Skip to main content

Advertisement

Discrete matrix delayed exponential for two delays and its property

Article metrics

  • 1523 Accesses

  • 8 Citations

Abstract

In recent papers, a discrete matrix delayed exponential for a single delay was defined and its main property connected with the solution of linear discrete systems with a single delay was proved. In the present paper, a generalization of the concept of discrete matrix delayed exponential is designed for two delays and its main property is proved as well.

Introduction

Throughout the paper, we use the following notation. For integers s, t, st, we define a set Z s t :={s,s+1,,t1,t}. Similarly, we define sets Z t :={,t1,t} and Z s :={s,s+1,}. The function used below is the floor integer function. We employ the following property of the floor integer function:

x1<xx,
(1)

where xR.

Define binomial coefficients as customary, i.e., for nZ and kZ,

( n k ) :={ n ! k ! ( n k ) ! if  n k 0 , 0 otherwise .
(2)

We recall that for a well-defined discrete function f(k), the forward difference operator Δ is defined as Δf(k)=f(k+1)f(k). In the paper, we also adopt the customary notation i = i 1 i 2 g i =0 if i 2 < i 1 . In the case of double sums, we set

i = i 1 , j = j 1 i 2 , j 2 g i j =0
(3)

if at least one of the inequalities i 2 < i 1 , j 2 < j 1 holds.

In [1, 2], a discrete matrix delayed exponential for a single delay mN was defined as follows.

Definition 1 For an r×r constant matrix B, kZ, and fixed mN, we define the discrete matrix delayed exponential e m B k as follows:

e m B k :={ Θ if  k Z m 1 , I + j = 1 B j ( k m ( j 1 ) j ) if  = 0 , 1 , 2 , , k Z ( 1 ) ( m + 1 ) + 1 ( m + 1 ) ,

where Θ is an r×r null matrix and I is an r×r unit matrix.

Next, the main property (Theorem 1 below) of discrete matrix delayed exponential for a single delay mN is proved in [1].

Theorem 1 Let B be a constant r×r matrix. Then, for k Z m ,

Δ e m B k =B e m B ( k m ) .
(4)

The paper is concerned with a generalization of the notion of discrete matrix delayed exponential for two delays and a proof of one of its properties, similar to the main property (4) of discrete matrix delayed exponential for a single delay.

Discrete matrix delayed exponential for two delays and its main property

We define a discrete r×r matrix function e m n B C k called the discrete matrix delayed exponential for two delays m,nN, mn and for two r×r commuting constant matrices B, C as follows.

Definition 2 Let B, C be constant r×r matrices with the property BC=CB and let m,nN, mn be fixed integers. We define a discrete r×r matrix function e m n B C k called the discrete matrix delayed exponential for two delays m, n and for two r×r constant matrices B, C:

e m n B C k :={ Θ if  k Z max { m , n } 1 , I if  k Z max { m , n } 0 , I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j + 1 ) if  k Z 1 ,

where

p ( k ) := k + m m + 1 , q ( k ) := k + n n + 1 .
(5)

The main property of e m n B C k is given by the following theorem.

Theorem 2 Let B, C be constant r×r matrices with the property BC=CB and let m,nN, mn be fixed integers. Then

Δ e m n B C k =B e m n B C ( k m ) +C e m n B C ( k n )
(6)

holds for k0.

Proof Let k1. From (1) and (5), we can see easily that, for an integer k0 satisfying

( p ( k ) 1)(m+1)+1k p ( k ) (m+1)( q ( k ) 1)(n+1)+1k q ( k ) (n+1),

the relation

Δ e m n B C k =Δ [ I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j + 1 ) ]

holds in accordance with Definition 2 of e m n B C k . Since ΔI=Θ, we have

Δ e m n B C k =Δ [ ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j + 1 ) ] .
(7)

Considering the increment by its definition, i.e.,

Δ e m n B C k = e m n B C ( k + 1 ) e m n B C k ,
(8)

we conclude that it is reasonable to divide the proof into four parts with respect to the value of integer k. In case one, k is such that

( p ( k ) 1)(m+1)+1k< p ( k ) (m+1)( q ( k ) 1)(n+1)+1k< q ( k ) (n+1),

in case two

k= p ( k ) (m+1)( q ( k ) 1)(n+1)+1k< q ( k ) (n+1),

in case three

( p ( k ) 1)(m+1)+1k< p ( k ) (m+1)k= q ( k ) (n+1)

and in case four

k= p ( k ) (m+1)k= q ( k ) (n+1).

We see that the above cases cover all the possible relations between k, p ( k ) and q ( k ) .

In the proof, we use the identities

( n + 1 k ) = ( n k ) + ( n k 1 ) ,
(9)

where n,kN and

( i i ) = ( i 1 i 1 ) , ( j 0 ) = ( j 1 0 ) , ( i + j i ) = ( i + j 1 i 1 ) + ( i + j 1 i ) ,
(10)

where i,jN, which are derived from (2) and (9).

I. ( p ( k ) 1)(m+1)+1k< p ( k ) (m+1)( q ( k ) 1)(n+1)+1k< q ( k ) (n+1)

From (1) and (5), we get

p ( k m ) = k m + m m + 1 k m + 1 < p ( k ) , p ( k m ) = k m + m m + 1 > k m + 1 1 = k m 1 m + 1 > p ( k ) 2 .

Therefore, p ( k m ) = p ( k ) 1 and, by Definition 2,

e m n B C ( k m ) =I+(B+C) i = 0 , j = 0 p ( k ) 2 , q ( k m ) 1 B i C j ( i + j i ) ( k m m i n j i + j + 1 ) .
(11)

Similarly, omitting details, we get (using (1), and (5)) q ( k n ) = q ( k ) 1 and

e m n B C ( k n ) =I+(B+C) i = 0 , j = 0 p ( k n ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k n m i n j i + j + 1 ) .
(12)

Let q ( k m ) 1. We show that

( k m m i n j i + j + 1 ) =0if i0,j q ( k m ) .
(13)

In accordance with (1),

q ( k m ) = k m + n n + 1 > k m + n n + 1 1= k m 1 n + 1

or

km<(n+1) q ( k m ) +1(m+1)i+(n+1)j+1if i0,j q ( k m ) .

From the last inequality, we get

kmminj<i+j+1if i0,j q ( k m )

and (13) holds by (2). For that reason and since q ( k m ) q ( k ) , we can replace q ( k m ) by q ( k ) in (11). Thus, we have

e m n B C ( k m ) =I+(B+C) i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) .
(14)

It is easy to see that, due to (3), formula (14) can be used instead of (11) if q ( k m ) <1 also.

Let p ( k n ) 1. Similarly, we can show that

( k n m i n j i + j + 1 ) =0if i p ( k n ) ,j0

and, since p ( k n ) p ( k ) , we can replace p ( k n ) by p ( k ) in (12). Thus, we have

e m n B C ( k n ) =I+(B+C) i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) .
(15)

It is easy to see that, due to (3), formula (15) can be used instead of (12) if p ( k n ) <1, too.

Due to (1), we also conclude that

p ( k + 1 ) = p ( k ) , q ( k + 1 ) = q ( k )
(16)

because

p ( k + 1 ) = k + 1 + m m + 1 k m + 1 +1< p ( k ) +1

and

p ( k + 1 ) = k + 1 + m m + 1 > k + 1 + m m + 1 1= k m + 1 p ( k ) 1+ 1 m + 1 .

The second formula can be proved similarly.

Now we are able to prove that

Δ e m n B C k = B e m n B C ( k m ) + C e m n B C ( k n ) = B [ I + ( B + C ) i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) ] + C [ I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) ] .
(17)

With the aid of (7), (8), (9) and (16), we get

Δ e m n B C k = e m n B C ( k + 1 ) e m n B C k = I + ( B + C ) i = 0 , j = 0 p ( k + 1 ) 1 , q ( k + 1 ) 1 B i C j ( i + j i ) ( k + 1 m i n j i + j + 1 ) I ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j + 1 ) = I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k + 1 m i n j i + j + 1 ) I ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j + 1 ) = ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) [ ( k + 1 m i n j i + j + 1 ) ( k m i n j i + j + 1 ) ] = ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j ) = ( B + C ) [ I + i = 1 p ( k ) 1 B i C 0 ( i i ) ( k m i i ) + j = 1 q ( k ) 1 B 0 C j ( j 0 ) ( k n j j ) + i = 1 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j ) ] .

By (10), we have

Δ e m n B C k = ( B + C ) [ I + i = 1 p ( k ) 1 B i C 0 ( i 1 i 1 ) ( k m i i ) + j = 1 q ( k ) 1 B 0 C j ( j 1 0 ) ( k n j j ) + i = 1 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i 1 ) ( k m i n j i + j ) + i = 1 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i ) ( k m i n j i + j ) ] = ( B + C ) [ I + i = 1 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i 1 ) ( k m i n j i + j ) + i = 0 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i ) ( k m i n j i + j ) ] .

Now in the first sum we replace the summation index i by i+1 and in the second sum we replace the summation index j by j+1. Then

Δ e m n B C k = ( B + C ) [ I + i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i + 1 C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) + i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j + 1 ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) ] = B + B ( B + C ) i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) + C + C ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) = B [ I + ( B + C ) i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) ] + C [ I + C ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) ] = B e m n B C ( k m ) + C e m n B C ( k n ) .

Due to (14) and (15), we conclude that formula (17) is valid.

II. k= p ( k ) (m+1)( q ( k ) 1)(n+1)+1k< q ( k ) (n+1)

In this case,

p ( k m ) = k m + m m + 1 = k m + 1 = p ( k ) , p ( k + 1 ) = k + 1 + m m + 1 k + 1 + m m + 1 = k m + 1 + 1 = p ( k ) + 1 , p ( k + 1 ) = k + 1 + m m + 1 > k + 1 + m m + 1 1 = k m + 1 = p ( k )

and p ( k + 1 ) = p ( k ) +1. In addition to this (see relevant computations performed in case I), we have q ( k n ) = q ( k ) 1 and q ( k + 1 ) = q ( k ) .

Then

e m n B C k = I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j + 1 ) , e m n B C ( k + 1 ) = I + ( B + C ) i = 0 , j = 0 p ( k ) , q ( k ) 1 B i C j ( i + j i ) ( k + 1 m i n j i + j + 1 )

and

e m n B C ( k m ) =I+(B+C) i = 0 , j = 0 p ( k ) 1 , q ( k m ) 1 B i C j ( i + j i ) ( k m m i n j i + j + 1 ) ,
(18)
e m n B C ( k n ) =I+(B+C) i = 0 , j = 0 p ( k n ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k n m i n j i + j + 1 ) .
(19)

Like with the computations performed in the previous part of the proof, we get

( k m m i n j i + j + 1 ) =0if i0,j q ( k m )

and

( k n m i n j i + j + 1 ) =0if i p ( k n ) ,j0.

So, we can substitute q ( k m ) by q ( k ) in (18) and p ( k n ) by p ( k ) in (19).

Accordingly, we have

e m n B C ( k m ) =I+(B+C) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) ,
(20)
e m n B C ( k n ) =I+(B+C) i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) .
(21)

It is easy to see that, due to (3), formula (20) can also be used instead of (18) if q ( k m ) <1 and formula (21) can also be used instead of (19) if p ( k n ) <1.

We have to prove

Δ e m n B C k = B e m n B C ( k m ) + C e m n B C ( k n ) = B [ I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) ] + C [ I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) ] .
(22)

Therefore,

Δ e m n B C k = e m n B C ( k + 1 ) e m n B C k = I + ( B + C ) i = 0 , j = 0 p ( k ) , q ( k ) 1 B i C j ( i + j i ) ( k + 1 m i n j i + j + 1 ) I ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j + 1 ) = ( B + C ) [ i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) [ ( k + 1 m i n j i + j + 1 ) ( k m i n j i + j + 1 ) ] + j = 0 q ( k ) 1 B p ( k ) C j ( p ( k ) + j p ( k ) ) ( k + 1 m p ( k ) n j p ( k ) + j + 1 ) ] .

With the aid of the equation k= p ( k ) (m+1), we get

( k + 1 m p ( k ) n j p ( k ) + j + 1 ) = ( p ( k ) + 1 n j p ( k ) + 1 + j ) =0if j>0

and, by (9), we have

Δ e m n B C k = ( B + C ) [ i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j ) + B p ( k ) ] = ( B + C ) [ I + i = 1 p ( k ) 1 B i C 0 ( i i ) ( k m i i ) + j = 1 q ( k ) 1 B 0 C j ( j 0 ) ( k n j j ) + i = 1 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j ) + B p ( k ) ] .

By (10), we have

Δ e m n B C k = ( B + C ) [ I + i = 1 p ( k ) 1 B i C 0 ( i 1 i 1 ) ( k m i i ) + j = 1 q ( k ) 1 B 0 C j ( j 1 0 ) ( k n j j ) + i = 1 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i 1 ) ( k m i n j i + j ) + i = 1 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i ) ( k m i n j i + j ) + B p ( k ) ] = ( B + C ) [ I + i = 1 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i 1 ) ( k m i n j i + j ) + i = 0 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i ) ( k m i n j i + j ) + B p ( k ) ] .

Now we replace in the first sum the summation index i by i+1 and in the second sum we replace the summation index j by j+1. Then

Δ e m n B C k = ( B + C ) [ I + i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i + 1 C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) + i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j + 1 ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) + B p ( k ) ] = B + B ( B + C ) i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) + B p ( k ) ( B + C ) + C + C ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) = B [ I + ( B + C ) ( i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) + B p ( k ) 1 ) ] + C [ I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) ] .

For k= p ( k ) (m+1), we have

B p ( k ) 1 = j = 0 q ( k ) 1 B p ( k ) 1 C j ( p ( k ) 1 + j p ( k ) 1 ) ( k m ( p ( k ) 1 + 1 ) n j p ( k ) 1 + j + 1 ) ,

where

( k m ( p ( k ) 1 + 1 ) n j p ( k ) 1 + j + 1 ) = ( k m p ( k ) n j p ( k ) + j ) =0if j>0.

Thus,

Δ e m n B C k = B [ I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) ] + C [ I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) ] = B e m n B C ( k m ) + C e m n B C ( k n )

and formula (22) is proved.

III. ( p ( k ) 1)(m+1)+1k< p ( k ) (m+1)k= q ( k ) (n+1)

In this case, we have (see relevant computations in cases I and II)

p ( k m ) = p ( k ) 1, p ( k + 1 ) = p ( k )

and

q ( k n ) = q ( k ) , q ( k + 1 ) = q ( k ) +1.

Then

e m n B C k = I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j + 1 ) , e m n B C ( k + 1 ) = I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) B i C j ( i + j i ) ( k + 1 m i n j i + j + 1 )

and

e m n B C ( k m ) =I+(B+C) i = 0 , j = 0 p ( k ) 2 , q ( k m ) 1 B i C j ( i + j i ) ( k m m i n j i + j + 1 ) ,
(23)
e m n B C ( k n ) =I+(B+C) i = 0 , j = 0 p ( k n ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k n m i n j i + j + 1 ) .
(24)

Like with the computations performed in case I, we can get

( k m m i n j i + j + 1 ) =0if i0,j q ( k m )

and

( k n m i n j i + j + 1 ) =0if i p ( k n ) ,j0.

So, we can substitute q ( k ) for q ( k m ) in (23) and p ( k ) for p ( k n ) in (24).

Thus, we have

e m n B C ( k m ) =I+(B+C) i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) ,
(25)
e m n B C ( k n ) =I+(B+C) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) .
(26)

It is easy to see that, due to (3), formula (25) can also be used instead of (23) if q ( k m ) <1 and formula (26) can also be used instead of (24) if p ( k n ) <1.

Now we have to prove

Δ e m n B C k = B e m n B C ( k m ) + C e m n B C ( k n ) = B [ I + ( B + C ) i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) ] + C [ I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) ] .
(27)

Considering the difference by its definition, we get

Δ e m n B C k = e m n B C ( k + 1 ) e m n B C k = I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) B i C j ( i + j i ) ( k + 1 m i n j i + j + 1 ) I ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j + 1 ) = ( B + C ) [ i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) [ ( k + 1 m i n j i + j + 1 ) ( k m i n j i + j + 1 ) ] + i = 0 p ( k ) 1 B i C q ( k ) ( i + q ( k ) i ) ( k + 1 m i n q ( k ) i + q ( k ) + 1 ) ] .

With the aid of relation k= q ( k ) (n+1), we get

( k + 1 m i n q ( k ) i + q ( k ) + 1 ) = ( q ( k ) + 1 m i q ( k ) + 1 + i ) =0if i>0

and

Δ e m n B C k = ( B + C ) [ i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j ) + C q ( k ) ] = ( B + C ) [ I + i = 1 p ( k ) 1 B i C 0 ( i i ) ( k m i i ) + j = 1 q ( k ) 1 B 0 C j ( j 0 ) ( k n j j ) + i = 1 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j ) + C q ( k ) ] .

By (10), we have

Δ e m n B C k = ( B + C ) [ I + i = 1 p ( k ) 1 B i C 0 ( i 1 i 1 ) ( k m i i ) + j = 1 q ( k ) 1 B 0 C j ( j 1 0 ) ( k n j j ) + i = 1 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i 1 ) ( k m i n j i + j ) + i = 1 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i ) ( k m i n j i + j ) + C q ( k ) ] = ( B + C ) [ I + i = 1 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i 1 ) ( k m i n j i + j ) + i = 0 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i ) ( k m i n j i + j ) + C q ( k ) ] .

Now we replace in the first sum the summation index i by i+1 and in the second sum we replace the summation index j by j+1. Then

Δ e m n B C k = ( B + C ) [ I + i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i + 1 C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) + i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j + 1 ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) + C q ( k ) ] = B + B ( B + C ) i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) + C + C ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) + C q ( k ) ( B + C ) = B [ I + ( B + C ) i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) ] + C [ I + ( B + C ) ( i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) + C q ( k ) 1 ) ] .

For k= q ( k ) (n+1), we have

C q ( k ) 1 = i = 0 p ( k ) 1 B i C q ( k ) 1 ( i + q ( k ) 1 i ) ( k m i n ( q ( k ) 1 + 1 ) i + q ( k ) 1 + 1 ) ,

where

( k m i n ( q ( k ) 1 + 1 ) i + q ( k ) 1 + 1 ) = ( k m i n q ( k ) i + q ( k ) ) =0if i>0.

Thus,

Δ e m n B C k = B [ I + ( B + C ) i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) ] + C [ I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) ] = B e m n B C ( k m ) + C e m n B C ( k n )

and formula (27) is proved.

IV. k= p ( k ) (m+1)k= q ( k ) (n+1)

In this case, we have (see similar combinations in cases II and III)

p ( k m ) = p ( k ) , p ( k + 1 ) = p ( k ) +1

and

q ( k n ) = q ( k ) , q ( k + 1 ) = q ( k ) +1.

Then

e m n B C k = I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j + 1 ) , e m n B C ( k + 1 ) = I + ( B + C ) i = 0 , j = 0 p ( k ) , q ( k ) B i C j ( i + j i ) ( k + 1 m i n j i + j + 1 )

and

e m n B C ( k m ) =I+(B+C) i = 0 , j = 0 p ( k ) 1 , q ( k m ) 1 B i C j ( i + j i ) ( k m m i n j i + j + 1 ) ,
(28)
e m n B C ( k n ) =I+(B+C) i = 0 , j = 0 p ( k n ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k n m i n j i + j + 1 ) .
(29)

As before,

( k m m i n j i + j + 1 ) =0if i0,j q ( k m )

and

( k n m i n j i + j + 1 ) =0if i p ( k n ) ,j0.

So, we can substitute q ( k ) for q ( k m ) in (28) and p ( k ) for p ( k n ) in (29) and

e m n B C ( k m ) =I+(B+C) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) ,
(30)
e m n B C ( k n ) =I+(B+C) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) .
(31)

It is easy to see that, due to (3), formula (30) can also be used instead of (28) if q ( k m ) <1 and formula (31) can also be used instead of (29) if p ( k n ) <1.

Now it is possible to prove the formula

Δ e m n B C k = B e m n B C ( k m ) + C e m n B C ( k n ) = B [ I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) ] + C [ I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) ] .
(32)

By definition, we get

Δ e m n B C k = e m n B C ( k + 1 ) e m n B C k = I + ( B + C ) i = 0 , j = 0 p ( k ) , q ( k ) B i C j ( i + j i ) ( k + 1 m i n j i + j + 1 ) I ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j + 1 ) = ( B + C ) [ i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) [ ( k + 1 m i n j i + j + 1 ) ( k m i n j i + j + 1 ) ] + j = 0 q ( k ) B p ( k ) C j ( p ( k ) + j p ( k ) ) ( k + 1 m p ( k ) n j p ( k ) + j + 1 ) + i = 0 p ( k ) B i C q ( k ) ( i + q ( k ) i ) ( k + 1 m i n q ( k ) i + q ( k ) + 1 ) ] .

With the aid of equations k= p ( k ) (m+1), k= q ( k ) (n+1), we get

( k + 1 m p ( k ) n j p ( k ) + j + 1 ) = ( p ( k ) + 1 n j p ( k ) + 1 + j ) = 0 if  j > 0 , ( k + 1 m i n q ( k ) i + q ( k ) + 1 ) = ( q ( k ) + 1 m i q ( k ) + 1 + i ) = 0 if  i > 0

and

Δ e m n B C k = ( B + C ) [ i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j ) + B p ( k ) + C q ( k ) ] = ( B + C ) [ I + i = 1 p ( k ) 1 B i C 0 ( i i ) ( k m i i ) + j = 1 q ( k ) 1 B 0 C j ( j 0 ) ( k n j j ) + i = 1 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n j i + j ) + B p ( k ) + C q ( k ) ] .

By (10), we have

Δ e m n B C k = ( B + C ) [ I + i = 1 p ( k ) 1 B i C 0 ( i 1 i 1 ) ( k m i i ) + j = 1 q ( k ) 1 B 0 C j ( j 1 0 ) ( k n j j ) + i = 1 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i 1 ) ( k m i n j i + j ) + i = 1 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i ) ( k m i n j i + j ) + B p ( k ) + C q ( k ) ] = ( B + C ) [ I + i = 1 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i 1 ) ( k m i n j i + j ) + i = 0 , j = 1 p ( k ) 1 , q ( k ) 1 B i C j ( i + j 1 i ) ( k m i n j i + j ) + B p ( k ) + C q ( k ) ] .

We replace in the first sum the summation index i by i+1 and in the second sum we substitute the summation index j by j+1. Then

Δ e m n B C k = ( B + C ) [ I + i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i + 1 C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) + i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j + 1 ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) + B p ( k ) + C q ( k ) ] = B + B ( B + C ) i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) + B p ( k ) ( B + C ) + C + C ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) + C q ( k ) ( B + C ) = B [ I + ( B + C ) ( i = 0 , j = 0 p ( k ) 2 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) + B p ( k ) 1 ) ] + C [ I + ( B + C ) ( i = 0 , j = 0 p ( k ) 1 , q ( k ) 2 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) + C q ( k ) 1 ) ] .

Because k= p ( k ) (m+1)= q ( k ) (n+1), we can express B p ( k ) 1 and C q ( k ) 1 in the form

B p ( k ) 1 = j = 0 q ( k ) 1 B p ( k ) 1 C j ( p ( k ) 1 + j p ( k ) 1 ) ( k m ( p ( k ) 1 + 1 ) n j p ( k ) 1 + j + 1 ) , C q ( k ) 1 = i = 0 p ( k ) 1 B i C q ( k ) 1 ( i + q ( k ) 1 i ) ( k m i n ( q ( k ) 1 + 1 ) i + q ( k ) 1 + 1 ) ,

where

( k m ( p ( k ) 1 + 1 ) n j p ( k ) 1 + j + 1 ) = ( k m p ( k ) n j p ( k ) + j ) = 0 if  j > 0 , ( k m i n ( q ( k ) 1 + 1 ) i + q ( k ) 1 + 1 ) = ( k m i n q ( k ) i + q ( k ) ) = 0 if  i > 0 .

Thus,

Δ e m n B C k = B [ I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m ( i + 1 ) n j i + j + 1 ) ] + C [ I + ( B + C ) i = 0 , j = 0 p ( k ) 1 , q ( k ) 1 B i C j ( i + j i ) ( k m i n ( j + 1 ) i + j + 1 ) ] = B e m n B C ( k m ) + C e m n B C ( k n ) .

Therefore, formula (32) is valid.

We proved that formula (6) holds in each of the considered cases I, II, III and IV for k1. If k=0, the proof can be done directly because p ( 0 ) = q ( 0 ) =0, p ( 1 ) = q ( 1 ) =1,

Δ e m n B C 0 = e m n B C 1 e m n B C 0 = I + ( B + C ) i = 0 , j = 0 0 , 0 B i C j ( i + j i ) ( 1 m i n j i + j + 1 ) I ( B + C ) i = 0 , j = 0 1 , 1 B i C j ( i + j i ) ( m i n j i + j + 1 ) = I + B + C I = B + C

and

B e m n B C ( m ) +C e m n B C ( n ) =BI+CI=B+C.

Formula (6) holds again. Theorem 2 is proved. □

Open problems and concluding remarks

Formula (4) is valid for k Z m . However, formula (6) holds for k Z 0 only. Therefore, there is a difference between the definition domains of the formulas, and it is a challenge how to modify Definition 2 of discrete matrix delayed exponential for two delays in such a way that formula (6) will hold for k Z max { m , n } . In [1] formula (4) is used to get a representation of the solution of the problems (both homogeneous and nonhomogeneous)

Δ y ( k ) = B y ( k m ) + f ( k ) , k Z 0 , y ( k ) = φ ( k ) , k Z m 0 ,

where f: Z 0 R r , y: Z m R r and φ: Z m 0 R r .

It is an open problem how to use formula (6) to get a representation of the solution of the homogeneous and nonhomogeneous problems

Δ y ( k ) = B y ( k m ) + C y ( k n ) + f ( k ) , k Z 0 , y ( k ) = φ ( k ) , k Z s 0 , s = max { m , n }

if BC=CB.

Let us note that the first concept of matrix delayed exponential was given in [3] and the first concept of discrete matrix delayed exponential was given in [1]. Further development of the delayed matrix exponentials method and its utilization to various problems can be found, e.g., in [416].

References

  1. 1.

    Diblík J, Khusainov DY: Representation of solutions of linear discrete systems with constant coefficients and pure delay. Adv. Differ. Equ. 2006., 2006: Article ID 80825

  2. 2.

    Diblík J, Khusainov DY:Representation of solutions of discrete delayed system x(k+1)=Ax(k)+Bx(km)+f(k) with commutative matrices. J. Math. Anal. Appl. 2006, 318(1):63-76. 10.1016/j.jmaa.2005.05.021

  3. 3.

    Khusainov DY, Shuklin GV: Linear autonomous time-delay system with permutation matrices solving. Stud. Univ. Žilina, Math. Ser. 2003, 17: 101-108.

  4. 4.

    Boichuk A, Diblík J, Khusainov D, Růžičková M: Boundary value problems for delay differential systems. Adv. Differ. Equ. 2010., 2010: Article ID 593834

  5. 5.

    Boichuk A, Diblík J, Khusainov D, Růžičková M: Fredholm’s boundary-value problems for differential systems with a single delay. Nonlinear Anal., Theory Methods Appl. 2010, 72: 2251-2258. 10.1016/j.na.2009.10.025

  6. 6.

    Boichuk A, Diblík J, Khusainov D, Růžičková M: Boundary-value problems for weakly nonlinear delay differential systems. Abstr. Appl. Anal. 2011., 2011: Article ID 631412

  7. 7.

    Diblík J, Fečkan M, Pospíšil M: Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices. Ukr. Mat. Zh. 2013, 65(1):58-69.

  8. 8.

    Diblík J, Khusainov D, Kukharenko O, Svoboda Z: Solution of the first boundary-value problem for a system of autonomous second-order linear PDEs of parabolic type with a single delay. Abstr. Appl. Anal. 2012., 2012: Article ID 219040

  9. 9.

    Diblík J, Khusainov DY, Lukáčová J, Růžičková M: Control of oscillating systems with a single delay. Adv. Differ. Equ. 2010., 2010: Article ID 108218

  10. 10.

    Diblík J, Khusainov DY, Růžičková M: Controllability of linear discrete systems with constant coefficients and pure delay. SIAM J. Control Optim. 2008, 47: 1140-1149. 10.1137/070689085

  11. 11.

    Khusainov DY, Diblík J, Růžičková M, Lukáčová J: Representation of a solution of the Cauchy problem for an oscillating system with pure delay. Nonlinear Oscil. 2008, 11(2):276-285. 10.1007/s11072-008-0030-8

  12. 12.

    Khusainov DY, Shuklin GV: Relative controllability in systems with pure delay. Int. Appl. Mech. 2005, 41: 210-221. 10.1007/s10778-005-0079-3

  13. 13.

    Medved’ M, Pospíšil M: Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts defined by pairwise permutable matrices. Nonlinear Anal., Theory Methods Appl. 2012, 75: 3348-3363. 10.1016/j.na.2011.12.031

  14. 14.

    Medved’ M, Pospíšil M, Škripková L: Stability and the nonexistence of blowing-up solutions of nonlinear delay systems with linear parts defined by permutable matrices. Nonlinear Anal., Theory Methods Appl. 2011, 74: 3903-3911. 10.1016/j.na.2011.02.026

  15. 15.

    Medved’ M, Škripková L: Sufficient conditions for the exponential stability of delay difference equations with linear parts defined by permutable matrices. Electron. J. Qual. Theory Differ. Equ. 2012., 2012: Article ID 22

  16. 16.

    Pospíšil M: Representation and stability of solutions of systems of functional differential equations with multiple delays. Electron. J. Qual. Theory Differ. Equ. 2012., 2012: Article ID 4

Download references

Acknowledgements

The first author was supported by Operational Programme Research and Development for Innovations, No. CZ.1.05/2.1.00/03.0097, as an activity of the regional Centre AdMaS.

Author information

Correspondence to Josef Diblík.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have made the same contribution. Both authors have read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Keywords

  • Main Property
  • Constant Matrix
  • Constant Matrice
  • Binomial Coefficient
  • Relevant Computation