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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-14-30

The structure of determining matrices for a class of double – delay 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 a class of double – delay autonomous linear differential systems through a sequence of lemmas, theorems, corollaries and the exploitation of key facts about permutations. The proofs were achieved using ingenious combinations of summation notations, the multinomial distribution, the greatest integer function, change of variables technique and compositions of signum and max functions. The paper has extended the results on single–delay models, with more complexity in the structure of the determining matrices.

KEYWORDS: Delay, Determining, Double, 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, 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 delay control systems. Also see Ukwu (2013a). However up-to-date review of literature on this subject reveals that there is currently no result on the structure of determining matrices for double-delay systems. This could be attributed to the severe difficulty in identifying recognizable mathematical patterns needed for inductive proof of any claimed result. Thus, this paper makes a positive contribution to knowledge by correctly establishing the structure of such determining matrices in this area of acute research need.

II.

MATERIALS AND METHODS

The derivation of necessary and sufficient condition for the Euclidean controllability of system (1) on the interval [0, t 1 ], using determining matrices depends on 1)

obtaining workable expressions for the determining equations of the

n  n matrices

for

j : t1  jh  0, k  0, 1,  2) 3)

showing that where

4)

showing that

=

Q (t1 )

( h),for j:

is a linear combination of

Q0 ( s ), Q1 ( s ),  , Qn 1 ( s ); s  0, h, (n  1)h. See Ukwu (2013a). Our objective is to prosecute task (i) in all ramifications. Tasks (ii) and (iii) will be prosecuted in other papers. 2.1 Identification of Work-Based Double-Delay Autonomous Control System We consider the double-delay autonomous control system:

x  t   A0 x  t   A1 x  t  h   A2 x  t  2h   B u  t  ; t  0 x  t     t  , t   2h, 0 , h  0

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(1) (2)

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The structure of determining matrices for a class of… Where A0 , A1 , A2 are n  n constant matrices with real entries,

B

is an

n m

constant matrix with

real entries. The initial function  is in C

2h, 0 , R  , the space of continuous functions from [2h, 0]

into the real n-dimension Euclidean space,

Rn with norm defined by   sup   t  , (the sup norm). The

control u is in the space L into

n

t2 h , 0

0, t  , R  , the space of essentially bounded measurable functions taking 0, t  n

1

1

Rn with norm   ess sup u(t ) . t  [ 0, t1 ]

Any control u  L the spaces 2.2

0, t , R  will be referred to as an admissible control. For full discussion on n

1

C p1 and Lp (or Lp ) , p {1, 2,..., } , see Chidume (2003 and 2007) and Royden (1988).

Preliminaries on the Partial Derivatives

 k X ( , t )  k

, k  0,1, 

Let t ,    0, t1  . For fixed t, let     , t  satisfy the matrix differential equation:

 X   , t    X  , t  A0  X   h, t  A1  X   2h, t  A2  for 0    t ,   t  k h, k  0,1,...

where X  , t  

(3)

In ;   t 0;   t

See Chukwu (1992), Hale (1977) and Tadmore (1984) for properties of   t ,   . Of particular importance

is

the

fact

    , t 

that

is

intervals t1   j  1 h, t1  j h , j  0,1,..., t1   j  1 h  0 . Any such

analytic

on

the

  t1   j  1 h, t1  j h 

is

called a regular point of     t ,   . See also Analytic function (2010) for a discussion on analytic functions. Let 

k 

 , t 

t   j  1 h, 1

Write 

 k 1

denote

k   , t1  , the k th partial derivative of   ,t1  with respect to , where  is in k 

t1  j h  ; j  0,1,..., r ,

 , t1  

for

some

integer

r

such

that

t1   r  1 h  0 .

 k   , t1  . 

Define:

  k   t1  jh, t1     k  t1 ,  t1  j h  , t1    k  

 t  j h , t  , 

1

1

(4)

for k  0,1,...; j  0, 1,...; t1  jh  0, where X limits of

X

k

k 

t  j h  , t  and X t , t  j h  , t  denote respectively the left and right hand k 

1

 , t1  at 

X ( k )  ( t1  jh) , t1  

1

1

1

1

 t1  j h . Hence:

lim

  t1  jh

X

k 

 ,t  1

(5)

t1  ( j  1)h    t1  jh

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The structure of determining matrices for a class of…

k 

 ,t1    t1  jh t1  jh    t1  ( j  1)h

X ( k )  ( t1  jh) , t1  

lim

X

(6)

2.3 Definition, Existence and Uniqueness of Determining Matrices for System (1) Let Q k (s) be then n  n matrix function defined by:

Qk  s   A0Qk 1  s   AQ 1 k 1  s  h  A2Qk 1  s  2h

for k  1, 2,; s  0, with initial conditions:

Q0  0  In

(7) (8)

Q0  s   0; s  0

(9)

These initial conditions guarantee the unique solvability of (7). Cf. [1]. The stage is now set for the establishment of 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. 2.4 Lemma on permutation products and sums (k )

Let

r0 , r1 , r2 be nonnegative integers and let P0 r ,1 r ,2( r ) denote the set of all permutations of 0 1 2

0,  2, 0,...0  1,1,...1 2,...2  : the permutations of the objects 0,1, and 2 in which i appears ri times, i  0,1, 2. r1 times

r0 times

r2 times

iL

Let P0 r ,1 r ,2( r ) denote the subset of 0 1 2 iT

P0 r0 ,1 r1 ,2( r2 ) with

position. Let P0 r ,1 r ,2( r ) denote the subset of 0 1 2 last position. Set

a 

i , that is, those with i occupying the first

leading

P0 r0 ,1 r1 ,2( r2 ) with trailing i , that is, those with i occupying the

r  r0  r1  r2 . Then for any fixed r0 , r1 , r2 ,

 v 1 ,...vr   P0  r 0 , 1 r1, 1r 2 

2

Av 1 ..., Avr  Sr (r0 , r1 , r2 )   i 0

iL

 v 1 ,...vr   P0 r 0 , 1r1, 1r 2 

2

Av 1 ..., Avr   SriL (r0 , r1 , r2 ) i 0

     A0  Av 1 ..., Avr 1  sgn(r0 )  A1  Av 1 ..., Avr 1  sgn(r1 )     v 1 ,...vr 1   P0  max0,r 0 1 , 1r1, 1 r 2     v 1 ,...vr 1   P0 ( r 0 ) , 1max0,r1 1 , 1 r 2           A2 Av1 ..., Avr1  sgn(r2 )  v ,...v  P    1 r1  0( r 0 ), 1r1, 2  max0, r 2 1 

 b

 v 1 ,...vr   P0  r 0 , 1 r1, 1r 2 

2

Av 1 ..., Avr  Sr (r0 , r1 , r2 )   i 0

iT

 v 1 ,...vr   P0 r 0 , 1r1, 1r 2 

2

Av 1 ..., Avr   SriT (r0 , r1 , r2 ) i 0

     Av 1 ..., Avr 1  A0 sgn(r0 )   Av 1 ..., Avr 1  A1 sgn(r1 )     v 1 ,...vr 1   P0  max0,r 0 1 , 1 r1, 1 r 2     v 1 ,...vr 1   P0 ( r 0 ) , 1max0,r1 1 , 1 r 2           Av1 ..., Avr1  A2 sgn(r2 )  v ,...v  P    1 r 1  0( r 0 ), 1r1, 2  max0, r 2 1  (c) Hence for all nonnegative integers

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The structure of determining matrices for a class of…

2

v 1 ,...vr  P

     

Av1 ..., Avr  Sr (r0 , r1 , r2 )  

iL i 0 v 1 ,...vr  P0 r

0 r 0 , 1 r1 ,1 r 2 r 0 r1r 2  r

 0 , 1r1,1r 2 

Av1 ..., Avr

r 0  r1  r 2  r

2



i  0 r 0  r1  r 2  r

SriL (r0 , r1 , r2 )

Similar statements hold with respect to the remaining relations. Note that 2

 i 0

2

r

r  r0

Av1 ..., Avr    

iL i 0 r0 0 r1 0 v 1 ,...vr  P0 r

v1 ,...vr  PiL

0  r 0 , 1 r1, 2  r 2        r 0 r1r 2  r

 0 , 1r1, 2r 2 

Av1 ..., Avr

sgn(ri ) ensures that the corresponding expression vanishes if ri  0  Ai does not appear and so cannot be factored out. max{0, ri  1} ensures that the resulting permutations are well-defined. In order not to clutter the work with ‘ max{0, ri  1} ’ and ‘ sgn(ri ) ’, the standard convention of letting

 v1 ,...vr   P0 r0 , 1r1,1r2 

Av 1 ..., Avr  0, for any fixed r0 , r1 , r2 ; r0  r1  r2  r : ri  0, for some i {0,1, 2}

would be adopted, as needed. Proofs of (a), (b) and (c) Every permutation involving 0, 1, and 2 must be led by one of those objects. If 0, 1 and 2 appear at least once, then each of them must lead at least once. Equivalent statements hold with ‘led’ replaced by ‘trailed’ and ‘lead’ replaced by ‘trail’. Hence the sum of the products of the permutations must be the sum of the products of those permutations led (trailed) by A0 , A1 , and A2 respectively. Consequently, r

r r0



r

r 0  0 r1  0 v 1 ,...vr  P0 r , 1 r , 2 r r r  0   1  0 1

r r 0

Av 1 ..., Avr    Sr (r0 , r1 , r  r0  r1 ) r 0  0 r1  0

r 0 r1r 2  r

2  r r  r0       Sr (r0 , r1 , r  r0  r1 )  restricted to those permutations with leading(trailing Ai )   i  0  r0  0 r1  0 

  A S ( r , r , max 0, r  1) sgn( r )  S (max 0, r  1 , r , r ) A sgn( r )  S  S ( r , r , max 0, r  1) A sgn( r )

 A0 S r 1 (max 0, r0  1 , r1 , r2 ) sgn( r0 )  A1 S r 1 ( r0 , max 0, r1  1 , r2 ) sgn( r1 ) r 1

2

0

r 1

0

r 1

2.5

1

0

1

2

1

2

2

2

0

0

2

r 1

( r0 , max 0, r1  1 , r2 ) A1 sgn( r1 )

2

Preliminary Lemma on Determining Matrices

Qk ( s ), s  R

(i) (ii) (iii) (iv)

(v) Q1 ( jh)  A j sgn(max{0, 3  j}), j  0. (vi)

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The structure of determining matrices for a class of… (vii) (viii)

I n , if j  0

 X ( t1  jh, t1 )   

0, otherwise

Proof (i) Then, We need to prove that Assertion: Proof: So the assertion is true for k = 1. Assume that

for some integer n. Then ,

by the induction hypothesis, since Therefore, Hence (ii)

Let k = 1 and let s

proving that

for any integer r. Then , since s  for some integer . Then

Assume

by the induction hypothesis. Hence (iii) This has already been proved. (iv)

for any integer r

v1P011,1(1)

Assume (iv) is true for

by the definition of

Av1 .

So (iv) is true for k = 1.

for some integer . Then by the induction hypothesis.

Therefore,

( v 1 ,,vn1 ) P0( n 1 1),1(1)

So (iv) is true for (v)

Av 1  Avn1 

( v 1 ,,vn1 ) P0( n 1 1),1(1)

Av 1  Avn1

. by (i) and (ii) respectively

For

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The structure of determining matrices for a class of… Now (by the definition of

), proving (v).

(vi) For k = l, this yields

Therefore

. Note that for sufficiently, close to

Assume

that

3  k  n, for

for



 

some

integer

n.

Then X  n 1 t1 , t 1   X n   , t 1   t    X n  t1 , t 1 A0   



 X ( n ) t1  h)  , t1 A1  X n  ( t1  2h)  , t1 A2 Therefore (vii) X (viii)

 11 A0n A0  0 A1  0 A2  1

n1

n

A0n1 .

X k  t1 , t1   1 A0k , proving (vi)

k 

k

t

1

, t1

lim t1  t1  h

(k )  X k   , t1   0, since   t . Therefore X ( t1 , t1 )  0 , proving

1  X  , t1    X  , t1 A0  X   h, t1 A1  X   2h, t1 A2 , 

for

, where

Let j be a non-negative number such that . Then we integrate the system (3), apply the above initial matrix function condition and the fundamental theorem of calculus, (F.T.C.) to get: (by the F.T.C.)

Similarly, +

,

)

Therefore,

since

is bounded and integrable (being of bounded

variation) and the fact that for any bounded integrable function, f. Therefore For we have completing the proof of (viii). See Bounded variation (2012) for detailed discussion on functions of bounded variation.

2.6

Lemma on Qk ( jh); j  {2k  2, 2k  1, }, k  1

For

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The structure of determining matrices for a class of…

Proof

Note that the first summations in (iv), yield

( v1 ,, v k ) 

p1 2 , 2  k 2 

Av1  Avk

Qk ([ 2k  1]h)  A0Qk 1 ([ 2k  1]h)  A1Qk 1 (2kh)  A2Qk 1 ([ 2k  1]h) by lemma 1.4. Clearly for So the lemma is valid for when Assume that the lemma is valid for for every integer, k such that . Then

Qn 2n  1h 

(10) (by the induction hypothesis), since

Equivalently, on the right-hand side of (1.10) set in the first term; Then clearly k < n and j > 2 k + 1 Hence the induction hypothesis applies to the right-hand side of (1.10), yielding 0 in each term and consequently 0 for the sum of the terms. Therefore , Qn 2n  1h   0 For any Now Hence

, and Combine this with the case proving that

(ii)

Consider So (ii) is valid for k =1. Assume the validity of (ii) for

. to conclude that as required in (i) of lemma 2.5. by lemma 1.5.

this yields

for some integer n. Then

Qn1 2n  1h 

and

of lemma 2.5,

and

.

(iii)

and For k = 1,

Now

by the induction hypothesis; therefore, Hence , by lemma 1.5 .

( v1 ,, vk )  P1(1), 2( k 1)

Av1  Avk 

v1P1(1)

Av1  A1 , for k  1.

, proving So (iii) is valid for k = 1

Assume the validity of (iii) for 1 < k < n, for some integer n. Then = Now

,

Therefore,

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The structure of determining matrices for a class of…

with leading

Av j s , j  1,n  1, in the above summation.

in each permutation of the

Since

appears only once in each permutation it can only lead in one and only one permutation, in this case In all other permutations will occupy positions 2, 3, … up the last position So the above expression for is the same as: proving that

k 1

To prove the second part, note that

A AA r 2

r 0

1

k 1 r 2

is the sum of the permutations of A1 and A2

which A1 appears once and A2 appears k – 1 times in each permutation. In the first permutation, corresponding to r = 0, A1 occupies the first postion (A1 leads), …, in the last permutation,

corresponding

to

r

=

k

1,

A1

occupies

the

last

position

(A1

trails).

the term under the summation represents the permutation in which A1 occupies the (r  1) position. st

Thus

(iv) Consider Qk ([2k  2]h), for k  1; this yields Q1 (0)  A0 (by lemma 2.5). Let us look at the right-hand side of (iv) in lemma 1.6.

If k  1, and j  2k  2, then

Now

 A ... A

v1 ( v 1 , ...,vk ) P0 1 , 2  k 1 

vk

A

A

...Avk 

v1 ( v 1 , ...,vk ) P1 2 , 2  1 

v 1 P0 1 

v1

A

v1 v 1 P1 2 , 2  1 

 0, by the summation infeasibil ity.

 A0 , for k  1. So, (iv) is valid for k  1.

Assume the validity of (iv) for 1 < k < n for some integer n. Then

(

)=

, by (ii)

Qn ([2n  1]h) 

n 1

( v1 ,, vn ) P1(1), 2( n1)

Qn ([2n  2]h) 

Av1  Avn   A2r A1 A2n1r ,

by (iii).

r 0

( v1 ,, vn )P1( 2 n[ 2 n2]), 2( 2 n2n )

Av1  Avn 

( v1 ,, vn )P0(1), 2( n1)

Av1  Avn ,

(by the induction hypothesis)

( v1 ,,vn )P1( 2), 2( n2)

Av1  Avn 

Qn 1 (2nh)  A0 A2n  A1  A2

( v1 ,, vn )P0(1), 2( n1)

Av1  Avn . Consequently,

( v1 ,,vn )P1( 2), 2( n1)

( v1 ,, vn )P1(1), 2( n2)

Av1  Avn

Av1  Avn  A2

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( v1 ,, vn )P0(1), 2( n1)

Av1  Avn . 21 | P a g e


The structure of determining matrices for a class of…

 A0 A2n 

( v1 ,,vn1 )P0(1), 2( n )

( v1 ,,vn1 )P1( 2), 2( n1)

Av1  Avn1 (with a leading A1 )

( v1 ,,vn1 )P1( 2), 2( n1)

( v1 ,,vn1 )P0(1), 2( n )

Av1  Avn1 (with a leading A2 )

Av1  Avn1 (with a leading A2 )

Av1  Avn1 

( v1 ,,vn1 )P1( 2), 2( n1)

Av1  Avn1

Notice that if k  n  1 and j  2(n  1)  2  2n, then 2k  j  2 and j  k  n  1. So (iv) is proved for

k  n 1 , and hence (iv) is valid . This completes the proof of the lemma. Lemma 2.6 can be restated in an equivalent form, devoid of explicit piece-wise representation as follows: 2.7

Lemma on Qk ( jh); j  {2k  2, 2k  1, }, k  1 using a composite function

For all nonnegative integers j and k , such that j  2k  2, k  1,

Qk ( jh)     Av1  Avk  sgn(max{0, 2k  1  j})   ( v1 ,,vk )P1( 2 k  j ), 2( jk )      Av Av  sgn(max{0, 2k  1  j}).  1 k  ( v1 ,,vk )P0(1), 2(k 1)  Proof: If j  2k  1 , both signum functions vanish, proving (i) of lemma 2.6. If j  2k , the second signum vanishes and the first yields 1, proving (ii). If j  2k  1 , the second signum vanishes and the first yields 1, proving (iii). If j  2k  2 , both signum functions yields 1, proving (iv).

III. 3.1

RESULTS AND DISCUSSIONS

Theorem on Qk ( jh); 0  j  k , k  0

For 0  j  k , j , k integers, k  0, Qk ( jh) 

 j    2     

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

Proof

k  1  Qk (0)  A0k , Qk (h) 

j  0  r  0  rhs 

( v1 ,, vk )  P0 ( k  j ),1( j )

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

j  1  r  0  rhs 

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

Av1  Avk

Av1  Avk , (by lemma 2.5)

Av1  Avk  A0k

Av1  Avk 

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( v1 ,,vk )  P0 ( k 1),1(1 )

Av1  Avk

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The structure of determining matrices for a class of…

j  2, k  2  Q2 (2h)  A0Q1 (2h)  A1Q1 (h)  A2Q1 (0)  A0 A2  A1 A1  A2 A0 (by lemma 2.5 )

j  2  r  {0,1}  rhs 

( v1 ,v2 ) P0(000),1(20),2(0)

Av1 Av2 

( v1 ,v2 ) P0(122).1(22),2(1)

Av1 Av2

 A12  A0 A2  A2 A0 So, the theorem is true for j  {0, 1}, k  1 and for j  2  k. Assume that the theorem is valid for all triple pairs j , k , Qk ( jh); j , k , Qk ( jh) for which j  k  j  k , for some j , k : k  j  3. Then

Qk 1 ( jh)  A0Qk ( jh)  A1Q k ([ j  1]h)  A2Qk ([ j  2]h) Now, j  k  1  j  1  k and j  2  k  1  k. So, we may apply the induction hypothesis to the righthand side of

(

to get:

Qk 1 ( jh)  A0

 j        2  

  r 0 ( v ,,v ) P 1

 A1

 A2

k

  j 1      2  

 r 0

Av1  Avk

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

 j 2      2  

 r 0

Av1  Avk

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

Two cases arise: even and odd Case 1: even. Then is odd are

(11)

0( r k  j ),1( j  2 r ),2( r )

Av1  Avk

(12)

(13)

is even; thus

  j  1    j  2   j  j   j   2      2    2  1 and   2    2         j  The summations in (11) are all feasible, since noting that r  1, 2,  ,  . 2  So the right hand side of (11) can be rewritten as:  j   2   

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

Av1  Avk ,

(14)

Av1  Avk

(15)

with a leading A0 (12) can be rewritten in the form: j 1 2

A1

We need to incorporate

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

j j in the range of r. If r  , then 2 2 www.ijmsi.org

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The structure of determining matrices for a class of… Therefore, the summation ( v1 ,,vk ) P

Av1  Avk is infeasible; hence it is set equal to 0. Thus the

j j 0( k 1 )),1( 1), 2( ) 2 2

case

j may be included in the expression (2.5) to yield: 2

r

j 2

A1

 (v ,,v ) P  r 0

1

Av1  Avk 

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

k

j 2

 (v ,,v r 0

k 1 )  P0( r ( k 1) j )),1( j 12 r ),2( r )

1

Av1  Avk 1 ,

(16)

with a leading A1 (2.3) may be rewritten in the form: j 1 2

A2 j

If r 

,

 (v ,,v ) P  r 0

1

k

Av1  Avk

(17)

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

then

so the summations with

r

2 being infeasible, yielding:

 (v ,,v ) P  r 0

1

k

, may be set equal to 0,

2

j 2

A2

j

Av1  Avk  A2

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

j 1 2

 (v ,,v ) P  r 1

1

k

Av1  Avk

(18)

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

(We used the change of variables technique: r  r 1 in the summand, r  r  1 in the limits). If + 1, then -2r = so the summations with r = + 1 may be equated to 0 and dropped. If r = 0, then r 1  1 . Therefore the summations with r = 0 are infeasible and hence set equal to 0. Thus (18) is the same as:

j 2

A2

 (v ,,v ) P  r 0

1

Av1  Avk 

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

k

j 2

 (v ,,v ) P  r 0

1

k

Av1  Avk ,

(19)

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

with a leading A2 . Therefore

Qk 1 ( jh) 

 j    2     

 (v ,,v r 0

1

 j   2   

 (v ,,v r 0

1

Av1  Avk 1 , with a leading A0

Av1  Avk 1 , with a leading A1

k 1 ) P0( r ( k 1) j ),1( j 2 r ),2( r )

k 1 ) P0( r ( k 1) j ),1( j 2 r ),2( r )

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The structure of determining matrices for a class of…  j   2   

 (v ,,v

r 0

j 2

 (v ,,v

r 0

k 1 ) P0( r ( k 1) j ),1( j 2 r ),2( r )

1

k 1 ) P0( r ( k 1) j ),1( j 2 r ),2( r )

1

Av1  Avk 1 

This concludes the proof of the theorem for even. If k = 0, then since 0  j  k , yielding (0) = Case 2: j odd. Then is even, is odd and

 1  1  1   2 ( j  1)    2 ( j  1)    2   

Av1  Avk 1 , with a leading A2

 j    2     

 (v ,,v r 0

1

k 1 ) P0( r ( k 1) j ),1( j 2 r ),2( r )

Av1  Avk 1

the n  n identity. is even. Hence

 1  1    1  1 j   , and   ( j  2)      ( j  3)    ( j  3)       2  2  2  2

 j   1  A

gain (11) is the same as:  j    2     

 (v ,,v r 0

1

k 1 ) P0( r ( k 1) j ),1( j 2 r ),2( r )

Av1  Avk 1 , (with a leading A0 )

(20)

(2.5) is the same as:  j    2     

A1

 (v ,,v r 0

k 1 ) P0( r ( k 1) j ),1( j 12 r ),2( r )

1

Av1  Avk 

 j    2     

 (v ,,v r 0

k 1 ) P0( r ( k 1) j ),1( j 2 r ),2( r )

1

Av1  Avk 1 ,

(21)

with a leading A1 , since r  ( k  1)  j , j  1  2r and r are all nonnegative for

  1 r  0,1,,    2  (2.7)

can be rewritten in the form:

 j      1   2  

A2

 1    j     0,1,,   ( j  1)    .     2

 r 0

 A2

 j        2  

 r 1

Av  Av

Av  Av

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

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

1

1

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k

k

(22)

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The structure of determining matrices for a class of… If r = 0, then r – 1 = -1 < 0. Therefore, the summations with r = 0 vanish, with (2.12) transforming to:  j    2     

A2

 (v ,,v ) P  r 0

1

Av1  Avk

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

k

 j    2     

 (v ,,v r 1

1

k 1 ) P0( r 1( k 1) j ),1( j 2 r )),2( r )

Av1  Avk 1 ,

(23)

with leading A2 . Finally,

Qk 1 ( jh) = (20) + (21) + (23), the same expression in each summation, but with

leading A0 , A1 and A2 respectively. Consequently,

Qk 1 ( jh) 

 j   2   

 (v ,,v r 0

1

k 1 ) P0( r ( k 1) j ),1( j 2 r ),2( r )

Av1  Avk 1 ,

completing the proof of the theorem for j odd. Hence the theorem has been proved for both cases; therefore, the validity of the theorem is established.

3.2

Theorem on Qk ( jh); j  k  1

For j  k  1, j , k integers,

   2 k2 j        Av1  Avk , 1  j  2k  Qk ( jh)    r 0 ( v1 ,,vk ) P0( r ),1(2k  j 2r ),2(r  j k )   0, j  2k  1 Proof

Qk ( jh) , for j  k  1.

Consider

 A1 , if j  1  For k  1, we appeal to lemma 1.4 to obtain Qk ( jh)  Q1 ( jh)   A2 , if j  2  0, if j  3  Hence, Q1 ( jh)  A j sgn(max{0, 3  j}), j  1. If j = 1, then If

j

2k  j 2 =

 2,

1 2

; so r = 0 and the rhs summation  A1. then

2k  j 2

0;

so

r

=

0,

and

the

rhs

summation

2k  j

1   ;so r is infeasible  the rhs summation  0, for j  3. 2 2 Therefore, in the stated formula, Q1 ( jh)  A j sgn(max{0, 3  j}), in agreement with lemma 2.5. Therefore

 A 2 . If j  3, then

the theorem is valid for k  1, j  k .

Assume that the theorem is valid for 1  k  n  j , for some integer n. Then, for j  n  1,

Qn 1 ( jh)  A0Qn ( jh)  A1Qn ([ j  1]h)  A2Qn ([ j  2]h). www.ijmsi.org

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The structure of determining matrices for a class of… We may apply the induction hypothesis to

Qn ( jh) 

Qn ( jh) to get

 2n j   2   

 r 0

( v1 ,,vn ) P0( r ),1(2 n j 2 r ),2(r  j n )

Av1  Avn , since j  n.

Now, j  n  1  j  1  n, or n  j  1. So, we may apply the induction principle to Qn ([ j  1]h ) to get Qn ([ j  1]h) 

  2 n [ j 1]     2  

( v1 ,,vn ) P0( r ),1(2 n[ j 1]2 r ),2( r [ j 1]n )

r 0

Av1  Avn ,

where all permutations are feasible. If j  2  n, apply the induction hypothesis to Qn ([ j  2]h) , to get

Qn ([ j  2]h)  Hence,

 A0

  2 n [ j  2]     2  

 r 0

( v1 ,,vn ) P0( r ),1(2 n[ j 2]2 r ),2( r [ j 2]n )

Av1  Avn .

Qn 1 ( jh)  2 n j    2    

 r 0

 A2

( v1 ,,vn ) P0( r ),1(2 n j 2 r ),2( r  j n )

  2 n [ j  2]     2  

Av1  Avn  A1

  2 n [ j 1]     2  

( v1 ,,vn ) P0( r ),1(2 n[ j  2] 2r ),2(r [ j  2]n )

r 0

Case:

j

even.

 r 0

Av1  Avn

Av1  Avn 2n  j

Then

 1 j    2 (2n  j )    n  2 ; 2n  ( j  2) is even,so   2n  [ j  1] is odd. So

( v1 ,,vn ) P0( r ),1(2 n[ j 1]2 r ),2( r [ j 1]n )

is

even.

So

 1 j    2 (2n  [ j  2])    n  2  n  1  j  

 1 j    1    2 (2n  [ j  1])      2 (2n  [ j  1]  1)    n  2 .       1 j  2(n  1)  j is even; so   (2[n  1]  j )    n  1  . Hence: 2   2 n

j 2

Qn1 ( jh)  A0 

r 0 ( v1 ,,vn ) P0( r ),1(2 n j 2 r ),2(r  j n )

n

 A1

Av1  Avn

j 2

  r 0 ( v ,,v ) P 1

n

Av1  Avn

(24)

(25)

0( r ),1(2 n1 j 2 r ),2( r  j 1n )

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The structure of determining matrices for a class of…

n 1

 r 0

 A2 Use

j 2

(v1,,vn ) P0( r ),1(2[ n1] j2r ),2(r  j[n1]1)

the 1 n 

A0

change

of

j 2

1 n 

 A0

1

j 2

  r 0 ( v ,,v ) P 1

n

j 2

  r 1 ( v ,,v ) P

Av1  Avn  A0

0( r 1),1(2 n j 2[ r 1]),2( r 1  j n )

n

(26) r  r  1, in (2.14) to get

variables 1 n 

  r 1 ( v ,,v ) P 1

Av1  Avn

n

Av1  Avn

0( r 1),1(2[ n1] j 2 r ),2( r  j [ n1])

Av1  Avn ,

0( r 1),1(2[ n1] j 2 r ),2( r  j [ n1])

(since the summation with r  0 is infeasible and hence equals 0).

  2( n 1)  j     2    

 r 0

( v1 ,,vn1 ) P0( r ),1(2[ n1] j 2 r ),2( r  j [ n1])

Av1  Avn1 , with a leading A0 .

(27)

j If we set r  n  1  ,in (25), then 2n  1  j  2r  2n  1  j  2n  2  j  1; so the 2 j summations with r  n  1  vanish, being infeasible.Therefore (2.15) is the same expression as: 2 n 1

A1

j 2

 (v ,,v ) P  r 0

1

n

  2( n 1)  j     2  

0( r ),1(2 n1 j 2 r ),2( r  j [ n1])

Av1  Avn

( v1 ,,vn1 ) P0( r ),1(2 n1 j2 r ),2( r  j [ n1])

r 0

with a leading

Av1  Avn1 ,

(28)

A1 .

Clearly (2.16) is the same expression as:   2( n 1)  j       2   

 r 0

with a leading

(v1,,vn1) P0( r ),1(2 n1 j2 r ),2( r  j[ n1])

Av1  Avn1 ,

(29)

A2 .

Add up (27), (28) and (29) to obtain:

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The structure of determining matrices for a class of…   2( n 1)  j       2   

 r 0

Qn1( jh) 

(v1,,vn1) P0( r ),1(2 n1 j2 r ),2( r  j[ n1])

Av1  Avn1 .

Hence, the theorem is valid for all j  n  1; this completes the proof for the case j even. Now consider the case: j odd. Then 2n – is odd.

Therefore,

 1 1   1    2 (2n  j )      2 (2n  j  1)    n  2 ( j  1),      1 1    1  2n  ( j  2) is odd;so,   (2n  ( j  2)     (2n  ( j  2)  1)   n  1 ( j  1) 2    2   2  1 1    1    2 (2n  j )      2 (2n  j  1)    n  2 ( j  1). Clearly, 2n  ( j  1) is even;       1 1 1  1 so,   (2n  ( j  1)    (2n  ( j  1)  (2[n  1]  1  j )  n  1  ( j  1) 2 2  2  2

 1 1    1  2(n  1)  j is odd; so,   (2[n  1]  j )     (2[n  1]  j  1)   n  1 ( j  1). 2    2   2 Hence:

Qn 1 ( jh) n

 A0

( j 1) 2

 r 0

( v1 ,,vn ) P0( r ),1(2 n j 2 r ),2( r  j n ) n 1

 A1

 r 0

n 1

 A2

Note that n  1 

( j 1) 2

( j 1) 2

 r 0

Av1  Avn

( v1 ,,vn ) P0( r ),1(2 n1 j 2 r ),2( r  j 1n )

(30)

Av1  Avn

( v1 ,,vn ) P0( r ),1( 2[ n1] j 2 r ),2( r  j[ n1]1)

Av1  Avn

(31)

(32)

   2(n  1)  j    1 ( j  1)      , as earlier established .Therefore using the 2 2    

change of variables

r  r 1,in (30),we see that (30) is exactly the same expression as

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The structure of determining matrices for a class of… n 1 1 ( j 1) 2

 r 0

A0

( v1 ,,vn ) P0( r 1),1(2[ n1] j 2 r ),2( r  j [ n1])

  2( n 1)  j       2   

 r 0

( v1 ,,vn1 ) P0( r ),1(2[ n1] j 2 r ),2( r  j [ n1])

Av1  Avn

Av1  Avn1 , with a leading A0 .

(33)

(31) is exactly the same expression as:   2( n 1)  j     2    

 r 0

( v1 ,,vn1 ) P0( r ),1(2[ n1] j 2 r ),2( r  j [ n1])

Av1  Avn1 , with a leading A1.

(34)

(32) is exactly the same expression as:   2( n 1)  j     2    

 r 0

( v1 ,,vn1 ) P0( r ),1(2[ n1] j 2 r ),2( r  j [ n1])

Av1  Avn1 , with a leading A2 .

(35)

Add up (33), (34) and (35) to obtain:

Qn1 ( jh) 

  2( n 1)  j     2    

 r 0

( v1 ,,vn1 ) P0( r ),1(2[ n1] j 2 r ),2( r  j [ n1])

Av1  Avn1 ,

(36)

proving the theorem for j odd, for the contingency j  2  n. Last case: – 2 < n . Then j  n  2; but j  n  1,forcing j  n  1. We invoke theorem 3.1 to conclude that

Qn1 ([n  1]h) 

  n 1)       2    

Now set j  n  1 , in the expression for

Qn1 ([n  1]h) 

  n 1)       2    

 r 0

 r 0

( v1 ,,vn1 ) P0( r n1( n1)),1( n12 r ),2( r )

Av1  Avn1 .

Qn 1 ( jh) , in theorem 3.2, to get

( v1 ,,vn1 ) P0( r )),1( n12 r ),2( r )

Av1  Avn1 ,

exactly the same expression as in theorem 3.1. This completes the proof of theorem 3.2. Remarks The expressions for

Qk ( jh) in theorems 3.1 and 3.2 coincide when j  k  0, as should be expected. IV.

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

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The structure of determining matrices for a class of… unfolding of intricate inter–play of the greatest integer function and the permutation objects in the course of deriving the expressions for the determining matrices. By using the greatest integer function analysis, change of variables technique and deft application of mathematical induction principles we were able to obtain the structure of the determining matrices for the double–delay control model, without which the computational investigation of Euclidean controllability would be impossible. The mathematical icing on the cake was our deft application of the max and sgn functions and their composite function sgn (max {.,.}) in the expressions for determining matrices. Such applications are optimal, in the sense that they obviate the need for explicit piece–wise representations of those and many other discrete mathematical objects and some others in the continuum.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

Gabasov, R. and Kirillova, F. The qualitative theory of optimal processes ( Marcel Dekker Inc., New York, 1976). 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. Ukwu, C. 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, 2013a (In progress). Chidume, C. An introduction to metric spaces. The Abdus Salam, International Centre for Theoretical Physics, Trieste, Italy, (2003). Chidume, C. Applicable functional analysis (The Abdus Salam, International Centre for Theoretical Physics, Trieste, Italy, 2007). Royden, H.L. Real analysis (3rd Ed. Macmillan Publishing Co., New York, 1988). Chukwu, E. N. Stability and time-optimal control of hereditary systems (Academic Press, New York, 1992). Hale, J. K. Theory of functional differential equations. Applied Mathematical Science, Vol. 3, 1977, Springer-Verlag, New York. 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. Wikipedia, Analytic function. Retrieved September 11, 2010 from http://en.wikipedia.org/wiki/Analytic_function. Wikipedia, Bounded variation, Retrieved December 30, 2012 from http://en.wikipedia.org/wiki/Bounded_variation.

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