P a g e


The structure of determining matrices for single-delay…

 1  r      X ((t1  (1  (r  i ))h) , t1 ) Ai  A1 ; r  0, i  0 yield  ( X (t1 , (t1  h) , t1 ) A0 r 0  i 0   r  0, i  1 yield  ( X (t1 , t1 ) A 1  0; r  1, i  0 yield  X (t1 , t1 ) A0 A 1  0; r  1, i  1 yield 0. Let us examine 

1

Sum the results for these contingencies to get 1  1     X ((t1  (1  (r  i ))h)  , t1 ) Ai  Ar1  ( X ((t1  h)  , t1 ) A0 r 0  i 0 

So, (ii) of the lemma is valid for j  1 . Let us explore j  2 .

j  2  lhs  X (1) ((t  2h)  , t1 )   X ((t  2h)  , t1 ) A0  X ((t  h)  , t1 ) A1  X (1) ((t  h)  , t1 ) A1   X ((t  2h) , t1 ) A0  X ((t  h)  , t1 ) A1  X ((t  h)  , t1 ) A0 A1  X (t1 , t1 ) A 1 A1  X (1) (t  , t1 ) A21   X ((t  2h)  , t1 ) A0  X ((t  h)  , t1 ) A1  X ((t  h)  , t1 )A0 A1  1  r     X ((t1  (2  (r  i ))h) , t1 ) Ai  A1 ; r 0  i 0   r  0, i  0 yield  ( X ((t1  2h) , t1 ) A0 ; r  0, i  1 yield  ( X ((t1  h)  , t1 ) A 1; Let us examine 

2

r  1, i  0 yield  ( X ((t1  h)  , t1 ) A0 A 1; r  1, i  1 yield 0; r  2, i  0 yield 0; r  2, i  1 yield 0. Sum the results for these contingencies to get 2  1     X ((t1  (2  (r  i ))h)  , t1 ) Ai  Ar1 r 0  i 0    ( X ((t1  2h) , t1 ) A0  ( X ((t1  h)  , t1 ) A 1  ( X ((t1  h)  , t1 ) A0 A 1. So, (ii) of the lemma is valid for j {0,1, 2}. Assume that (ii) of the lemma is valid for

3  j  n , for some integer n. Then n 1  X (1) ((t1  nh) , t1 )    X ((t1  (n  (r  i))h)  , t1 ) Ai Ar1, by the induction hypothesis. r 0  i 0  (1)   Now, X ((t1  ( n  1) h) , t1 )    X ((t1  ( n  1) h) , t1 ) A0  X ((t1  nh)  , t1 ) A1 

 X (1) ((t1  nh)  , t1 ) A1 1

 X (1) ((t1  nh)  , t1 ) A1   X ((t1  (n  1  i )h)  , t1 ) Ai i 0

1       X ((t1  (n  (r  i ))h)  , t1 ) Ai  Ar11   X ((t1  (n  1  i )h)  , t1 ) Ai r 0  i 0 i 0  n 1  1     X ((t1  (n  1  (r  i ))h)  , t1 ) Ai  Ar1 , as desired. So (ii) is true for j  n  1, and r 0  i 0  hence true for all j. n

1

Proof of (iii) The proof follows by noting that

X (1) (t1  jh, t1 )  X (1) ((t1  jh) , t1 )  X (1) ((t 1  jh) , t1 ) and then

using (i) and (ii) to deduce that

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39 | P a g e


The structure of determining matrices for single-delay… j  1  X (1) (t1 , (t1  jh))      X ((t1  ( j  (r  i ))h)  , t1 )  X ((t1  ( j  (r  i ))h)  , t1 )  Ai  Ar1 r 0  i 0  j  1  r  1       X ((t1  ( j  (r  i ))h), t1 ) Ai  A1     Aj1( r i ) Ai Ar1  r 0  i 0 r 0  i 0   (k ) Lemma on X (t1 , ((t1  jh)) recursions j

2.7

j 1  X ( k ) (t1  jh, t1 )     X ( k 1) ((t1  ( j  (r  i))h), t1 ) Ai Ar1 r 0  i 0 

Proof

 1  If j  0, then r  0; lhs  X ( k ) (t1 , t1 )  (1) k A0k , rhs     X ( k 1) (t1  ih, t1 ) Ai   i 0  ( k 1) ( k 1) k 1 k 1 k k  X (t1 , t1 ) A0  X (t1  h, t1 ) A1  (1)(1) A0 A0  0  (1) A0 (by lemma 2.4) If j  1, then using lemma 2.4 ,

lhs  X ( k ) (t1  (1  1)h, t1 ) A1   X ( k 1) (t1  h, t1 ) A0  X ( k 1) (t1 , t1 ) A1   X ( k ) (t1  (1  1)h, t1 ) A1   X ( k 1) (t1  h, t1 ) A0  (1) k 1 A0k 1 A1   (1)k Ak0 A1  [X ( k 1) (t1  h, t1 ) A0  (1)k 1 A0k 1 A1 (by 2.4).  1  Rhs      X ( k 1) ((t1  ( j  (r  i ))h), t1 ) Ai  Ar1 r 0  i 0     X ( k 1) ((t1  h), t1 ) A0 A01  X ( k 1) (t1 , t1 ) A1    X ( k 1) (t1 , t1 ) A0 A11  X ( k 1) (t1  h, t1 ) A1 A1  j

   X ( k 1) ((t1  h), t1 ) A0  (1) k 1 A0k 1 A1   (1) k 1 A0k 1 A0 A11  0   (1) k Ak0 A1   X ( k 1) ((t1  h), t1 ) A0  (1) k 1 A0k 1 A1  So, lhs = rhs. Therefore the lemma is valid for j = 1. Assume the validity of the lemma for 2  j  m for m 1  ((t1  mh), t1 )     X ( k 1) ((t1  (m  (r  i))h), t1 ) Ai Ar1 r 0  i 0  (k ) Now X ((t1  (m  1)h), t1 )

some integer m .

Then X

(k )

   X ( k 1) ((t1  (m  1)h), t1 ) A0  X ( k 1) (t1  mh, t1 ) A 1   X ( k ) (t1  mh, t1 ) A1 1 m  1    X ( k 1) ((t1  (m  1  i )h), t1 ) Ai    X ( k 1) ((t1  (m  (r  i ))h, t1 ) Ai Ar11  i 0 r 0  i 0  1 m1 1     X ( k 1) ((t1  (m  1  i)h), t1 ) Ai    X ( k 1) ((t1  (m  (r  1  i ))h, t1 ) Ai Ar111  i 0 r 1  i 0  1 m 1 1     X ( k 1) ((t1  (m  1  i )h), t1 ) Ai    X ( k 1) ((t1  (m  1  (r  i ))h, t1 ) Ai Ar1  i 0 r 0  i 0  1 m 1 1     X ( k 1) ((t1  (m  1  i )h), t1 ) Ai     X ( k 1) ((t1  (m  1  (r  i ))h, t1 ) Ai Ar1  i 0 r 0  i 0 

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40 | P a g e


The structure of determining matrices for single-delay…

r  r 1 in the r  summand, r  r  1 in the r  limits). So the lemma is valid for j  m  1 . This completes the desired proof. (We used the change of variables

The following major theorem gives explicit computable expressions for Q k ( jh ), based on painstakingly and rigorously observed Q k ( jh ) patterns for various j , k contingencies.

III.

THEOREM ON EXPLICIT COMPUTABLE EXPRESSION FOR DETERMINING MATRICES OF (1)

Let j and k be nonnegative integers. If j  k  1, then

Qk ( jh)

( v1 ,,v j k )  P1( j ),0( k )

Av1  Av

k 1

 (v ,,v

r 1

1

j k

Av1  Av j

( v1 ,,v j )  P1( j  k ),1( k )

Av1  Av jr

j r )  P1( r  j k ),0( r ),1( k r )

If k  j  1, then Q k ( jh)

( v1 ,,v j k )  P1( j ), 0( k )

Av1  Av jk 

j 1

 r 1 ( v ,,v

( v1 ,,v k )  P0( k  j ),1( j )

k r )  P1( r ), 0( r k  j ),1( j r )

1

Av1  Av k

Av1  Avk r

Proof

Case j  k . If j  k  0, then Qk ( jh)  Q0 (0)  I n , by ( 4); k  0  Qk ( jh)  Q0 ( jh)  A1 (by (ii) of lemma 2.4). j

j 1

j  1, k  1  Qk ( jh)  Q 1 ( jh)  A1 A 0   A1 ( A 0 A 1  A1 ) A1 j

j  ( r 1)

r

, (by (iv) of lemma 2.4)

r 0

j 1

j 1

r 0

r 0

 Aj1 A0   Ar1 A0 Aj1r   Ar1 A1 Aj1( r 1) 

( v1 ,,v j1 )  P1( j ),0(1)

Av1  Av j1 

( v 1 ,,v j )  P1( j 1),1(1)

Av1  Av j

If j  k  2, then Qk ( jh)  Q2 (2h)  A1Q2 ( h)  A0Q1 (2h)  A1Q1 (h)  A1

( v 1 ,,v3 )  P1(1), 0(2)

 A0

Av  Av  A1 1

( v 1 ,,v3 )  P1(2), 0(1)

( v1 ,,v 4

 ) P

 ( v1 ,,v 4

3

Av 1  Av 3  A0

Av 1  Av 4  A12 

1(2),0(2)

 ) P

1(2),0(2)

Av 1  Av 4 

( v 1 ,v2 )  P0(1),1(1)

Av Av

( v 1 ,v2 )  P1(1),1(1)

1

2

Av Av  A1 1

2

( v1 ,,v3 )  P1(1),0(1),1(1)

 ( v ,v )  P 1 2

Av Av  A1 1

2

v 1  P1(1)

Av

1

Av 1  Av 3 2 1

Av1 Av 2  

1(22),1(2)

( v 1 ,v2 )  P1(1), 0(1)

r 1

( v1 ,,v2 r s )  P1( r ),0( r ),1(2r )

Av1  Av 2r

Therefore the theorem is valid for j , k {0, 1, 2}. In particular it is valid for 0  k  j  2.

The cases : Q1 ( jh) and Q k ( h) have been established already. We need only prove the remaining cases.

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41 | P a g e


The structure of determining matrices for single-delay…

Case j  k : Assume that the theorem is valid for all pairs of triples j , k , Qk ( jh); j , k, Qk ( jh) for which j  k  j  k , for some j  3, and k  2. Then apply the induction principle to the right-hand side of the determining equation:

Qk ([ j  1]h)  A0Qk 1 ([ j  1]h)  A1Qk 1 ( jh)  A1Qk ( jh), noting that j  k  j  1  k ,

,

j  1  k  1, j  k  1. Therefore:

Q k ([ j  1]h)  A0 Av1  Av jk  A0 Av1  Av j1    ( v1 ,,v j k )  P1( j 1), 0( k 1) ( v1 ,,v j 1 )  P1( j 1 ( k 1)),1( k 1)  k 11  A Av1  Av j1r   0  r 1 ( v1 ,,v j  1 r )  P1( r  j 1( k 1), 0( r ),1( k 1r )   A1 Av  Av jk 1  A1 Av 1  Av j    ( v1 ,,v jk 1 )  P1( j ), 0( k 1) 1 ( v1 ,,v j )  P1( j  ( k 1)),1( k 1)  k 1��� r  Av1  Av jr   A1   s 1 ( v1 ,,v j r )  P1( r  j ( k 1),0( r ),1( k 1r )   Av1  Av jk  A1 Av1  Av j  A1  P ( v ,  , v )  P ( v ,  , v )  j 1 1 j  k  1( j ), 0( k )  1( j  k ),1( k )   k 1   A1  Av1  Av jr   r 1 ( v1 ,,v j r )  P1( r  j k ),0( r ),1( k r )

0

i 1 ( v1 ,,v j 1 )  PiL1( j 1), 0( k )

 A0 A1

( v1 ,,v j1 )  P1( j 2k ),1( k 1)

1( j ),0( k 1)

k 1

 (v ,,v r 1

1

The expression

i{1,1} ( v1 ,,v j 1 )  PiL1( j 1k ),1( k ) k 2

 (v ,,v

Av1  Av j1  A0

r 1

1

k 2

 ) P

( v1 ,,v j k 1

 A1

Av1  Av j1k 

Av1  Av

j k 1

j r )  P1( r  j k ),0( r ),1( k r )

9

( v1 ,,v j 1 )  P1( j 1), 0( k )

r 1

j  1 r )  P1( r  j 2 k ), 0( r ),1( k 1r )

( v1 ,,v jr )  P1( r  j 1k ),0( r ),1( k 1r )

Av1  Av jr

(9) Av1  Av j1r (10)

Av1  Av jr

(11)

(12)

yields:

Av1  Av j1k  k 1

The expression:

 A1 

Av1  Av j1

( v1 ,,v j 1 )  P1( j 1k ),1( k )

1L r 1 ( v1 ,,v j r )  P1( r  j 1k ), 0( r ),1( k r )

Av1  Av j1

Av1  Av j1r

(13)

(14)

is immediate from (12)

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42 | P a g e


The structure of determining matrices for single-delay… We use the change of variables technique in the expression 10  to get:

A0 ( v1 ,,v j 1

 A0  A0  A0

 )P

Av1  Av j1  A0

Av1  Av j1  A0

k 2

 (v ,,v r 1

1( j 2k ),1( k 1)

( v1 ,,v j 1 )  P1( j 2k ),1( k 1)

( v1 ,,v j 1 )  P1( j 2k ),1( k 1)

( v1 ,,v j 1 )  P1( j 2k ),1( k 1)

 A0 ( v1 ,,v j 111

j  1 r )  P1( r  j 2 k ), 0( r ),1( k 1r )

k 1

 (v ,,v r 2

 (v ,,v r 2

 (v ,,v r 1

Av1  Av j111

1

1(1  j 1 k ), 0(11),1( k 1)

Av1  Av j1r 1

k 1

 (v ,,v r 1

Av1  Av j1r1

Av1  Av j1r 1

j  1 r 1 )  P1( r  j 1 k ), 0( r 1),1( k r )

 A0

(15)

j  1 r 1 )  P1( r  j 1 k ), 0( r 1),1( k r )

1

k 1

Av 1  Av j1  A0

Av1  Av j1r

j  1 r 1 )  P1( r 1  j 2 k ), 0( r 1),1( k 1( r 1))

1

k 1

Av1  Av j1  A0

 )P

1

1

k 1

j  1 r 1 )  P1( r  j 1 k ), 0( r 1),1( k r )

r 1 ( v1 ,,v j  1 r )  P01(L r  j 1 k ), 0( r ),1( k r )

Av1  Av j1r1

Av1  Av j1r

(16)

We are left with expression 11 , which yields:

A1

k 2

( v1 ,,v j k 1 )  P1( j ), 0( k 1)

 A1

Av1  Av jk 1  A1 

Av1  Av jr

k 1

k 1

Av1  Av jr

r 1 ( v1 ,,v j r )  P1( r  j 1k ), 0( r ),1( k 1r )

( v1 ,,v j r )  P1( k 1 j 1k ), 0( k 1),1( k 1( k 1))



r 1 ( v1 ,,v j r )  P1( r  j 1k ), 0( r ),1( k 1r ) k 1

( v1 ,,v j k 1 )  P1( j ), 0( k 1)

 A1

Av1  Av jk 1  A1 

L r 1 ( v1 ,,v j 1 r )  P11( r  j 1k ), 0( r ),1( k r )

Av1  Av jr  A1 

r 1 ( v1 ,,v j r )  P1( r  j 1k ), 0( r ),1( k 1r )

Av1  Av j1r

(17)

Av1  Av jr (18)

The resulting expressions (13), (14), (16) and (18) add up to yield:

Q k ([ j  1]h)

( v1 ,,v j 1k )  P1( j 1),0( k ) k 1

 (v ,,v

r 1

1

Av1  Av jk 

( v1 ,,v j )  P1( j 1 k ),1( k )

j 1r )  P1( r  j 1k ),0( r ),1( k r )

Av1  Av j1r

Av 1  Av j (19)

This concludes the inductive proof for the case j  k. We proceed to furnish the inductive proof for the case k  j. Theorem 3 is already proved for j, k {1, 2,3, 4.}, k  j. Therefore the theorem is valid for j, k {0, 1, 2, 3, 4}. Assume that the theorem is valid for all pairs of triples j , k , Q ( jh); j , k , Q  ( jh) for which j  k  j  k , for some j  3, and k  4. Then apply the induction k

k

principle to the right-hand side of the determining equation:

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43 | P a g e


The structure of determining matrices for single-delay… Qk 1 ( jh)  A0Qk ( jh)  A1Qk ([ j  1]h)  A 1Qk 1 ([ j  1]h), noting that k  j  k  1  j , k  1  j  1, k  j  1. Therefore:

Q k 1 ( jh)

 A0 Av1  Av jk  A0 Av 1  Av k    ( v1 ,,v j k )  P1( j ), 0( k ) ( v1 ,,vk )  P0( k  j )),1( j )  j 1  A Av1  Av k r  0   r 1 ( v1 ,,v j r )  P1( r ), 0( r k  j ),1( j r )   A1 Av  Av jk 1  A1 Av1  Av k    ( v1 ,,v j1k )  P1( j1), 0( k ) 1 ( v1 ,,v j 1 )  P0( k  ( j 1)),1( j 1)  j 1 r    A1  Av1  Avk r   r 1 ( v1 ,,vk r )  P1( r ), 0( r k ( j 1)),1( j 1 r )  Av1  Av jk  A1 Av1  Av k 1  A1   ( v1 ,,v j k )  P1( j 1), 0( k 1) ( v1 ,,v j )  P0( k 1 ( j 1)),1( j 1)   j 2   A1  Av1  Avk 1r   r 1 ( v1 ,,vk 1r )  P1( r ), 0( r k 1( j 1)),1( j 1r )

0

i 1 ( v1 ,,v j k 1 )  PiL1( j ), 0( k 1)

 A0 A1

j 1

 (v ,,v r 1

k r )  P1( r ), 0( r k  j ),1( j r )

1

( v1 ,,vk 1

i 0

iL ( v1 ,,v j 1 )  P0( k 1 j ),1( j )

( v1 ,,v j 1 )  P1( j ), 0( k 1)

Av1  Av k 2 j  A1

 r 1

j 2

 (v ,,v r 1

1

Av1  Avk r

k 1 r )  P1( r ), 0( r k 2 j ),1( j 1r )

(22)

Av1  Avk 1r

(23)

yields:

Av1  Av j1k 

j 1

The expression:

(20) (21)

r 1 ( v1 ,,vk r )  P1( r ), 0( r k 1 j ),1( j 1r )

0( k 2 j ), 0( j 1)

 20 

Av1  Av k 1

Av1  Av k r

Av1  Av j1 k  A1 

 )P

The expression

1

j 2

( v1 ,,v j 1 k )  P1( j 1), 0( k )

 A1

Av1  Av jk 1 

( v1 ,,v j 1 )  P0( k 1 j ),1( j )

( v1 ,,vk 1r )  P01(L r ),0( r k 1 j ),1( j r )

Av1  Av j1

Av1  Av

k 1r

(24)

(25)

is immediate from (21)

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44 | P a g e


The structure of determining matrices for single-delay…

We use the addition and subtraction technique in (22) to get:

A1

( v1 ,,v j 1k )  P1( j 1),0( k )

= A1

( v1 ,,v j 1k

 A1 

j 2

 A1  r 1

( v1 ,,vk r )  P1( r ),0( r k 1 j ),1( j 1r )

Av1  Av j1 k  A1  r 1

1( j 1),0( k )

Av1  Av

( v1 ,,vk r )  P1( r ),0( r k 1 j ),1( j 1r )

k r

Av1  Avk r

Av1  Av

k  j 1

( v1 ,,vk  j 1 )  P1( j 1),0( j 1k 1 j ),1( j 1( j 1))

L ( v1 ,,vk r )  P11( r ),0( r k 1 j ),1( j r )

r 1

j 1 k

j 1

 ) P

j 1

Av1  Av

Av1  Av

(26)

k 1r

We are left with expression (23), which, using the change of variables technique yields:

A1

( v1 ,,vk 1 )  P0( k 2 j ),0( j1)

=A1

( v1 ,,vk 1 )  P0( k 2 j ),0( j1)

 A1  A1

( v1 ,,vk 1 )  P0( k 2 j ),0( j1)

Av1  Av

k 2 j

k 2 j

 A1  A1

Av1  Av k 2 j  A1

( v1 ,,vk 111 )  P1(11),0(11k 2 j ),1( j 1(11))

j 1

 r 1

Av1  Av

1L ( v1 ,,vk 1r )  P1( r ),0( r k 1 j ),1( j r )

j 2

 (v ,,v r 1

j 1

 (v ,,v r 2

1

 (v ,,v 1

Av1  Av

k 1r

Av1  Av

Av1  Av

k 1 r 1 )  P1( r 1), 0( r 1k 2 j ),1( j 1( r 1))

j 1

r 1

k 1r )  P1( r ),0( r k 2 j ),1( j 1r )

1

k 1 r 1 )  P1( r 1), 0( r 1k 2 j ),1( j 1( r 1))

k 1r 1

k 1r 1

Av1  Av

k 111

Av1  Avk 1r

(27)

The resulting expressions (24), (25), (26) and (27) add up to yield:

Q k 1 ( jh)

( v1 ,,v j k 1 )  P1( j ), 0( k 1)

j 1

 (v ,,v r 1

1

Av1  Av jk 

( v1 ,,vk 1 )  P0( k 1  j ),1( j )

j 1r )  P1( r ), 0( r k 1 j ),1( j r )

Av1  Av k 1

Av1  Avk 1r

(28)

This concludes the inductive proof for the case k  j, completing the proof of the theorem. The cases j  k and k  j , in the preceding theorem can be unified by using a composition of the max and the signum functions as follows:

IV.

INDUCTION PITFALLS IN THE DERIVATION OF THE DETERMINING MATRICES

In the development of Qk ( jh ) for the single – delay neutral system (1) we had proposed, based on the observed pattern for Qk ( jh) for 0  min{ j , k}  2, that for j  k  1,

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The structure of determining matrices for single-delay…

Q k ( jh)

( v1 ,,v j k )  P1( j ),0( k ) k 1 k  r

  r 1 s 1

and for

k  j  1,

Qk ( jh) 

Av1  Av jk 

( v1 ,,v2 r  j k s )  P1( r  jk ),0( r ),1( s )

(v1,,v jk )  P1( j ),0( k )

Av1  Av

j 1 j  r

  r 1 s 1

( v1 ,,v j )  P1( j  k ),1( k )

j k

j

Av1  Av jr ; k  2

( v1 ,,v k )  P0( k  j ),1( j )

Av1  Av ; k  1

(v1,,v2 r k  js )  P1( r ),0( r k  j ),1( s )

Av  Av ; j  1 1

k

Av1  Av2 r k  j s ; j  2

This formula is valid for all j , k :1  min{ j, k}  2. However it turns out to be false for all

j, k : min{ j, k}  3. This revelelation came to the fore following the discovery of some fallacy in the proof of the expressions for Qk ( jh), which could never be overcome; it became imperative to examine Q3 (3h ) from the determining equations for Qk ( jh). The formula was found to be inconsistent with the true result from the determining equations. Therefore the formula had to be abandoned after so much effort. Further long and sustained research effort led to the conjecture

Qk ( jh) 

(v1,,v jk )  P1( j ),0( k )

Av  Av 1

j k

( v1 ,,v j )  P1( j  k ),1( k )

Av  Av j ; k  1 1

 k 1 k r Av1  Av jr ; k  2, j  k ,   r 1 s 1 ( v1 ,,v2 r  j k s )  P1( r  j k ),0( r ),1( s )    k 1 k r  Av1  Av jr ; j  k  2    r 1 s  max{1, k 1 2 r } ( v1 ,,v2 r  j k s )  P1( r  j k ),0( r ),1( s )  as well as the analogous expression for Qk ( jh) for k  j  1. The formula was found to be valid for j , k :1  min{ j , k}  2, as in the aborted formula. Furthermore

Q3 (3h) obtained from the postulated formula agreed with the corresponding result (sum of 63 permutation products) obtained directly from the determining equations. Unfortunately, the proof for arbitrary j and k could not push through; there was no way to apply induction on the lower limit for s, not to talk of dealing with the piece–wise nature of the third component summations. Above pitfalls provided object lessons that integer–based mathematical claims should never be accepted without error-free proofs by mathematical induction. There is no other way to ascertain that a pattern will persist –no matter the number of terms for which it has remained valid.

V.

CONCLUSION

The results in this article bear eloquent testimony to the fact that we have comprehensively extended the previous single-delay result by Ukwu (1992) together with appropriate embellishments through the unfolding of intricate inter–play multiple summations and permutation objects in the course of deriving the expressions for the determining matrices. By using the change of variables technique, deft application of mathematical induction principles and careful avoidance of some induction pitfalls, we were able to obtain the structure of the determining matrices for the single–delay neutral control model, without which the computational investigation of Euclidean controllability would be impossible.

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The structure of determining matrices for single-delay‌ REFERENCES [1] [2] [3] [4]

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