Principles of Mathematics in Operations Research

14.4 Problems 201

Remark 14.4.6 In order to solve the linear difference system

y(k + l) = Ay(k) + f(k); y(0) = y 0 ,

we will take the Z transform of the components of y(k), then we have

(zI-A)rj(z) = zy 0 + (t>{z),

where n(z) = [771(2),-•• ,r]n(z)}T is the vector of Z transforms of the com-

ponents of y. If z is not an eigenvalue of A, then the coefficient matrix is

nonsingular. Thus, for sufficiently large \z\, the unique solution is

V(z) = z(zl - A)-

l

Vo + (zl - i4)" V(«),

where we have

Z{Ak}=z(zI-A)-^1.

In order to find a particular solution, we solve (zl — A)^1 p(z) = 4>(z) forp(z)

and find its inverse Z transform.

Example 14.4.7 Let us take our previous example problem:

yi(k + 1) = y 2 (k) + 1, 2/i(0) = 3,

y 2 (k + l) = yi(k) + l, 2/2(0) = 1.

V(z) =

z 1

(z-l)(z + l) [lz_

Z{Aky 0 }

z-\

2

2

1 zl

(*-l)(z + l) [if

z 1

-1

C^IFJ

n(z) =

z

(z-ir

z

+ (z-iy

=> y(k) = k^2  1

L 4 J

• k

' 1'

4

3

.4.

"0"

1

. 2 _

z+1

z

(z-1)

+ (-l)fc

(*+l)

19

8

13

Problems

14.1. Solve y"(t) - y(t) = e2t; y(0) = 2, j,'(0) = 0.

14.2. Solve y(k + 1) = y(k) + 2ek; y(0) = 1.