(Luận án tiến sĩ) stability and robust stability of singular linear difference equations , tính ổn định và ổn định vững của phương trình sai phân tuyến tính suy biến 624601

❱■❊❚◆❆▼ ◆❆❚■❖◆❆▲ ❯◆■❱❊❘❙■❚❨✱ ❍❆◆❖■

❱◆❯ ❯◆■❱❊❘❙■❚❨ ❖❋ ❙❈■❊◆❈❊

◆❣ô ❚❤✐✳ ❚❤❛♥❤ ◆❣❛

❙❚❆❇■▲■❚❨ ❆◆❉ ❘❖❇❯❙❚ ❙❚❆❇■▲■❚❨
❖❋ ❙■◆●❯▲❆❘ ▲■◆❊❆❘ ❉■❋❋❊❘❊◆❈❊ ❊◗❯❆❚■❖◆❙

❚❍❊❙■❙ ❋❖❘ ❚❍❊ ❉❊●❘❊❊ ❖❋

❉❖❈❚❖❘ ❖❋ P❍■▲❖❙❖P❍❨ ■◆ ▼❆❚❍❊▼❆❚■❈❙

❍❆◆❖■ ✕ ✷✵✶✽


❱■❊❚◆❆▼ ◆❆❚■❖◆❆▲ ❯◆■❱❊❘❙■❚❨✱ ❍❆◆❖■
❱◆❯ ❯◆■❱❊❘❙■❚❨ ❖❋ ❙❈■❊◆❈❊

◆❣ô ❚❤✐✳ ❚❤❛♥❤ ◆❣❛

❙❚❆❇■▲■❚❨ ❆◆❉ ❘❖❇❯❙❚ ❙❚❆❇■▲■❚❨
❖❋ ❙■◆●❯▲❆❘ ▲■◆❊❆❘ ❉■❋❋❊❘❊◆❈❊ ❊◗❯❆❚■❖◆❙
❙♣❡❝✐❛❧✐t②✿ ❉✐✛❡r❡♥t✐❛❧ ❛♥❞ ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s
❙♣❡❝✐❛❧✐t② ❈♦❞❡✿ ✻✷ ✹✻ ✵✶ ✵✸

❚❍❊❙■❙ ❋❖❘ ❚❍❊ ❉❊●❘❊❊ ❖❋

❉❖❈❚❖❘ ❖❋ P❍❨▲❖❙❖P❍❨ ■◆ ▼❆❚❍❊▼❆❚■❈❙

Supervisors: ASSOC. PROF. DR. HABIL. VŨ HOÀNG LINH

and PROF. DR. NGUYỄN HỮU DƯ

❍❆◆❖■ ✕ ✷✵✶✽


ĐẠI HỌC QUỐC GIA HÀ NỘI
TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN

Ngơ Thị Thanh Nga

TÍNH ỔN ĐỊNH VÀ ỔN ĐỊNH VỮNG CỦA
PHƯƠNG TRÌNH SAI PHÂN TUYẾN TÍNH SUY BIẾN

Chun ngành: Phương trình Vi phân và Tích phân
Mã số: 62 46 01 03

LUẬN ÁN TIẾN SĨ TOÁN HỌC

Người hướng dẫn khoa học:
PGS.TSKH. VŨ HOÀNG LINH
GS.TS. NGUYỄN HỮU DƯ

HÀ NỘI – 2018


❉❡❝❧❛r❛t✐♦♥
This work has been completed at the Faculty of Mathematics, Mechanics and
Informatics, University of Science, Vietnam National University, Hanoi, under
the supervision of Assoc.Prof.Dr.habil. Vu Hoang Linh and Prof.Dr. Nguyen
Huu Du. I hereby declare that the results presented in the thesis are new and

have never been published fully or partially in any other thesis/work.
Author: Ngô Thị Thanh Nga

✶


❆❝❦♥♦✇❧❡❞❣♠❡♥ts
Firstly, I would like to thank my two supervisors Prof.Dr. Nguyễn Hữu Dư
and especially Assoc.Prof.Dr.habil. Vũ Hoàng Linh for the continuous support
of my PhD study and related research; for their patience, motivation and immense knowledge. Without their help I could not have overcome the difficulties
in research and study.
I would like to express sincere thanks to Assoc.Prof.Dr. Lê Văn Hiện and Dr.
Nguyễn Trung Hiếu for their useful comments and suggestions that led to the
improvement of the thesis. I would also like to thank Dr. Đỗ Đức Thuận for his
collaboration in research. My deepest appreciation goes to Prof. Phạm Kỳ Anh
and other members of "Seminar on Computational and Applied Mathematics",
and also to the members of "Seminar on Differential Equations and Dynamical
Systems" at the Faculty of Mathematics, Mechanics and Informatics, VNU
University of Science, Hanoi, for their valuable comments and discussions.
I am grateful to my parents, brother, my beloved daughters, my husband and
other members in my big family, who have provided me moral and emotional
support throughout my life.
A very special gratitude goes to all Thang Long University, National Foundation for Science and Technology Development, the MOET project 911 for
providing the funding for me in the period of my study.
Last but not least, I would like to thank my colleagues in Thang Long
University, the staffs of Vietnam Institute for Advanced Study in Mathematics,
my friends, and many other people beside me for their love, motivation and
constant guidance.
Thanks all for your love and support!

❆❜str❛❝t
❚❤✐s ✇♦r❦ ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ ❧✐♥❡❛r s✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✭▲❙❉❊s✮ ♦❢
✜rst ♦r❞❡r ❛♥❞ s❡❝♦♥❞ ♦r❞❡r✳ ❋♦r ▲❙❉❊s ♦❢ ✜rst ♦r❞❡r✱ ❜② ✉s✐♥❣ t❤❡ ♣r♦❥❡❝t♦r✲
❜❛s❡❞ ❛♣♣r♦❛❝❤ ✇❡ ❝❤❛r❛❝t❡r✐③❡ t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ s②st❡♠ ✉♥❞❡r ♣❡rt✉r❜❛✲
t✐♦♥s ❛♥❞ ❡st❛❜❧✐s❤ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❜♦✉♥❞❡❞♥❡ss ♦❢ s♦❧✉t✐♦♥s ♦❢ ♥♦♥✲
❤♦♠♦❣❡♥❡♦✉s s②st❡♠s ❛♥❞ t❤❡ ❡①♣♦♥❡♥t✐❛❧✴ ✉♥✐❢♦r♠ st❛❜✐❧✐t② ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✲
✐♥❣ ❤♦♠♦❣❡♥❡♦✉s s②st❡♠s✳ ❲❡ ❛❧s♦ ❡①t❡♥❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ❇♦❤❧ ❡①♣♦♥❡♥t ❢r♦♠
r❡❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s t♦ ▲❙❉❊s ❛♥❞ ✐♥✈❡st✐❣❛t❡ ✐ts ♣r♦♣❡rt✐❡s✳
❋♦r ▲❙❉❊s ♦❢ s❡❝♦♥❞✲♦r❞❡r✱ ✇❡ ✉s❡ t❤❡ str❛♥❣❡♥❡ss✲✐♥❞❡① ❛♣♣r♦❛❝❤✳

❯♥✲

❞❡r t❤❡ str❛♥❣❡♥❡ss✲❢r❡❡ ❛ss✉♠♣t✐♦♥ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ s♦❧✈❛❜✐❧✐t② ♦❢ ■❱Ps✱ t❤❡
❝♦♥s✐st❡♥❝② ♦❢ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✱ ❛♥❞ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s♦❧✉t✐♦♥ s❡ts
♦❢ t❤❡ s②st❡♠s ❛♥❞ t❤♦s❡ ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ r❡❞✉❝❡❞ r❡❣✉❧❛r s②st❡♠s✳

❇② ❛

❝♦♠♣❛r✐s♦♥ ♣r✐♥❝✐♣❧❡✱ s♦♠❡ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❝r✐t❡r✐❛ ❛r❡ ♦❜t❛✐♥❡❞✳ ❆ ❇♦❤❧✲
P❡rr♦♥✲t②♣❡ t❤❡♦r❡♠ ✐s ❛❧s♦ ❣✐✈❡♥ t♦ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ✐♥♣✉t✲s♦❧✉t✐♦♥ r❡❧❛t✐♦♥
♦❢ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥s✳ ❋✐♥❛❧❧②✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ r♦❜✉st st❛❜✐❧✐t② ✉♥❞❡r
r❡str✐❝t❡❞ str✉❝t✉r❡❞ ♣❡rt✉r❜❛t✐♦♥s ✐s ✐♥✈❡st✐❣❛t❡❞✳ ❆❧s♦ ✉s✐♥❣ t❤❡ ❝♦♠♣❛r✐s♦♥
♣r✐♥❝✐♣❧❡✱ ❛♥ ❡①♣❧✐❝✐t ❜♦✉♥❞ ❢♦r ♣❡rt✉r❜❛t✐♦♥s ✉♥❞❡r ✇❤✐❝❤ t❤❡ s②st❡♠s ♣r❡s❡r✈❡
t❤❡✐r ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ✐s ♦❜t❛✐♥❡❞✳

✐


Tóm tắt

Trong cơng trình này chúng tơi nghiên cứu về phương trình sai phân suy
biến tuyến tính cấp một và cấp hai. Đối với phương trình sai phân suy biến
tuyến tính cấp 1, chúng tơi sử dụng cách tiếp cận bằng phép chiếu và đưa ra
được các kết quả như: đặc trưng hóa tính ổn định của hệ dưới tác động của
nhiễu; thiết lập mối quan hệ giữa tính ổn định mũ/ ổn định đều của hệ thuần
nhất và tính chất nghiệm của hệ không thuần nhất; mở rộng khái niệm số mũ
Bohl cho hệ sai phân suy biến và chỉ ra một số tính chất.
Đối với phương trình sai phân suy biến cấp hai, chúng tôi sử dụng cách tiếp
cận dùng chỉ số lạ. Dưới giả thiết chỉ số lạ bằng khơng, chúng tơi nghiên cứu
tính giải được của bài toán giá trị ban đầu và các điều kiện đầu tương thích,
mối quan hệ giữa tập nghiệm của hệ ban đầu và tập nghiệm của hệ được đưa
về dạng chính quy. Bằng cách sử dụng nguyên lý so sánh, tiêu chuẩn cho sự ổn
định mũ được thiết lập. Một định lý dạng Bohl-Perron được đưa ra nhằm đặc
trưng mối quan hệ đầu vào-nghiệm của hệ không thuần nhất. Cuối cùng, bài
tốn về tính ổn định vững dưới tác động của nhiễu có cấu trúc được chỉ ra.
Tiếp tục sử dụng nguyên lý so sánh một lần nữa, chúng tôi đưa ra được một
chặn trên cho nhiễu để hệ bị nhiễu vẫn bảo tồn được chỉ số cũng như tính ổn
định mũ.

✐✐


▲✐st ♦❢ ◆♦t❛t✐♦♥s
t❤❡ s❡t ♦❢ r❡❛❧ ✭❝♦♠♣❧❡①✮ ♥✉♠❜❡rs

(C)

R
N

t❤❡ s❡t ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs

N(n0 )

t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs t❤❛t ❛r❡ ❣r❡❛t❡r t❤❛♥ ♦r ❡q✉❛❧ t♦ ❛ ❣✐✈❡♥ ✐♥t❡❣❡r

K

R

C
R

♦r

C

d

t❤❡

d−

❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣❧❡① ✈❡❝t♦r s♣❛❝❡

d

t❤❡

d−

❞✐♠❡♥s✐♦♥❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡

Cd,d

t❤❡ s❡t ♦❢

d×d

♠❛tr✐❝❡s ✇✐t❤ ❡♥tr✐❡s ✐♥

GL(K )

t❤❡ s❡t ♦❢

d×d

✐♥✈❡rt✐❜❧❡ ♠❛tr✐❝❡s ✇✐t❤ ❡♥tr✐❡s ✐♥

❦❡rE

t❤❡ ❦❡r♥❡❧ s♣❛❝❡ ♦❢

E

✐♠E

t❤❡ ✐♠❛❣❡ s♣❛❝❡ ♦❢

E

rank E

t❤❡ r❛♥❦ ♦❢ ♠❛tr✐①

E

d

x

♥♦r♠ ♦❢ ✈❡❝t♦r

∆

♥♦r♠ ♦❢ ♠❛tr✐①

C
K

x
∆

B(0, 1)

✉♥✐t ❞✐s❦ ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡

det A

t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ ♠❛tr✐①

A

t❤❡ ♠❛tr✐① t✉♣❧❡

rK

t❤❡ ✭str✉❝t✉r❡❞✮ st❛❜✐❧✐t② r❛❞✐✉s

H

n0

A

(A, B, C)

A

t❤❡ ❝♦♥❥✉❣❛t❡ tr❛♥s♣♦s❡ ♦❢ ♠❛tr✐①

lp (n0 )

t❤❡ s♣❛❝❡ ♦❢ s❡q✉❡♥❝❡s

{qn }n

n0

A

⊂ Kd

s✉❝❤ t❤❛t

qn

p

< ∞✱ p

1

n n0

s♣❡❝tr❛❧ r❛❞✐✉s ♦❢ ♠❛tr✐①

ρ(A)
❞✐❛❣(σ1 , · · ·

, σp )

t❤❡ ♠❛tr✐① ✐♥

C

m,n

A

✇❤♦s❡

u>0

❡❛❝❤ ❝♦♠♣♦♥❡♥t ♦❢ ✈❡❝t♦r

u>v

♠❡❛♥s t❤❛t

ii
u

u−v >0

✐✐✐

❡♥tr② ✐s

σi

✐s ♣♦s✐t✐✈❡

❢♦r ❛♥②

i = 1, ..., p

❛♥❞ t❤❡ ♦t❤❡rs ❛r❡ ③❡r♦

❈♦♥t❡♥ts
P❛❣❡
❆❜str❛❝t

✐

Tóm tắt

ii

▲✐st ♦❢ ◆♦t❛t✐♦♥s

✐✐✐

■♥tr♦❞✉❝t✐♦♥

✶

❈❤❛♣t❡r ✶ Pr❡❧✐♠✐♥❛r✐❡s
✶✳✶

✶✳✷

▲✐♥❡❛r s✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❜② tr❛❝t❛❜✐❧✐t②✲✐♥❞❡① ❛♣♣r♦❛❝❤ ✶✶
✶✳✶✳✶✳

❉❡✜♥✐t✐♦♥ ♦❢ ✐♥❞❡①✲✶ s②st❡♠s ❛♥❞ t❤❡✐r ♣r♦♣❡rt✐❡s

✳ ✳ ✳ ✳

✶✶

✶✳✶✳✷✳

❙♦❧✉t✐♦♥s ♦❢ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

▲✐♥❡❛r s✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❜② str❛♥❣❡♥❡ss✲✐♥❞❡① ❛♣♣r♦❛❝❤ ✶✺
✶✳✷✳✶✳

❉❡✜♥✐t✐♦♥ ♦❢ str❛♥❣❡♥❡ss ✐♥❞❡① ❛♥❞ ❇rü❧❧✬s r❡s✉❧ts ✳ ✳ ✳ ✳

✶✺

✶✳✷✳✷✳

❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ t②♣❡s ♦❢ ✐♥❞❡① ❞❡✜♥✐t✐♦♥s

✳

✷✶

✶✳✷✳✸✳

▲✐♥❡❛r t✐♠❡✲✐♥✈❛r✐❛♥t s✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ♦❢ s❡❝✲
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✷

♦♥❞ ♦r❞❡r
✶✳✸

✶✶

❋✉rt❤❡r ❛✉①✐❧✐❛r② r❡s✉❧ts

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✐✈

✷✺


❈❤❛♣t❡r ✷ ❙✐♥❣✉❧❛r s②st❡♠s ♦❢ ✜rst✲♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s

✷✽

✷✳✶

❙t❛❜✐❧✐t② ♥♦t✐♦♥s ❢♦r s✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

✷✳✷

❙t❛❜✐❧✐t② ♦❢ ♣❡rt✉r❜❡❞ ❡q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✷

✷✳✷✳✶✳

❚❤❡ ❝❛s❡ ♦❢ ♦♥❡✲s✐❞❡❞ ♣❡rt✉r❜❛t✐♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✸

✷✳✷✳✷✳

❚❤❡ ❝❛s❡ ♦❢ t✇♦✲s✐❞❡❞ ♣❡rt✉r❜❛t✐♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✶

✷✳✸

✷✳✹

✷✳✺

❇♦❤❧✲P❡rr♦♥✲t②♣❡ st❛❜✐❧✐t② t❤❡♦r❡♠s
✷✳✸✳✶✳

❇♦✉♥❞❡❞♥❡ss ♦❢ s♦❧✉t✐♦♥s ♦❢ ♥♦♥❤♦♠♦❣❡♥♦✉s ❡q✉❛t✐♦♥s

✳

✹✶

✷✳✸✳✷✳

❇♦❤❧✲P❡rr♦♥✲t②♣❡ t❤❡♦r❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✻

❇♦❤❧ ❡①♣♦♥❡♥ts ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✺

✷✳✹✳✶✳

❇♦❤❧ ❡①♣♦♥❡♥ts ❛♥❞ t❤❡✐r ❜❛s✐❝ ♣r♦♣❡rt✐❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✺

✷✳✹✳✷✳

❘♦❜✉st♥❡ss ♦❢ ❇♦❤❧ ❡①♣♦♥❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✵

❚❤❡ ❝❛s❡ ♦❢ ✉♥❜♦✉♥❞❡❞ ❝❛♥♦♥✐❝❛❧ ♣r♦❥❡❝t♦r ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✺

✷✳✺✳✶✳

❯♥✐❢♦r♠ st❛❜✐❧✐t② ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ♦❢ ♣❡rt✉r❜❡❞
❡q✉❛t✐♦♥s

✷✳✺✳✷✳
✷✳✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✺

❇♦❧❤ ❡①♣♦♥❡♥t ♦❢ s♦❧✉t✐♦♥s ❛♥❞ ❇♦❤❧ ❡①♣♦♥❡♥t ♦❢ t❤❡ s②st❡♠ ✻✼

❈♦♥❝❧✉s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✶

❈❤❛♣t❡r ✸ ❙✐♥❣✉❧❛r s②st❡♠s ♦❢ s❡❝♦♥❞✲♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s
✼✷
✸✳✶

■♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✷

✸✳✷

❊①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t②

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✸

✸✳✷✳✶✳

◆♦t✐♦♥ ♦❢ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✸

✸✳✷✳✷✳

❈r✐t❡r✐❛ ❢♦r ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✺

✸✳✷✳✸✳

❇♦❤❧✲P❡rr♦♥ t❤❡♦r❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✶

✸✳✸

❘♦❜✉st st❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✹

✸✳✹

❈♦♥❝❧✉s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵✾

✈


❈♦♥❝❧✉s✐♦♥

✶✶✶

❚❤❡ ❛✉t❤♦r✬s ♣✉❜❧✐❝❛t✐♦♥s r❡❧❛t❡❞ t♦ t❤❡ t❤❡s✐s

✶✶✸

❆♣♣❡♥❞✐①

✶✶✹

❇✐❜❧✐♦❣r❛♣❤②

✶✶✽

✈✐

■♥tr♦❞✉❝t✐♦♥
❚❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❝❡rt❛✐♥ ♣❤❡♥♦♠❡♥❛ ✐♥ r❡❛❧✲✇♦r❧❞ ♦✈❡r t❤❡ ❝♦✉rs❡ ♦❢ t✐♠❡
✐s ✉s✉❛❧❧② ❞❡s❝r✐❜❡❞ ❜② ❞✐✛❡r❡♥t✐❛❧ ❛♥❞ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✳ ■♥ ❞✐s❝r❡t❡✲t✐♠❡
s❝❛❧❡✱ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧s ❧❡❛❞ t♦ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✳ ❉✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s
♣❧❛② ✐♠♣♦rt❛♥t r♦❧❡s ✐♥ ♠❛♥② ❛r❡❛s s✉❝❤ ❛s ❝♦♥tr♦❧✱ ❜✐♦❧♦❣②✱ ❡❝♦♥♦♠✐❝s✱✳✳✳ ❚②♣✲
✐❝❛❧❧②✱ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ❢♦r♠

F (x(n + k), x(n + k − 1), · · · , x(n + 1), x(n)) = 0,

✭✵✳✶✮

k ✐s ❛ ✜①❡❞ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r❀ n ∈ N❀ x : Z −→ Kd ❛♥❞ F : Kd × Kd ×
· · · × Kd −→ Kd . ■❢ k = 1✱ ❡q✉❛t✐♦♥ ✭✵✳✶✮ ✐s s❛✐❞ t♦ ❜❡ ♦❢ ✜rst✲♦r❞❡r✳ ❖t❤❡r✇✐s❡✱
❡q✉❛t✐♦♥ ✭✵✳✶✮ ✐s s❛✐❞ t♦ ❜❡ ♦❢ ❤✐❣❤✲♦r❞❡r✳ ■❢ t❤❡ ❤✐❣❤❡st ♦r❞❡r t❡r♠ x(n + k) ✐s
s♦❧✈❡❞ ❢r♦♠ ✭✵✳✶✮ ❢♦r ❡❛❝❤ n✱ t❤❡♥ ✇❡ ❤❛✈❡ ❡①♣❧✐❝✐t ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✱ ✇❤✐❝❤

✇❤❡r❡

❛r❡ ❣✐✈❡♥ ✐♥ t❤❡ ❢♦r♠

x(n + k) = f (x(n + k − 1), · · · , x(n + 1), x(n)).

✭✵✳✷✮

❊①♣❧✐❝✐t ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s ❤❛✈❡ ❜❡❡♥ ❡①t❡♥s✐✈❡❧② ✐♥✲
✈❡st✐❣❛t❡❞ ✐♥ ♠❛♥② ♣❛♣❡rs ❛♥❞ ♠♦♥♦❣r❛♣❤s❀ s❡❡ ❬✶❪✱ ❬✸✷❪ ❛♥❞ t❤❡ r❡❢❡r❡♥❝❡s
t❤❡r❡✐♥✳ ❍♦✇❡✈❡r✱ ✐♥ ♠❛♥② s✐t✉❛t✐♦♥s s②st❡♠ ✭✵✳✶✮ ✐s ♥♦t s♦❧✈❛❜❧❡ ❢♦r

x(n + k)✳

❚❤❡♥✱ ✇❡ s❛② ✭✵✳✶✮ ✐s ❛♥ ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✭■❉❊✮ ♦r ❛ s✐♥❣✉❧❛r ❞✐✛❡r✲
❡♥❝❡ ❡q✉❛t✐♦♥ ✭❙❉❊✮✳ ❚❤❡ s✐♠♣❧❡st ❝❛s❡ ♦❢ ❙❉❊s ✐s ❧✐♥❡❛r ❙❉❊s ♦❢ ✜rst✲♦r❞❡r✱
✇❤✐❝❤ ❛r❡ ❣✐✈❡♥ ❜②

En y(n + 1) = An y(n) + qn ,

n ∈ N(n0 ),

En , An ∈ Kd×d ❀ y(n), qn ∈ Kd ✱ N(n0 ) ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs
❛r❡ ❣r❡❛t❡r t❤❛♥ ♦r ❡q✉❛❧ t♦ ❛ ❣✐✈❡♥ ✐♥t❡❣❡r n0 ✱ ❛♥❞ t❤❡ ❧❡❛❞✐♥❣ ♠❛tr✐① En
✇❤❡r❡

✭✵✳✸✮
t❤❛t
♠❛②

❜❡ s✐♥❣✉❧❛r✳ ❚❤❡ ❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✭✵✳✸✮ ✐s

En x(n + 1) = An x(n),
✶

n ∈ N(n0 ).

✭✵✳✹✮


❯♥❧✐❦❡ ❡①♣❧✐❝✐t ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✱ t❤❡ ❛♥❛❧②s✐s ♦❢ ❙❉❊s ✐s ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞✳
❊✈❡♥ t❤❡ s♦❧✈❛❜✐❧✐t② ❛♥❛❧②s✐s ✐s ♥♦t tr✐✈✐❛❧✳ ❋♦r ❡①❛♠♣❧❡✱ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣
s②st❡♠

1 0
0 0

x(n + 1)
y(n + 1)

=

1 0
0 0

fn
gn

,n ∈ N

x(n), y(n), fn , gn ✱ ❢♦r ❡❛❝❤ n ❛r❡ r❡❛❧❀ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s
x(0) = x0 , y(0) = y0 ✳ ❊q✉❛t✐♦♥ ✭✵✳✺✮ ✐s ❡q✉✐✈❛❧❡♥t t♦
✇❤❡r❡

✭✵✳✺✮

❛r❡ ❞❡✜♥❡❞ ❜②

x(n + 1) = x(n) + fn ,
0 = gn .
❚❤❡♥✱ ✐t ✐s ❡❛s② t♦ s❡❡ t❤❛t s♦❧✉t✐♦♥s ♦❢ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ✭■❱P✮ ♦❢ ❙❉❊
✭✵✳✺✮ ❡✐t❤❡r ❤❛s ✐♥✜♥✐t❡❧② ♠❛♥② s♦❧✉t✐♦♥s ✐❢

gn ≡ 0

♦r ♦t❤❡r✇✐s❡ ♥♦ s♦❧✉t✐♦♥✳

❲❡ ❣✐✈❡ ❤❡r❡ ❛♥ ❡①❛♠♣❧❡ t♦ ✐❧❧✉str❛t❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ s✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡
❡q✉❛t✐♦♥s ✐♥ ♣r❛❝t✐❝❛❧ ❛r❡❛s ✭s❡❡ ❊①❛♠♣❧❡ ✶✲✶✳✷ ✐♥ ❬✷✶❪✮✳

❊①❛♠♣❧❡ ✵✳✵✳✶✳

❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❞②♥❛♠✐❝ ▲❡♦♥t✐❡❢ ♠♦❞❡❧ ♦❢ ❡❝♦♥♦♠✐❝ s②s✲

t❡♠s ✐s ❛ s✐♥❣✉❧❛r s②st❡♠✳ ■ts ❞❡s❝r✐♣t✐♦♥ ♠♦❞❡❧ ✐s ✭❬✺✷❪✮✿

x(k) = Ax(k) + B[x(k + 1) − x(k)] + d(k)
✇❤❡r❡

x(k)

✐s t❤❡

n

❞✐♠❡♥s✐♦♥❛❧ ♣r♦❞✉❝t✐♦♥ ✈❡❝t♦r ♦❢

✐s ❛♥ ✐♥♣✉t✲♦✉t♣✉t ✭♦r ♣r♦❞✉❝t✐♦♥✮ ♠❛tr✐①❀

Ax(k)

n

s❡❝t♦rs❀

B[x(k + 1) − x(k)]

A ∈ Rn×n

st❛♥❞s ❢♦r t❤❡ ❢r❛❝t✐♦♥ ♦❢

♣r♦❞✉❝t✐♦♥ r❡q✉✐r❡❞ ❛s ✐♥♣✉t ❢♦r t❤❡ ❝✉rr❡♥t ♣r♦❞✉❝t✐♦♥✱
❝❛♣✐t❛❧ ❝♦❡✣❝✐❡♥t ♠❛tr✐①✱ ❛♥❞

✭✵✳✻✮

B ∈ Rn×n

✐s t❤❡

✐s t❤❡ ❛♠♦✉♥t ❢♦r ❝❛♣❛❝✐t②

❡①♣❛♥s✐♦♥✱ ✇❤✐❝❤ ♦❢t❡♥ ❛♣♣❡❛rs ✐♥ t❤❡ ❢♦r♠ ♦❢ ❝❛♣✐t❛❧✱

d(k)

✐s t❤❡ ✈❡❝t♦r t❤❛t

✐♥❝❧✉❞❡s ❞❡♠❛♥❞ ♦r ❝♦♥s✉♠♣t✐♦♥✳ ❊q✉❛t✐♦♥ ✭✵✳✻✮ ♠❛② ❜❡ r❡✇r✐tt❡♥ ❛s

Bx(k + 1) = (I − A + B)x(k) − d(k).
■♥ ♠✉❧t✐s❡❝t♦r ❡❝♦♥♦♠✐❝ s②st❡♠s✱ ♣r♦❞✉❝t✐♦♥ ❛✉❣♠❡♥t✐♦♥ ✐♥ ♦♥❡ s❡❝t♦r ♦❢t❡♥
❞♦❡s♥✬t ♥❡❡❞ t❤❡ ✐♥✈❡st♠❡♥t ❢r♦♠ ❛❧❧ ♦t❤❡r s❡❝t♦rs✱ ❛♥❞ ♠♦r❡♦✈❡r✱ ✐♥ ♣r❛❝t✐❝❛❧

❝❛s❡s ♦♥❧② ❛ ❢❡✇ s❡❝t♦rs ❝❛♥ ♦✛❡r ✐♥✈❡st♠❡♥t ✐♥ ❝❛♣✐t❛❧ t♦ ♦t❤❡r s❡❝t♦rs✳ ❚❤✉s✱
♠♦st ♦❢ t❤❡ ❡❧❡♠❡♥ts ✐♥

B

❛r❡ ③❡r♦ ❡①❝❡♣t ❢♦r ❛ ❢❡✇✳

B

✐s ♦❢t❡♥ s✐♥❣✉❧❛r✳ ■♥

t❤✐s s❡♥s❡ t❤❡ s②st❡♠ ✭✵✳✻✮ ✐s ❛ t②♣✐❝❛❧ ❞✐s❝r❡t❡✲t✐♠❡ s✐♥❣✉❧❛r s②st❡♠✳
❚❤❡ ✜rst ✇♦r❦s ♦♥ ❙❉❊s ✇❡r❡ ❞♦♥❡ ❜② ❙✳▲✳ ❈❛♠♣❜❡❧❧ ❬✶✼✱ ✶✽❪✱ ▲✳ ❉❛✐ ❬✷✶❪
❛♥❞ ❉✳●✳ ▲✉❡♥❜❡❣❡r ❬✺✷✱ ✺✹❪ ✐♥ ❧❛t❡ ✼✵✬s ❛♥❞ ✽✵✬s✳ ■❢
♠❛tr✐❝❡s✱ ✐✳❡✳

En ≡ E, An ≡ A✱

En

❛♥❞

An

❛r❡ ❝♦♥st❛♥t

t❤❡♥ ✇❡ ❤❛✈❡ ❧✐♥❡❛r t✐♠❡✲✐♥✈❛r✐❛♥t ✭❝♦♥st❛♥t

✷

❝♦❡✣❝✐❡♥t✮ ❙❉❊s✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ♣❡♥❝✐❧

0✱

{E, A} ✐s r❡❣✉❧❛r✱ ✐✳❡✳ det(λE−A) ≡

t❤❡♥ t❤❡r❡ ❡①✐sts t❤❡ s♦✲❝❛❧❧❡❞ ❲❡✐❡rstr❛ss✲❑r♦♥❡❝❦❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡r❡

❛r❡ ♥♦♥✲s✐♥❣✉❧❛r ♠❛tr✐❝❡s

T

❛♥❞

W

I 0
0 N

T EW =

s✉❝❤ t❤❛t

,

T AW =

A1 0
0 I

,

r ✐✳❡✳ N r = 0, N k = 0, k < r. ❲❡
❛❧s♦ s❛② t❤❛t t❤❡ ✐♥❞❡① ♦❢ ♠❛tr✐① ♣❡♥❝✐❧ (E, A) ✐s r ✳ ❇② ✐♥tr♦❞✉❝✐♥❣ ❛ ✈❛r✐❛❜❧❡
❝❤❛♥❣❡ xn = W yn ❛♥❞ ♠✉❧t✐♣❧②✐♥❣ ❜♦t❤ s✐❞❡s ♦❢ ✭✵✳✸✮ ❜② T ✱ ✇❡ ♦❜t❛✐♥

✇❤❡r❡

N

✐s ❛ ♥✐❧♣♦t❡♥t ♠❛tr✐① ♦❢ ✐♥❞❡①

y 1 (n + 1) = A1 y 1 (n) + qn1 ,
N y 2 (n + 1) = y 2 (n) + qn2 .
r = 1 t❤❡♥ N ≡ 0✱ ❛♥❞ t❤❡ s❡❝♦♥❞
y (n) = −qn2 ✱ ✇❤✐❝❤ ✐s ❛♥ ❛❧❣❡❜r❛✐❝

❚❤❡ ✜rst ❡q✉❛t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✐s ❡①♣❧✐❝✐t✳ ■❢
❡q✉❛t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✐s ❡q✉✐✈❛❧❡♥t t♦

2

❡q✉❛t✐♦♥✳ ■♥ ❬✶✼✱ ✶✽❪✱ ❈❛♠♣❜❡❧❧ ❝♦♥s✐❞❡r❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛ss ♦❢ ❧✐♥❡❛r t✐♠❡✲
✐♥✈❛r✐❛♥t ❙❉❊s

Ax(n + 1) = Bx(n) + fn ,
A, B ∈ Cm,m ✱ A

✭✵✳✼✮

c✱ t❤❡
■❱P ♦❢ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥✱ Ax(n + 1) = Bx(n), x(0) = c, n = 1, 2, ...
❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts ❛ λ ∈ C s✉❝❤ t❤❛t λA + B ✐s
♥♦♥✲s✐♥❣✉❧❛r✳ ■♥ t❤✐s ❝❛s❡ t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ s✉❜❥❡❝t t♦ x(0) = c ✐s ❣✐✈❡♥ ❜②

✇❤❡r❡

✐s s✐♥❣✉❧❛r✳

❋♦r ❡❛❝❤ ❝♦♥s✐st❡♥t ✐♥✐t✐❛❧ ✈❡❝t♦r

k−1

n−1

ˆ D )i B
ˆ D fˆn+i ,
(AˆB

ˆ n−i−1 fˆi − (I − AˆAˆD )
(AˆD B)

ˆ n AˆAˆD q + AˆD
x(n) = (AˆD B)

i=0

i=0

ˆ = (λA−B)−1 A, ✐♥❞(A)
ˆ = k, ❛♥❞
fˆi = (λA−B)−1 fi , Aˆ = (λA−B)−1 A, B
ˆ D ❛r❡ ❉r❛③✐♥ ✐♥✈❡rs❡s ♦❢ Aˆ ❛♥❞ B
ˆ ✱ r❡s♣❡❝t✐✈❡❧② ✭❢♦r t❤❡ ❞❡✜♥✐t✐♦♥
q ∈ Cm ❀ AˆD ✱ B

✇❤❡r❡

♦❢ ❉r❛③✐♥ ✐♥✈❡rs❡✱ s❡❡ ❬✶✼❪✮✳ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s ✇❛s ❛❧s♦ ❡st❛❜❧✐s❤❡❞ ❢♦r
t❤❡ ❞✐s❝r❡t❡ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

x(k + 1) = Ax(k) + Bu(k), k = 0, N − 1.
❙✐♠✉❧t❛♥❡♦✉s❧②✱ s♦♠❡ r❡s✉❧ts ♦♥ ❧✐♥❡❛r t✐♠❡✲✐♥✈❛r✐❛♥t ❙❉❊s ✇❡r❡ ❛♣♣❧✐❡❞ t♦ t❤❡
▲❡♦♥t✐❡❢ ❞②♥❛♠✐❝ ♠♦❞❡❧ ♦❢ ♠✉❧t✐s❡❝t♦r ❡❝♦♥♦♠②✳ ❋♦r ♥♦♥✲❛✉t♦♥♦♠♦✉s ❞✐s❝r❡t❡
s②st❡♠

Ak x(k + 1) + Bk x(k) = fk , k
✸

0, Ak , Bk ∈ Cn×n ,

✭✵✳✽✮


rank Ak ≡ r ❢♦r ❛❧❧ k ✳
♠❛tr✐① Pk ✱ ✇❡ ♦❜t❛✐♥

✇❡ ❛ss✉♠❡ t❤❛t
❛♥ ✐♥✈❡rt✐❜❧❡

Tk
0
✇❤❡r❡

x(k + 1) +

rank Tk ≡ r, Tk ∈ Cr×n .

♦♥❧② ❜❡ s❤♦✇♥ ✐♥ t✇♦ ❝❛s❡s✳

❚❤❡♥✱ ♠✉❧t✐♣❧②✐♥❣ ❜♦t❤ s✐❞❡s ♦❢ ✭✵✳✽✮ ❜②

Ck
Dk

gk
hk

,

✭✵✳✾✮

❚❤❡r❡ ❛r❡ ❢♦✉r ♣♦ss✐❜✐❧✐t✐❡s ❜✉t s♦❧✉t✐♦♥s ❝♦✉❧❞
❚❤❡ ✜rst ♦♥❡ ✐s t❤❡ ❝❛s❡ ✇❤❡r❡

✐♥✈❡rt✐❜❧❡✱ t❤❡ s❡❝♦♥❞ ♦♥❡ ✐s t❤❡ ❝❛s❡ ✇❤❡r❡

k✳

x(k) =

rank

Tk
Dk+1

Tk
Dk+1

= rank(Tk )

✐s

❢♦r ❛❧❧

❚❤❡ ♠❡t❤♦❞ ✉s❡❞ ❜② ❈❛♠♣❜❡❧❧ ✐s ❛ s❧✐❣❤t ❡①t❡♥s✐♦♥ ♦❢ t❤❛t ❡♠♣❧♦②❡❞ ❜②

▲✉❡♥❜❡r❣❡r ❬✺✷✱ ✺✹❪✳
❘❡❝❡♥t❧②✱ ❜② ❛❞♦♣t✐♥❣ t❤❡ ♣r♦❥❡❝t♦r✲❜❛s❡❞ ❛♥❛❧②s✐s ✭❞❡✈❡❧♦♣❡❞ ❜② ▼är③ ❡t✳❛❧✳
s✐♥❝❡ ✽✵✬s✮ ♦r t❤❡ str❛♥❣❡♥❡ss✲❢r❡❡ ❢♦r♠ ❢♦r♠✉❧❛t✐♦♥ ✭❑✉♥❦❡❧ ❛♥❞ ▼❡❤r♠❛♥♥
s✐♥❝❡ ✾✵✬s✮ ❢♦r ❉❆❊s✱ s♦♠❡ r❡s✉❧ts ♦♥ t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ st❛❜✐❧✐t② ♦❢ s♦❧✉t✐♦♥s
❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❢♦r ❙❉❊s ❜② ❑② ❆♥❤ ❡t✳❛❧✳ ❬✷✱ ✸✱ ✹✱ ✺✱ ✻❪ ❛♥❞ ❜② ❇rü❧❧ ✐♥
❬✶✸✱ ✶✹❪✱ r❡s♣❡❝t✐✈❡❧②✳ ❋✐rst✱ ❛ ❞❡✜♥✐t✐♦♥ ♦❢ ✐♥❞❡①✲✶ ✇❛s ❡①t❡♥❞❡❞ t♦ ❛ ❝❧❛ss ♦❢
♥♦♥❧✐♥❡❛r ❙❉❊s ✐♥ ❬✺❪✱

fn (x(n + 1), x(n)) = 0,

✇❤❡r❡

0, x(n) ∈ Rd , fn : Rd × Rd −→ Rd

n

t❤❡♥ t❤❡ s♦❧✈❛❜✐❧✐t② ♦❢ ■❱Ps ❢♦r ♥♦♥❧✐♥❡❛r ❙❉❊s ✇❛s ✐♥✈❡st✐❣❛t❡❞✳ ❚❤❡ ❋❧♦q✉❡t
t❤❡♦r② ❢♦r ❙❉❊s ✇❛s ❞❡✈❡❧♦♣❡❞ ✐♥ ❬✻❪✳ ■t ✇❛s ♣r♦✈❡❞ t❤❛t ❛♥② ✐♥❞❡①✲✶ ❙❉❊s
❝♦✉❧❞ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦ ✐ts ❑r♦♥❡❝❦❡r ♥♦r♠❛❧ ❢♦r♠✱ t❤❡♥ ❋❧♦q✉❡t t❤❡♦r❡♠
♦♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ♠❛tr✐① ♦❢ ✐♥❞❡①✲✶ ♣❡r✐♦❞✐❝ ❙❉❊s ❤❛s
❜❡❡♥ ❡st❛❜❧✐s❤❡❞✳

❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ t❤❡ ▲②❛♣✉♥♦✈ r❡❞✉❝t✐♦♥ t❤❡♦r❡♠ ✇❛s

♣r♦✈❡❞✳ ■♥ ❬✹❪✱ t❤❡ ▲②❛♣✉♥♦✈ ❢✉♥❝t✐♦♥ ♠❡t❤♦❞ ✇❛s ❛♣♣❧✐❡❞ t♦ st✉❞② st❛❜✐❧✐t②
♦❢ s✐♥❣✉❧❛r q✉❛s✐✲❧✐♥❡❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠

An x(n + 1) + Bn x(n) = fn (x(n)),
✇❤❡r❡

An , Bn ∈ Rd,d , fn (x)

n

0, x(n) ∈ Rd ,

✐s ❛ ❣✐✈❡♥ ▲✐♣s❝❤✐t③ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✇✐t❤ ❛

s✉✣❝✐❡♥t❧② s♠❛❧❧ ▲✐♣s❝❤✐t③ ❝♦❡✣❝✐❡♥t✱

fn : Rd × Rd −→ Rd ✳

❘❡❝❡♥t❧②✱ ✐♥ ❬✺✾❪✱

◆❣❛ ❞❡r✐✈❡❞ ❝♦♥❞✐t✐♦♥s ✉♥❞❡r ✇❤✐❝❤ t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ s♦❧✉t✐♦♥s ✐s
♣r❡s❡r✈❡❞ ❢♦r ♣❡rt✉r❜❡❞ ❧✐♥❡❛r ❙❉❊s✳ ❚❤✐s ❝❛♥ ❜❡ r❡❣❛r❞❡❞ ❛s t❤❡ ✜rst ❛tt❡♠♣t
♦❢ ❡①t❡♥❞✐♥❣ t❤❡ st❛❜✐❧✐t② r❡s✉❧ts ❢♦r r❡❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✐♥ ❬✶✱ ✸✷❪ t♦
❙❉❊s✳

✹


■♥ t❤❡ ❧❛st ❞❡❝❛❞❡✱ ✐♥t❡r❡st ✐♥ ❙❉❊s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s ✇❛s ❝♦♥t✐♥✉❡❞
✐♥t❡♥s✐✈❡❧② ❜② ♦t❤❡r r❡s❡❛r❝❤ ❣r♦✉♣s✱ ❛s ✇❡❧❧✳

❚❤❡r❡ ❤❛✈❡ ❜❡❡♥ ❛ ♥✉♠❜❡r ♦❢

♣❛♣❡rs t❤❛t ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞ t♦ t❤❡ t♦♣✐❝ ♦❢ t❤✐s t❤❡s✐s✱ ❡✳❣✳✱ s❡❡ ❬✼✱ ✸✸✱ ✸✹✱ ✸✺✱
✹✹✱ ✹✺✱ ✹✻✱ ✺✼✱ ✺✽✱ ✻✸✱ ✼✹✱ ✼✺❪✳ P❛rt✐❝✉❧❛r ❛tt❡♥t✐♦♥ ❤❛s ❜❡❡♥ ♣❛✐❞ t♦ st❛❜✐❧✐t②
❛♥❞ r♦❜✉st st❛❜✐❧✐t② ♦❢ s✐♥❣✉❧❛r ❞✐s❝r❡t❡✲t✐♠❡ s②st❡♠s ✇✐t❤ ♦r ✇✐t❤♦✉t ❞❡❧❛②✱
s❡❡ ❬✼✱ ✸✸✱ ✸✹✱ ✹✹✱ ✺✽✱ ✻✸✱ ✼✹✱ ✼✺❪✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ ❬✼❪ t❤❡ ❛✉t❤♦rs ✐♥✈❡st✐❣❛t❡❞
t❤❡ st❛❜✐❧✐t② ♦❢ ❧✐♥❡❛r s✇✐t❝❤❡❞ s✐♥❣✉❧❛r s②st❡♠s✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛
s♣❡❝✐❛❧ ❝❧❛ss ♦❢ ❧✐♥❡❛r t✐♠❡✲✈❛r②✐♥❣ s✐♥❣✉❧❛r s②st❡♠s✳
❆t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ✷✵t❤ ❝❡♥t✉r②✱ ❇♦❤❧✱ ❛♥❞ ❧❛t❡r P❡rr♦♥✱ ♣r♦✈❡❞ t❤❛t
t❤❡ ❜♦✉♥❞❡❞ ✐♥♣✉t ✲ ❜♦✉♥❞❡❞ st❛t❡ ✭❛❧s♦ ❝❛❧❧❡❞ P❡rr♦♥✬s ♣r♦♣❡rt②✮ ♦❢ ❛ ♥♦♥✲
❤♦♠♦❣❡♥❡♦✉s ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✭✉♥❞❡r s♦♠❡ ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡
❝♦❡✣❝✐❡♥t✮ ✐♠♣❧✐❡s t❤❡ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❤♦♠♦❣❡♥❡♦✉s
❡q✉❛t✐♦♥ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✐♥st❡❛❞ ♦❢ t❤❡ ❜♦✉♥❞❡❞♥❡ss ♣r♦♣❡rt②✱
✇❡ ❝❛♥ t❛❦❡ ❛ ♣❛✐r ♦❢ ❛♣♣r♦♣r✐❛t❡ ❇❛♥❛❝❤ s♣❛❝❡s

B1

❛♥❞

B2

❛♥❞ ❝♦♥s✐❞❡r t❤❡

B1 ✲✐♥♣✉t ✲ B2 ✲st❛t❡ ♣r♦♣❡rt② ✭❛❧s♦ ❝❛❧❧❡❞ P❡rr♦♥✬s ♣r♦♣❡rt②✮✱ ✐✳❡✳✱ ❢♦r ❛♥② ✐♥♣✉t
❜❡❧♦♥❣✐♥❣ t♦ B1 ✱ t❤❡r❡ ❡①✐sts ❛ s♦❧✉t✐♦♥ ❜❡❧♦♥❣✐♥❣ t♦ B2 ✳ ❚❤✐s ❝❤❛r❛❝t❡r✐③❛t✐♦♥
❤❛s ❜❡❡♥ ❣❡♥❡r❛❧✐③❡❞ t♦ ✈❛r✐♦✉s ❝❤♦✐❝❡s ♦❢ s♣❛❝❡ ♣❛✐rs ❛♥❞ ❞✐✛❡r❡♥t ❦✐♥❞s ♦❢
❡q✉❛t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ ❇♦❤❧✲P❡rr♦♥✲t②♣❡ st❛❜✐❧✐t② t❤❡♦r❡♠s ✇❡r❡ ❢♦r♠✉❧❛t❡❞
❢♦r ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ✐♥ ❬✷✷❪✱ ❢♦r ❞✐✛❡r❡♥❝❡ ❡q✉❛✲
t✐♦♥s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ✐♥ ❬✽✱ ✼✵❪✱ ❢♦r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ ❞❡❧❛② ✐♥ ❬✾✱ ✶✷❪✳
■♥ ❬✽❪✱ ❆✉❧❜❛❝❤ ❛♥❞ ❱❛♥ ▼✐♥❤ ❝♦♥s✐❞❡r❡❞ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✐♥ ❛ ❇❛♥❛❝❤
s♣❛❝❡

xn+1 = An xn + fn ,
✇❤❡r❡

∞✱

sup An < ∞✳

✭✵✳✶✵✮

■t ✇❛s ♣r♦✈❡❞ t❤❛t ✐❢ t❤❡ s♦❧✉t✐♦♥ ❜❡❧♦♥❣s t♦ lp , 1

p

n
❢♦r ❛♥② s❡q✉❡♥❝❡

fn

❜❡❧♦♥❣✐♥❣ t♦ t❤❡ s❛♠❡ s♣❛❝❡✱ t❤❡♥ ❛❧❧ s♦❧✉t✐♦♥s ♦❢

xn+1 = An xn

✭✵✳✶✶✮

❞❡❝❛② ❡①♣♦♥❡♥t✐❛❧❧②✳ ■t ✇❛s ❛❧s♦ ♣r♦✈❡❞ t❤❛t ✐❢ ❢♦r ❛♥② s❡q✉❡♥❝❡

fn

✐♥

l1

t❤❡

❝♦rr❡s♣♦♥❞✐♥❣ s♦❧✉t✐♦♥ ♦❢ ✭✵✳✶✵✮ ✐s ❜♦✉♥❞❡❞✱ t❤❡♥ t❤❡ ③❡r♦ s♦❧✉t✐♦♥ ♦❢ ✭✵✳✶✶✮ ✐s
✉♥✐❢♦r♠❧② st❛❜❧❡✳ ■♥ ❬✼✵❪✱ ▼✳ P✐t✉❦ ✐♠♣r♦✈❡❞ t❤❡ ❛❜♦✈❡ r❡s✉❧ts ❜② s❤♦✇✐♥❣ t❤❛t
✐❢ ❢♦r ❛♥②

fn ❢r♦♠ lp , 1 < p < ∞✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s♦❧✉t✐♦♥ ♦❢

l∞ t❤❡♥ t❤❡ tr✐✈✐❛❧ s♦❧✉t✐♦♥ ♦❢

✭✵✳✶✵✮ ❜❡❧♦♥❣s t♦

✭✵✳✶✶✮ ✐s ❡①♣♦♥❡♥t✐❛❧❧② st❛❜❧❡✳ ▲✳ ❇❡r❡③❛♥s❦② ❛♥❞

❊✳ ❇r❛✈❡r♠❛♥ ♣r♦♣♦s❡❞ s✐♠✐❧❛r ❇♦❤❧✲P❡rr♦♥✲t②♣❡ t❤❡♦r❡♠s ❢♦r ❞❡❧❛② ❞✐✛❡r❡♥❝❡

✺


❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠

n

A(n, k)x(k) + f (n).

x(n + 1) =

✭✵✳✶✷✮

k=−d
❆♥❛❧♦❣♦✉s ❇♦❤❧✲P❡rr♦♥ t❤❡♦r❡♠s ✇❡r❡ ❛❧s♦ ❢♦r♠✉❧❛t❡❞ ❢♦r ❞✐✛❡r❡♥t✐❛❧✲❛❧❣❡❜r❛✐❝
❡q✉❛t✐♦♥s ✐♥ ❬✶✵✱ ✶✾❪✳

❚❤❡ ❛♣♣r♦❛❝❤ ✉s❡❞ ✐♥ t❤♦s❡ ♣❛♣❡rs ✐s t♦ ❞❡❝♦✉♣❧❡ t❤❡

s②st❡♠ ✐♥t♦ ❛♥ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✭❖❉❊✮ ❛♥❞ ❛♥ ❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥✱
t❤❡♥ t♦ ❛♣♣❧② t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ ❇♦❤❧✲P❡rr♦♥ t❤❡♦r❡♠ ❢♦r ❖❉❊s ✐♥ ♦r❞❡r t♦ ❣❡t
❛♥ ❛♥❛❧♦❣♦✉s r❡s✉❧t ❢♦r ❉❆❊s✳
❇❡s✐❞❡s t❤❡ ♣r♦❜❧❡♠s ♦❢ st❛❜✐❧✐t②✱ r❡s❡❛r❝❤❡rs ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ r♦❜✉st
st❛❜✐❧✐t② ♣r♦❜❧❡♠s✳

❚❤❡ r❡❛s♦♥ ✐s t❤❛t ✐♥ ♠♦❞❡❧✐♥❣ ♦❢ r❡❛❧✲❧✐❢❡ ♣❤❡♥♦♠❡♥❛✱

✉♥❝❡rt❛✐♥t✐❡s s✉❝❤ ❛s ♠♦❞❡❧✐♥❣ ❡rr♦rs ❞✉❡ t♦ s✐♠♣❧✐❢②✐♥❣ ❛ss✉♠♣t✐♦♥s✱ ❞❛t❛
❡rr♦r✱ ❡t❝✳ ❛r✐s❡✳ ❚❤✉s✱ t❤❡ q✉❡st✐♦♥ ✇❤❡t❤❡r ❛ s②st❡♠ ♣r❡s❡r✈❡s st❛❜✐❧✐t② ✉♥❞❡r
s♠❛❧❧ ♣❡rt✉r❜❛t✐♦♥s ✐s ✈❡r② ✐♠♣♦rt❛♥t ❢♦r s✐♠✉❧❛t✐♦♥ ❛♥❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✳ ❚❤❡
❞✐st❛♥❝❡ t♦ ✐♥st❛❜✐❧✐t② ❝❛♥ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② t❤❡ s♦✲❝❛❧❧❡❞ st❛❜✐❧✐t② r❛❞✐✐✱
✇❤✐❝❤ ✇❛s ❢♦r♠✉❧❛t❡❞ ✐♥ s❡♠✐♥❛❧ ✇♦r❦s ❜② ❍✐♥r✐❝❤s❡♥ ❛♥❞ P✐t❝❤❛r❞ ❬✹✵✱ ✹✶❪✳
❚❤✐s ♣r♦❜❧❡♠ ✐s st❛t❡❞ ❛s ❢♦❧❧♦✇s✳ ●✐✈❡♥ ❛ ❧✐♥❡❛r ❝♦♥t✐♥✉♦✉s✲t✐♠❡ s②st❡♠

A ∈ Km×m .

x˙ = Ax,

✭✵✳✶✸✮

▲❡t ✉s s✉♣♣♦s❡ t❤❛t ✐t ✐s ❛s②♠♣t♦t✐❝❛❧❧② st❛❜❧❡✳ ❚♦❣❡t❤❡r ✇✐t❤ ✭✵✳✶✸✮✱ ❝♦♥s✐❞❡r
❛ ♣❡rt✉r❜❡❞ s②st❡♠

x˙ = (A + B∆C)x,
✇❤❡r❡

∆

✐s ❛♥ ✉♥❝❡rt❛✐♥ ♣❡rt✉r❜❛t✐♦♥✱

B

❛♥❞

C

✭✵✳✶✹✮

❛r❡ ♠❛tr✐❝❡s t❤❛t ❞❡s❝r✐❜❡ t❤❡

str✉❝t✉r❡ ♦❢ t❤❡ ♣❡rt✉r❜❛t✐♦♥✳ ❲❡ ❞❡✜♥❡ t❤❡ st❛❜✐❧✐t② r❛❞✐✉s ❜②

rK = inf{ ∆ ,
❍❡r❡

·

✭✵✳✶✹✮ ✐s ✉♥st❛❜❧❡}.

K = C(R r❡s✳)

✐s ❛ ♠❛tr✐① ♥♦r♠ ❛♥❞ ✐❢

t❤❡♥

rK

✐s ❝❛❧❧❡❞ t❤❡ ❝♦♠♣❧❡①

✭r❡❛❧ r❡s✳✮ str✉❝t✉r❡❞ st❛❜✐❧✐t② r❛❞✐✉s✳ ■t ✐s ♥♦t ❞✐✣❝✉❧t t♦ ♣r♦✈❡ t❤❛t

rC = sup C(sI − A)−1 B

−1

.

s∈iR

❋✐♥❞✐♥❣ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r t❤❡ r❡❛❧ st❛❜✐❧✐t② r❛❞✐✉s ✐s ♠♦r❡ ❞✐✣❝✉❧t✳
✶✾✾✺✱ ✐t ✇❛s s♦❧✈❡❞ ❜② ◗✐✉ ❡t✳

❛❧✳

❬✼✷❪✳

■♥

❙t❛❜✐❧✐t② r❛❞✐✐ ❢♦r ❧✐♥❡❛r ❞✐s❝r❡t❡✲

t✐♠❡ s②st❡♠s ❛r❡ ❞❡✜♥❡❞ ❛♥❛❧♦❣♦✉s❧②✳ ❙✐♥❝❡ ✾✵✬s✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ st❛❜✐❧✐t② r❛❞✐✐
❤❛s ❜❡❡♥ ✐♥✈❡st✐❣❛t❡❞ ❢♦r ✈❛r✐♦✉s s②st❡♠s✱ s✉❝❤ ❛s ♣♦s✐t✐✈❡ s②st❡♠s ❬✻✹❪✱ ❞❡❧❛②
s②st❡♠s ❛♥❞ ❤✐❣❤❡r✲♦r❞❡r s②st❡♠s ❬✹✷✱ ✻✺✱ ✻✻❪✳ ❚❤✐s ♣r♦❜❧❡♠ ✇❛s ❛❧s♦ st❛t❡❞ ✐♥

✻


♠♦r❡ ❣❡♥❡r❛❧ s❡tt✐♥❣s✱ ❢♦r ❡①❛♠♣❧❡✱ ❙♦♥ ❛♥❞ ❚❤✉❛♥ ❬✻✼✱ ✻✽❪ s♦❧✈❡❞ t❤❡ ♣r♦❜❧❡♠
♦❢ ❛ s✉r❥❡❝t✐✈✐t② r❛❞✐✉s ❢♦r ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✱ ✇❤✐❝❤ ✐♠♣❧✐❡s st❛❜✐❧✐t② r❛❞✐✐ r❡s✉❧ts✳
❆♥♦t❤❡r ❡①t❡♥s✐♦♥ ✐s t❤❡ r♦❜✉st st❛❜✐❧✐t② ♦❢ s✐♥❣✉❧❛r s②st❡♠s✱ ✐✳❡✳ ❉❆❊s ❛♥❞
❙❉❊s✳ ❇②❡rs ❛♥❞ ◆✐❝❤♦❧s ❬✶✻❪✱ ◗✐✉ ❛♥❞ ❉❛✈✐s♦♥ ❬✼✶❪ ♣r♦♣♦s❡❞ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s
❢♦r t❤❡ ❝♦♠♣❧❡① st❛❜✐❧✐t② r❛❞✐✉s ♦❢ t❤❡ ❉❆❊s ♦❢ t❤❡ ❢♦r♠

E x˙ = Ax,
✇❤❡r❡
✶✳

E

✐s ❛ s✐♥❣✉❧❛r ♠❛tr✐① ❛♥❞ t❤❡ ♣❡♥❝✐❧

{E, A}

✐s s✉♣♣♦s❡❞ t♦ ❤❛s ✐♥❞❡①✲

❉✉ ❝♦♥s✐❞❡r❡❞ ❤✐❣❤❡r✲✐♥❞❡① ❉❆❊s ❛♥❞ ♠❛❞❡ s♦♠❡ ❡①t❡♥s✐♦♥s t♦ ♣♦s✐t✐✈❡

s②st❡♠s ❬✷✸❪✳ ❚❤❡ ❜❡❤❛✈✐♦r ♦❢ ❝♦♠♣❧❡① st❛❜✐❧✐t② r❛❞✐✉s ❢♦r s✐♥❣✉❧❛r❧② ♣❡rt✉r❜❡❞
❉❆❊s ✇❛s ❛❧s♦ ❛♥❛❧②③❡❞ ❜② ❉✉ ❛♥❞ ▲✐♥❤ ❬✷✹✱ ✷✺❪✳ ●❡♥❡r❛❧❧②✱ t❤❡ ♣r♦❜❧❡♠ ♦❢
r♦❜✉st st❛❜✐❧✐t② ♦❢ ❉❆❊s ✐s ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ t❤❛♥ t❤❛t ♦❢ ❖❉❊s ❞✉❡ t♦ t❤❡
s✐♥❣✉❧❛r✐t② ♦❢

E✳

❚♦ t❤✐s ♣r♦❜❧❡♠✱ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ♣❡♥❝✐❧

{E, A}

♣❧❛②s ❛♥

✐♠♣♦rt❛♥t r♦❧❡✱ ✇❤✐❝❤ s❤♦✉❧❞ ❜❡ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✐♥ t❤❡ ♣r♦❜❧❡♠ ♦❢ st❛❜✐❧✐t②
❛♥❛❧②s✐s✳
■♥ ❬✸✶❪✱ t❤❡ ❛✉t❤♦rs ♣r♦♣♦s❡❞ ❛ ❢♦r♠✉❧❛ ❢♦r ❝♦♠♣❧❡① st❛❜✐❧✐t② r❛❞✐✉s ♦❢ s✐♥❣✉✲
❧❛r s②st❡♠s ♦❢ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✳ ❆s ❛♥ ❛♣♣❧✐❝❛t✐♦♥✱ t❤❡② ❝❤❛r❛❝t❡r✐③❡❞ t❤❡
❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ st❛❜✐❧✐t② r❛❞✐✉s ♦❢ t❤❡ s②st❡♠ r❡s✉❧t❡❞ ❜② ❞✐s❝r❡t✐③✐♥❣
❛ ❉❆❊✳ ❊①t❡♥s✐♦♥s ♦❢ t❤❡s❡ r❡s✉❧ts t♦ t❤❡ ❤✐❣❤❡r✲♦r❞❡r s✐♥❣✉❧❛r s②st❡♠s ❛♥❞
s✐♥❣✉❧❛r ❞②♥❛♠✐❝ s②st❡♠s ❤❛✈❡ ❜❡❡♥ ❞♦♥❡ r❡❝❡♥t❧② ✐♥ ❬✸✼✱ ✺✻❪✳ ❊①t❡♥❞✐♥❣ s✉❝❤
r❡s✉❧ts ❢♦r t✐♠❡✲✐♥✈❛r✐❛♥t s②st❡♠s t♦ t✐♠❡✲✈❛r②✐♥❣ s②st❡♠s ✐s ♠✉❝❤ ♠♦r❡ ❞✐❢✲

✜❝✉❧t✳ ❙✐♥❝❡ ❛s②♠♣t♦t✐❝✴❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ♦❢ ❛ ❧✐♥❡❛r t✐♠❡✲✈❛r②✐♥❣ s②st❡♠
❝❛♥♥♦t ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② t❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥t ♠❛tr✐❝❡s✱ t❤❡ ❛♣✲
♣r♦❛❝❤❡s ❞❡✈❡❧♦♣❡❞ ❢♦r ❧✐♥❡❛r t✐♠❡✲✐♥✈❛r✐❛♥t s②st❡♠s ❛r❡ ♥♦ ❧♦♥❣❡r ❛♣♣❧✐❝❛❜❧❡✳
❇② ✉s✐♥❣ ♥♦✈❡❧ r❡s✉❧ts ✐♥ t❤❡ ♦♣❡r❛t♦r t❤❡♦r②✱ ❏❛❝♦❜ s✉❝❝❡❡❞❡❞ ✐♥ ♣r♦✈✐♥❣ ❛♥
❡①❛❝t ❢♦r♠✉❧❛ ❢♦r t❤❡ st❛❜✐❧✐t② r❛❞✐✉s ♦❢ ❧✐♥❡❛r t✐♠❡✲✈❛r②✐♥❣ ❖❉❊s ❬✹✸❪✳ ■♥ t❤❡
s♣✐r✐t ♦❢ ❏❛❝♦❜✬s r❡s✉❧t✱ ✐♥ ❬✶✾✱ ✷✻❪ ❉✉ ❛♥❞ ▲✐♥❤ ❡①t❡♥❞❡❞ t❤❡ ❛♥❛❧②s✐s t♦ ❧✐♥❡❛r
t✐♠❡✲✈❛r②✐♥❣ ❉❆❊s✳

❆♥ ❛♥❛❧♦❣♦✉s r❡s✉❧t ✇❛s ♦❜t❛✐♥❡❞ ❢♦r s✐♥❣✉❧❛r s②st❡♠s

♦❢ ❧✐♥❡❛r t✐♠❡✲✈❛r②✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✐♥ ❬✻✶❪✳

❘❡❝❡♥t❧②✱ ▼❡❤r♠❛♥♥ ❛♥❞

❚❤✉❛♥ ❬✺✻❪ ❝❤❛r❛❝t❡r✐③❡❞ t❤❡ st❛❜✐❧✐t② r❛❞✐✐ ♦❢ s✐♥❣✉❧❛r s②st❡♠s ♦❢ ❤✐❣❤❡r✲♦r❞❡r
❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❜② ✉s✐♥❣ ❛♥ ❛♣♣r♦❛❝❤ t❤❛t ✇❛s ✉s❡❞ ✐♥ ♣r❡✈✐♦✉s ✇♦r❦ ❢♦r
❞❡❧❛② ❉❆❊s ✐♥ ❬✷✾❪✳ ❚♦ ♦✉r ❦♥♦✇❧❡❞❣❡✱ st❛❜✐❧✐t② ❛♥❞ r♦❜✉st st❛❜✐❧✐t② ♦❢ s✐♥❣✉❧❛r
s②st❡♠s ♦❢ ❧✐♥❡❛r t✐♠❡✲✈❛r②✐♥❣ ❤✐❣❤❡r✲♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❤❛✈❡ ♥♦t ❜❡❡♥
❞✐s❝✉ss❡❞ ✐♥ ❧✐t❡r❛t✉r❡✳
❚❤❡ ✜rst ♠❛✐♥ ❛✐♠ ♦❢ ♦✉r ✇♦r❦ ✐s t♦ ❡①t❡♥❞ s♦♠❡ ❡①✐st✐♥❣ st❛❜✐❧✐t② r❡s✉❧ts ❢♦r

✼


❖❉❊s t♦ ❧✐♥❡❛r ❙❉❊s ♦❢ t❤❡ ❢♦r♠s ✭✵✳✸✮ ❛♥❞ ✭✵✳✹✮✳ ❚❤❡s❡ r❡s✉❧ts ❝♦♠♣❧❡♠❡♥t
t❤♦s❡ ✐♥ ❬✹✱ ✺✱ ✻✶❪✳ ❚❤❡② ❝❛♥ ❛❧s♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s t❤❡ ❞✐s❝r❡t❡✲t✐♠❡ ❛♥❛❧♦❣✉❡s
♦❢ s♦♠❡ r❡❝❡♥t r❡s✉❧ts ❢♦r ❉❆❊s✱ s❡❡ ❬✶✵✱ ✶✶✱ ✶✾✱ ✹✾✱ ✺✵❪✳

❋♦r t❤❡ r✐❣♦r♦✉s

♣r♦♦❢s ♦❢ ♠❛✐♥ r❡s✉❧ts✱ ✇❡ ❤❛✈❡ t♦ ♦✈❡r❝♦♠❡ t❤❡ ❞✐✣❝✉❧t✐❡s t❤❛t ❛r❡ ❝❛✉s❡❞
s✐♠✉❧t❛♥❡♦✉s❧② ❜② t❤❡ s✐♥❣✉❧❛r✐t② ❛♥❞ t❤❡ ❞✐s❝r❡t❡✲t✐♠❡ ♥❛t✉r❡ ♦❢ t❤❡ s②st❡♠s✳
❚♦ t❤❡ ❜❡st ♦❢ ♦✉r ❦♥♦✇❧❡❞❣❡✱ t❤✐s r❡s❡❛r❝❤ ✐s t❤❡ ✜rst ✇♦r❦ t❤❛t ✉s❡s t❤❡
❝♦♥❝❡♣t ♦❢ ❇♦❤❧ ❡①♣♦♥❡♥t ❬✶✺❪ t♦ ❝❤❛r❛❝t❡r✐③❡ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❛♥❞ r♦❜✉st
❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ♦❢ s✐♥❣✉❧❛r ❞✐s❝r❡t❡✲t✐♠❡ s②st❡♠s✳ ❋✉rt❤❡r♠♦r❡✱ ✉♥❧✐❦❡ t❤❡
♣r♦❜❧❡♠ ❢♦r♠✉❧❛t✐♦♥ ✐♥ ❬✶✾✱ ✻✶❪✱ ❤❡r❡ ✇❡ ❝♦♥s✐❞❡r ❛ ❣❡♥❡r❛❧ ❝❧❛ss ♦❢ ❛❧❧♦✇❛❜❧❡
str✉❝t✉r❡❞ ♣❡rt✉r❜❛t✐♦♥s ❛r✐s✐♥❣ ✐♥ ❜♦t❤ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ s②st❡♠ ✭✵✳✹✮✳ ❲❡
❛❧s♦ ❡①t❡♥❞ ❇♦❤❧✲P❡rr♦♥✲t②♣❡ st❛❜✐❧✐t② t❤❡♦r❡♠s ✐♥ ❬✽✱ ✼✵❪ ❢r♦♠ r❡❣✉❧❛r ❡①♣❧✐❝✐t
❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s t♦ ❙❉❊s ✭✵✳✸✮✳
❆♥♦t❤❡r ♠❛✐♥ ❛✐♠ ♦❢ t❤✐s t❤❡s✐s ✐s t♦ st✉❞② s♦❧✈❛❜✐❧✐t②✱ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t②✱
❛♥❞ r♦❜✉st st❛❜✐❧✐t② ♦❢ ❧✐♥❡❛r t✐♠❡✲✈❛r②✐♥❣ ❙❉❊s ♦❢ t❤❡ ❢♦r♠

An x(n + 2) + Bn x(n + 1) + Cn x(n) = fn ,

n = n0 , n0 + 1, . . . ,

✭✵✳✶✺✮

n0 ∈ N ❛♥❞ ❝♦❡✣❝✐❡♥ts An , Bn , Cn ∈ Cd,d , n n0 ✳ ❚❤❡ ❧❡❛❞✐♥❣ ❝♦❡✣✲
❝✐❡♥t An ✐s s✉♣♣♦s❡❞ t♦ ❜❡ s✐♥❣✉❧❛r ✇✐t❤ rank An ≡ d1 < d ❢♦r ❛❧❧ n
n0 ✳ ❯♥❞❡r
✇❤❡r❡

t❤❡ str❛♥❣❡♥❡ss✲❢r❡❡ ❛ss✉♠♣t✐♦♥✱ ✇❡ ♣r♦♣♦s❡ ❛♥ ❡①♣❧✐❝✐t ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ s♦✲
❝❛❧❧❡❞ ❝♦♥s✐st❡♥t ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❜② ✇❤✐❝❤ t❤❡ ■❱P ❛❞♠✐ts ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥✳
❲❡ ❛❧s♦ s❤♦✇ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❙❉❊ ✭✵✳✶✺✮ ❛♥❞ ❛ ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❡①♣❧✐❝✐t
❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✳ ❈♦♠❜✐♥✐♥❣ t❤✐s ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❛♥❞ ❛ ❝♦♠♣❛r✐s♦♥ t❡❝❤✲
♥✐q✉❡✱ ✇❤✐❝❤ ✐s s✐♠✐❧❛r t♦ t❤❛t ♦❢ ❬✻✵❪✱ ✇❡ ♦❜t❛✐♥ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❝r✐t❡r✐❛
❢♦r ❙❉❊s ✭✵✳✶✺✮✳ ❊①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ✐s ❛❧s♦ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❛ ❇♦❤❧✲P❡rr♦♥✲
t②♣❡ t❤❡♦r❡♠✳ ◆❡①t✱ ✇❡ ♠❛❦❡ ✉s❡ ♦❢ ❛ r❡❝❡♥t r❡s✉❧t ❢♦r ❧✐♥❡❛r t✐♠❡✲✐♥✈❛r✐❛♥t

❙❉❊s ✐♥ ❬✺✻❪ ❛♥❞ ♦❜t❛✐♥ ❜♦✉♥❞s ❢♦r r♦❜✉st st❛❜✐❧✐t② ♦❢ ❙❉❊s ✭✵✳✶✺✮ ✇❤❡♥ t❤❡
❝♦❡✣❝✐❡♥ts ❛r❡ s✉❜❥❡❝t t♦ str✉❝t✉r❡❞ ♣❡rt✉r❜❛t✐♦♥s✳

❚❤❡ ❛♣♣r♦❛❝❤ ♣r♦♣♦s❡❞

✐♥ t❤✐s t❤❡s✐s ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❤✐❣❤❡r✲♦r❞❡r ❙❉❊s ❛♥❞ s✐♠✐❧❛r r❡s✉❧ts ❝❛♥
❜❡ ♦❜t❛✐♥❡❞✳ ❍♦✇❡✈❡r✱ ❢♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t②✱ ✐♥ t❤✐s ✇♦r❦✱ ✇❡ r❡str✐❝t t❤❡
✐♥✈❡st✐❣❛t✐♦♥ t♦ s❡❝♦♥❞✲♦r❞❡r ❙❉❊s✳ ❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐♥t♦ str❛♥❣❡♥❡ss✲❢r❡❡
❢♦r♠ ❜❛s❡❞ ♦♥ ♠❛tr✐① ❞❡❝♦♠♣♦s✐t✐♦♥s ♣❧❛②s ❛ ❦❡② r♦❧❡ ✐♥ ♦✉r ❛♥❛❧②s✐s✳ ❯♣ t♦ ♦✉r
❦♥♦✇❧❡❞❣❡✱ t❤✐s ✐s t❤❡ ✜rst ✇♦r❦ ❛❞❞r❡ss✐♥❣ t❤❡ st❛❜✐❧✐t② ❛♥❞ r♦❜✉st st❛❜✐❧✐t②
♦❢

❧✐♥❡❛r t✐♠❡✲✈❛r②✐♥❣ s✐♥❣✉❧❛r

❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ♦❢

❤✐❣❤❡r✲♦r❞❡r✳

❚❤❡ t❤❡s✐s ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✳

•

■♥ t❤❡ ✜rst ❝❤❛♣t❡r✱ ✇❡ r❡❝❛❧❧ t❤❡ tr❛❝t❛❜✐❧✐t②✲✐♥❞❡① ♥♦t✐♦♥ ❛♥❞ ❛ ❞❡❝♦✉♣❧✐♥❣

✽


t❡❝❤♥✐q✉❡ ❢♦r ❧✐♥❡❛r s✐♥❣✉❧❛r s②st❡♠s ♦❢ ✐♥❞❡①✲✶ ❜② ✉s✐♥❣ ♣r♦❥❡❝t♦rs✳ ❚❤❡♥✱
t❤❡ str❛♥❣❡♥❡ss✲❢r❡❡ ✐♥❞❡① ♥♦t✐♦♥ ✐s ✐♥tr♦❞✉❝❡❞ ❛♥❞ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦
✐♥❞❡① ♥♦t✐♦♥s ✐s ❞✐s❝✉ss❡❞✳ ❲❡ ❛❧s♦ ♠❡♥t✐♦♥ s♦♠❡ r❡s✉❧ts ♦♥ st❛❜✐❧✐t② ❛♥❞

st❛❜✐❧✐t② r❛❞✐✐ ✐♥ ❬✺✻❪ ❢♦r s❡❝♦♥❞✲♦r❞❡r ❧✐♥❡❛r t✐♠❡✲✐♥✈❛r✐❛♥t s②st❡♠s✳

•

■♥ t❤❡ s❡❝♦♥❞ ❝❤❛♣t❡r✱ ✇❡ st✉❞② t❤❡ ♣r❡s❡r✈❛t✐♦♥ ♦❢ ✉♥✐❢♦r♠ st❛❜✐❧✐t② ❛♥❞
❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ✇❤❡♥ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ s②st❡♠ ✭✶✳✷✮ ❛r❡ s✉❜❥❡❝t t♦
♣❡rt✉r❜❛t✐♦♥s✳ ❲❡ ❛❧s♦ ♣r❡s❡♥t ❇♦❤❧✲P❡rr♦♥✲t②♣❡ t❤❡♦r❡♠s t❤❛t ❡st❛❜❧✐s❤
❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❡①♣♦♥❡♥t✐❛❧✴✉♥✐❢♦r♠ st❛❜✐❧✐t② ♦❢ ❤♦♠♦❣❡♥❡♦✉s s②st❡♠
✭✵✳✹✮ ❛♥❞ t❤❡ ❜♦✉♥❞❡❞♥❡ss ♦❢ s♦❧✉t✐♦♥s t♦ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s s②st❡♠ ✭✵✳✸✮✳
◆❡①t✱ ✇❡ ❣✐✈❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❇♦❤❧ ❡①♣♦♥❡♥ts ❢♦r ❧✐♥❡❛r s✐♥❣✉❧❛r s②st❡♠s ✭✶✳✷✮
❛♥❞ ❛♥❛❧②③❡ ✐ts ♣r♦♣❡rt✐❡s ✐♥❝❧✉❞✐♥❣ s❡♥s✐t✐✈✐t② t♦ ♣❡rt✉r❜❛t✐♦♥s ♦❝❝✉rr✐♥❣
✐♥ t❤❡ s②st❡♠ ❝♦❡✣❝✐❡♥ts✳ ❲❡ ❛❧s♦ ❞✐s❝✉ss t❤❡ ❝❛s❡ ♦❢ ✉♥❜♦✉♥❞❡❞ ❝❛♥♦♥✐❝❛❧
♣r♦❥❡❝t♦r✳

•

■♥ t❤❡ t❤✐r❞ ❝❤❛♣t❡r✱ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ s♦❧✈❛❜✐❧✐t② ♦❢ ❙❉❊ ✭✵✳✶✺✮✳ ❚❤❡♥✱
✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❢♦r ❤♦♠♦❣❡♥❡♦✉s ❙❉❊s
❛ss♦❝✐❛t❡❞ ✇✐t❤ ✭✵✳✶✺✮ ❛♥❞ ❡st❛❜❧✐s❤ s♦♠❡ ❝r✐t❡r✐❛ ❢♦r ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t②✳
◆❡①t✱ ✇❡ ❝♦♥s✐❞❡r ❙❉❊ ✭✵✳✶✺✮ s✉❜❥❡❝t t♦ str✉❝t✉r❡❞ ♣❡rt✉r❜❛t✐♦♥s ❛♥❞
♦❜t❛✐♥ ❛ ❜♦✉♥❞ ❢♦r t❤❡ ♣❡rt✉r❜❛t✐♦♥s s✉❝❤ t❤❛t t❤❡ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t②
✐s ♣r❡s❡r✈❡❞✳

•

❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❞✐s❝✉ss✐♦♥ ♦❢ ❢✉t✉r❡ ✇♦r❦s ❛r❡ ❞r❛✇♥ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥✳

P❛rts ♦❢ t❤✐s t❤❡s✐s ❤❛✈❡ ❜❡❡♥ ♣✉❜❧✐s❤❡❞ ✐♥
✶✳ ◆❣✉②❡♥ ❍✉✉ ❉✉✱ ❱✉ ❍♦❛♥❣ ▲✐♥❤ ❛♥❞ ◆❣♦ ❚❤✐ ❚❤❛♥❤ ◆❣❛ ✭✷✵✶✻✮✱ ❖♥
st❛❜✐❧✐t② ❛♥❞ ❇♦❤❧ ❡①♣♦♥❡♥t ♦❢ ❧✐♥❡❛r s✐♥❣✉❧❛r s②st❡♠s ♦❢ ❞✐✛❡r❡♥❝❡ ❡q✉❛✲

t✐♦♥s ✇✐t❤ ✈❛r✐❛❜❧❡ ❝♦❡✣❝✐❡♥ts✱

❏✳ ❉✐✛❡r✳ ❊q✉❛t✐♦♥s ❆♣♣❧✳✱ ✷✷✱ ✶✸✺✵✕✶✸✼✼✳

✭❙❈■❊✮
✷✳ ❱✉ ❍♦❛♥❣ ▲✐♥❤✱ ◆❣♦ ❚❤✐ ❚❤❛♥❤ ◆❣❛✱ ❇♦❤❧✕P❡rr♦♥ ❚②♣❡ ❙t❛❜✐❧✐t② ❚❤❡♦✲
r❡♠s ❢♦r ▲✐♥❡❛r ❙✐♥❣✉❧❛r ❉✐✛❡r❡♥❝❡ ❊q✉❛t✐♦♥s✱

❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳

✭✷✵✶✼✮✳

❤tt♣s✿✴✴❞♦✐✳♦r❣✴✶✵✳✶✵✵✼✴s✶✵✵✶✸✲✵✶✼✲✵✷✹✺✲③✳ ✭❊❙❈■✱ ❙❝♦♣✉s✮
✸✳ ❱✉ ❍♦❛♥❣ ▲✐♥❤✱ ◆❣♦ ❚❤✐ ❚❤❛♥❤ ◆❣❛✱ ❉♦ ❉✉❝ ❚❤✉❛♥✱ ❊①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t②
❛♥❞ r♦❜✉st st❛❜✐❧✐t② ❢♦r ❧✐♥❡❛r t✐♠❡✲✈❛r②✐♥❣ s✐♥❣✉❧❛r s②st❡♠s ♦❢ s❡❝♦♥❞✲
♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✱

❙■❆▼ ❏✳ ▼❛tr✐① ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✾✲✶ ✭✷✵✶✽✮✱ ✷✵✹✲

✷✸✸✳✭❙❈■✮

✾


❛♥❞ ❛❧s♦ ♣r❡s❡♥t❡❞ ❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❢❡r❡♥❝❡s ❛♥❞ s❡♠✐♥❛rs
✶✳ ❚❤❡ ✷♥❞ PP■❈❚❆✱ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❙❡ss✐♦♥✱ ◆♦✈❡♠❜❡r ✶✸✲✶✼✱ ✷✵✶✼✱
❇✉s❛♥✱ ❑♦r❡❛✳
✷✳ ✧❱✐❡t♥❛♠✲❑♦r❡❛ ❏♦✐♥t ❈♦♥❢❡r❡♥❝❡ ♦♥ ❙❡❧❡❝t❡❞ ❚♦♣✐❝s ✐♥ ▼❛t❤❡♠❛t✐❝s✧✱
❋❡❜r✉❛r② ✷✵✲✷✹✱ ✷✵✶✼✱ ❉❛ ◆❛♥❣✱ ❱✐❡t♥❛♠✳
✸✳ ✧❱✐❡t♥❛♠✲❑♦r❡❛ ❏♦✐♥t ❲♦r❦s❤♦♣ ♦♥ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❘❡❧❛t❡❞ ❚♦♣✲

✐❝s✧✱ ▼❛r❝❤✱ ✷✵✶✻✱ ❱✐❡t♥❛♠ ■♥st✐t✉t❡ ❢♦r ❆❞✈❛♥❝❡❞ ❙t✉❞② ✐♥ ▼❛t❤❡♠❛t✐❝s✳
✹✳ ❈♦♥❢❡r❡♥❝❡ ♦♥ ▼❛t❤❡♠❛t✐❝s✱ ▼❡❝❤❛♥✐❝s ❛♥❞ ■♥❢♦r♠❛t✐❝s✱ ❱◆❯ ❯♥✐✈❡rs✐t②
♦❢ ❙❝✐❡♥❝❡✱ ❍❛♥♦✐✱ ✷✵✶✹ ❛♥❞ ✷✵✶✻✳
✺✳ ❙❡♠✐♥❛r ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ❋❛❝✉❧t② ♦❢ ▼❛t❤✲
❡♠❛t✐❝s✱ ▼❡❝❤❛♥✐❝s ❛♥❞ ■♥❢♦r♠❛t✐❝s✱ ❱◆❯ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡✱ ❍❛♥♦✐✱
✷✵✶✹✲✷✵✶✼✳
✻✳ ❙❡♠✐♥❛r ♦♥ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ❛♥❞ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s✱ ❋❛❝✉❧t② ♦❢
▼❛t❤❡♠❛t✐❝s✱ ▼❡❝❤❛♥✐❝s ❛♥❞ ■♥❢♦r♠❛t✐❝s✱ ❱◆❯ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡✱
❍❛♥♦✐✱ ✷✵✶✹✲✷✵✶✼✳

✶✵


❈❤❛♣t❡r ✶

Pr❡❧✐♠✐♥❛r✐❡s
■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ✐♥tr♦❞✉❝❡ s♦♠❡ ❢✉♥❞❛♠❡♥t❛❧ ♥♦t✐♦♥s ♦♥ s✐♥❣✉❧❛r ❧✐♥❡❛r
❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❛♥❞ ♦t❤❡r ❛✉①✐❧✐❛r② r❡s✉❧ts ✇❤✐❝❤ ✇✐❧❧ ❜❡ ✉s❡❞ ✐♥ ♥❡①t
❝❤❛♣t❡rs✳

✶✳✶ ▲✐♥❡❛r s✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❜② tr❛❝t❛❜✐❧✐t②✲✐♥❞❡①
❛♣♣r♦❛❝❤
■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ r❡❝❛❧❧ t❤❡ ✐♥❞❡① ♥♦t✐♦♥ ❛♥❞ ❛ ❞❡❝♦✉♣❧✐♥❣ t❡❝❤♥✐q✉❡ ❢♦r
❧✐♥❡❛r s✐♥❣✉❧❛r s②st❡♠s ♦❢ ✐♥❞❡①✲✶ ❜② ✉s✐♥❣ ❛♣♣r♦♣r✐❛t❡ ♣r♦❥❡❝t♦rs✳ ❚❤✐s ❝❛♥ ❜❡
❝♦♥s✐❞❡r❡❞ ❛s ❛ ❞✐s❝r❡t❡ ❛♥❛❧♦❣✉❡ ♦❢ ♣r♦❥❡❝t♦r ❛♣♣r♦❛❝❤ ❢♦r ❉❆❊s ❬✹✽❪✳ ❈♦♥s✐❞❡r
t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♥❡❛r s✐♥❣✉❧❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✭▲❙❉❊✮

En y(n + 1) = An y(n) + qn ,
✇❤❡r❡

En , An ∈ Kd,d

❛♥❞

qn ∈ Kd ✳

n ∈ N(n0 ),

✭✶✳✶✮

❚❤❡ ❤♦♠♦❣❡♥❡♦✉s s②st❡♠ ❛ss♦❝✐❛t❡❞ ✇✐t❤

✭✶✳✶✮ ✐s ❣✐✈❡♥ ❜②

En x(n + 1) = An x(n),

n ∈ N(n0 ).

✭✶✳✷✮

✶✳✶✳✶✳ ❉❡✜♥✐t✐♦♥ ♦❢ ✐♥❞❡①✲✶ s②st❡♠s ❛♥❞ t❤❡✐r ♣r♦♣❡rt✐❡s
Nn := ❦❡rEn ❛♥❞ ❧❡t Qn ❜❡ ❛ ♣r♦✲
d
P✉t Pn := I − Qn ✳ ▲❡t Tn ∈ GL(K ) ✭n ≥ n0 + 1✮ ❜❡ s✉❝❤
✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ Nn ❛♥❞ Nn−1 ✳ ❲❡ ✐♥tr♦❞✉❝❡ ❢♦❧❧♦✇✐♥❣

❈♦♥s✐❞❡r s✐♥❣✉❧❛r s②st❡♠ ✭✶✳✶✮✳ ❉❡♥♦t❡
❥❡❝t✐♦♥ ♦♥t♦
t❤❛t

T n |N n

Nn ✳

✐s ❛♥

✶✶


♠❛tr✐❝❡s ❛♥❞ s✉❜s♣❛❝❡s ❛ss♦❝✐❛t✐♥❣ ✇✐t❤ ✭✶✳✶✮

Gn := En −An Tn Qn

(n ≥ n0 +1),

Sn := {z ∈ Kd : An z ∈ ■♠En } (n ≥ n0 ).

❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛✳

▲❡♠♠❛ ✶✳✶✳✶✳

✭❬✸✱ ▲❡♠♠❛ ✷✳✸❪✮

❛♥② n ∈ N(n0 )✿

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t ❢♦r

✭✐✮ ❚❤❡ ♠❛tr✐① Gn := En − An Tn Qn ✐s ♥♦♥s✐♥❣✉❧❛r❀
✭✐✐✮ Nn−1 ⊕ Sn = Kd ❀
✭✐✐✐✮ Nn−1 ∩ Sn = {0}.

Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✶✳✶✳✶ ✐s ❛✈❛✐❧❛❜❧❡ ✐♥ ❆♣♣❡♥❞✐①✳
❇② ✈✐rt✉❡ ♦❢ ▲❡♠♠❛ ✶✳✶✳✶✱ ✇❡ ❝❛♥ ❞❡✜♥❡ ▲❙❉❊s ♦❢ tr❛❝t❛❜✐❧✐t② ✐♥❞❡①✲✶ ✭s❡❡
❬✸✱ ❉❡✜♥✐t✐♦♥ ✷✳✷❪✮✳

❉❡✜♥✐t✐♦♥ ✶✳✶✳✷✳ ❚❤❡ ▲❙❉❊
❢♦r s❤♦rt✮ ✐❢ ❢♦r ❛❧❧
✭✐✮ r❛♥❦En
✭✐✐✮

✭✶✳✶✮ ✐s s❛✐❞ t♦ ❜❡ ♦❢ tr❛❝t❛❜✐❧✐t② ✐♥❞❡①✲✶ ✭✐♥❞❡①✲✶

n ∈ N(n0 + 1)✱

t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❝♦♥❞✐t✐♦♥s ❤♦❧❞

= r✭❝♦♥st❛♥t✮❀

Nn−1 ∩ Sn = {0}✳

dim Sn0 = r ❛♥❞ ❧❡t En0 −1 ∈ Kd,d ❜❡ ❛
d
✜①❡❞ ♠❛tr✐① s❛t✐s❢②✐♥❣ t❤❡ r❡❧❛t✐♦♥ K = Sn0 ⊕ ❦❡rEn0 −1 ✳ ❚❤✉s✱ ❝♦♥❞✐t✐♦♥ ✭✐✐✮
✐♥ ❉❡✜♥✐t✐♦♥ ✶✳✶✳✷ ❤♦❧❞s ❢♦r ❛❧❧ n ∈ N(n0 ) ❛♥❞ t❤❡ ♦♣❡r❛t♦rs Tn ❛s ✇❡❧❧ ❛s
♠❛tr✐❝❡s Gn ❛r❡ ❞❡✜♥❡❞ ❢♦r ❛❧❧ n ∈ N(n0 )✳
❍❡r❡❛❢t❡r✱ ✇❡ ❛❧✇❛②s ❛ss✉♠❡ t❤❛t

▲❡♠♠❛ ✶✳✶✳✸✳ ✭❬✸❪✮ ❙✉♣♣♦s❡ t❤❡ ▲❙❉❊

✐s ♦❢ ✐♥❞❡①✲✶ ❛♥❞ Qn ❛r❡ ❛r❜✐tr❛r②
♣r♦❥❡❝t✐♦♥s ♦♥t♦ Nn ✱ n ≥ n0 ✳ ❚❤❡♥✱ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥s ❤♦❧❞✿

✭✐✮
✭✐✐✮

✭✶✳✶✮

Pn = G−1
n En , ✇❤❡r❡ Pn := I − Qn ;
−1
Pn G−1
n An = Pn Gn An Pn−1 ;

✭✶✳✸✮

−1
−1
Qn G−1
n An = Qn Gn An Pn−1 − Tn Qn−1 ;
✭✶✳✹✮

✭✐✐✐✮

Qn−1 := −Tn Qn G−1
n An ✐s t❤❡ ♣r♦❥❡❝t♦r ♦♥t♦ Nn−1 ❛❧♦♥❣ Sn .

Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✶✳✶✳✸ ✐s ❛✈❛✐❧❛❜❧❡ ✐♥ ❆♣♣❡♥❞✐①✳

✶✷

✭✶✳✺✮

❉✉❡ t♦ ✭✐✐✐✮✱ t❤❡ ♣r♦❥❡❝t♦r

Qn−1

❞❡✜♥❡❞ ✐♥ ▲❡♠♠❛ ✶✳✶✳✸ ✐s ✉♥✐q✉❡❧② ❞❡t❡r✲

♠✐♥❡❞✱ ✐✳❡✳✱ ✐t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢
s♣♦♥❞✐♥❣ ♣r♦❥❡❝t♦r

Pn := I − Qn ✐s ❛❧s♦ ✉♥✐q✉❡✳

Qn

❛♥❞

Tn ✳

❚❤✉s✱ t❤❡ ❝♦rr❡✲

❚❤❡② t♦❣❡t❤❡r ❢♦r♠ ❛ ❝❛♥♦♥✐❝❛❧

♣r♦❥❡❝t♦r ♣❛✐r✳ ❲❡ ❤❛✈❡ s♦♠❡ ♣r♦♣❡rt✐❡s ✐♥✈♦❧✈✐♥❣ t❤❡ ❝❛♥♦♥✐❝❛❧ ♣r♦❥❡❝t♦r ♣❛✐r
❛s ❢♦❧❧♦✇s✳

▲❡♠♠❛ ✶✳✶✳✹✳ ❚❤❡ ♠❛tr✐❝❡s

❝❤♦✐❝❡ ♦❢ Tn ❛♥❞ Qn ✳

−1

Pn G−1
n ❛♥❞ Tn Qn Gn ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡

Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✶✳✶✳✹ ✐s ❛✈❛✐❧❛❜❧❡ ✐♥ ❆♣♣❡♥❞✐①✳
❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ▲❡♠♠❛ ✶✳✶✳✹✱ ✐t ❢♦❧❧♦✇s ✐♠♠❡❞✐❛t❡❧② t❤❛t t❤❡ ♠❛tr✐❝❡s

Pn G−1
n

❛♥❞

Tn Qn G−1
n

❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢

s♣♦♥❞✐♥❣ s❝❛❧✐♥❣ ♠❛tr✐①

Gn := En − An Tn Qn

Tn ✳

❍❡r❡✱ t❤❡ ❝♦rr❡✲

✐s s❡t✳

✶✳✶✳✷✳ ❙♦❧✉t✐♦♥s ♦❢ ❈❛✉❝❤② ♣r♦❜❧❡♠
■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❜r✐❡✢② ♣r❡s❡♥t ❛ ❞❡❝♦✉♣❧✐♥❣ t❡❝❤♥✐q✉❡ ❢♦r ✐♥❞❡①✲✶ ▲❙❉❊s✳
❇② ✈✐rt✉❡ ♦❢ ▲❡♠♠❛
❛❧❧

n ≥ n0 ✳

Gn ❛r❡ ♥♦♥s✐♥❣✉❧❛r ❢♦r
Qn G−1
n ✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞

✶✳✶✳✶✱ ✇❡ s❡❡ t❤❛t t❤❡ ♠❛tr✐❝❡s

❍❡♥❝❡✱ ♠✉❧t✐♣❧②✐♥❣

✭✶✳✶✮ ❜②

Pn G−1
n

❛♥❞

❛♣♣❧②✐♥❣ ❢♦r♠✉❧❛s✭✶✳✸✮✲✭✶✳✹✮ ✐♥ ▲❡♠♠❛ ✶✳✶✳✸ ✇❡ ❞❡❝♦✉♣❧❡ t❤❡ ✐♥❞❡①✲✶ ▲❙❉❊
✐♥t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠

−1
Pn y(n + 1) = Pn G−1
n An Pn−1 y(n) + Pn Gn qn ,

✭✶✳✻✮

−1
0 = Qn G−1
n An y(n) + Qn Gn qn .

▼✉❧t✐♣❧②✐♥❣ ❜♦t❤ s✐❞❡s ♦❢ ❡q✉❛t✐♦♥ ✭✶✳✼✮ ❜②

Tn

✭✶✳✼✮

❛♥❞ ✉s✐♥❣ t❤❡ s❡❝♦♥❞ ❡q✉❛❧✐t②

✐♥ ✭✶✳✹✮✱ t❤✐s ❡q✉❛t✐♦♥ ✐s r❡✇r✐tt❡♥ ❛s

Qn−1 y(n) = −Qn−1 Pn−1 y(n) + Tn Qn G−1
n qn .
❚❤✉s✱ s♦❧✉t✐♦♥

Qn−1 y(n)✱

y(n)

✐s ❞❡❝♦♠♣♦s❡❞ ❛s ❛ s✉♠ ♦❢ t✇♦ ❝♦♠♣♦♥❡♥ts

✇❤❡r❡ t❤❡ ✏❞②♥❛♠✐❝✑ ❝♦♠♣♦♥❡♥t

Pn−1 y(n)

✭✶✳✽✮

Pn−1 y(n)

❛♥❞

✐s ❣♦✈❡r♥❡❞ ❜② ❡q✉❛t✐♦♥

✭✶✳✻✮✱ ✇❤✐❧❡ t❤❡ ✏❛❧❣❡❜r❛✐❝✑ ❝♦♠♣♦♥❡♥t ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥
✭✶✳✽✮✳ ■♥s♣✐r❡❞ ❜② t❤✐s ❞❡❝♦✉♣❧✐♥❣ ♣r♦❝❡❞✉r❡✱ ✇❡ ❢♦r♠✉❧❛t❡ t❤❡ ❝♦rr❡❝t❧② st❛t❡❞
✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❢♦r ✐♥❞❡①✲✶ ▲❙❉❊

Pn0 −1 (y(n0 ) − y0 ) = 0,

✭✶✳✶✮ ❛s

y0 ∈ Kd

✐s ❛r❜✐tr❛r✐❧② ❣✐✈❡♥✳

✭✶✳✾✮

❚❤❡r❡❢♦r❡✱ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✭✶✳✶✮✲✭✶✳✾✮ ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ❞❡✜♥❡❞ ♦♥

N(n0 )

❬✸❪✳

✶✸


❘❡♠❛r❦ ✶✳✶✳✺✳ ❚❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✭✶✳✾✮ ✐s ❛❝t✉❛❧❧② ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤♦✐❝❡

♦❢ Pn0 −1 ✳ ❆ ❣✐✈❡♥ ✐♥✐t✐❛❧ ✈❡❝t♦r y0 ✐s s❛✐❞ t♦ ❜❡ ❝♦♥s✐st❡♥t ✇✐t❤ ▲❙❉❊ ✭✶✳✶✮ ✐❢
Qn0 −1 y0 = Tn0 Qn0 G−1
n0 qn0 ✳ ❚❤❡♥✱ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r ✭✶✳✶✮ ✇✐t❤ ❝♦♥s✐st❡♥t

✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ y(n0 ) = y0 ❛❞♠✐ts ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥✳
◆❡①t✱ ✇❡ ❝♦♥s✐❞❡r ❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥ ✭✶✳✷✮✱ ✇❤❡r❡
▲❡t ✉s ❞❡✜♥❡

z(n) = Pn−1 x(n)✳

❚❤❡ r❡❣✉❧❛r ♦r❞✐♥❛r② ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥

z(n + 1) = Pn G−1
n An z(n)
✐s ❝❛❧❧❡❞ t❤❡

❢♦r ❛❧❧

n ∈ N(n0 )

✐♥❤❡r❡♥t ✭r❡❣✉❧❛r✮ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥

✇✐t❤ t❤❡ ♣r♦❥❡❝t♦r ♣❛✐r

qn ≡ 0, n ∈ N(n0 )✳

✭✶✳✶✵✮

♦❢ ▲❙❉❊ ✭✶✳✷✮ ❛ss♦❝✐❛t❡❞

Pn , Qn ✳

❋r♦♠ ♥♦✇ ♦♥✱ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t② ❛♥❞ ❥✉st ❢♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t②✱ ✇❡
✉s❡ t❤❡ ✉♥✐q✉❡❧② ❞❡✜♥❡❞ ❝❛♥♦♥✐❝❛❧ ♣r♦❥❡❝t♦r ♣❛✐r

Pn

❛♥❞

Qn

✐♥st❡❛❞ ♦❢ ❣❡♥❡r❛❧

♦♥❡s✳ ❇② ▲❡♠♠❛ ✶✳✶✳✸✱ ❡q✉❛t✐♦♥ ✭✶✳✶✮ ✐s ❞❡❝♦✉♣❧❡❞ ❛s

−1
Pn y(n + 1) = Pn G−1
n An Pn−1 y(n) + Pn Gn qn
Qn−1 y(n) = Tn Qn G−1
n qn .

❢♦r ❛❧❧

n ∈ N(n0 ),

✭✶✳✶✶✮

❋r♦♠ ✭✶✳✶✶✮✱ ✐t ✐s ❡❛s② t♦ s❡❡ t❤❛t ✭✶✳✷✮ ✐s ❡q✉✐✈❛❧❡♥t t♦

Pn x(n + 1) = Pn G−1
n An Pn−1 x(n)
Qn−1 x(n) = 0.

❢♦r ❛❧❧

n ∈ N(n0 ),

✭✶✳✶✷✮

❲❡ ♥♦✇ ❝♦♥str✉❝t t❤❡ ❈❛✉❝❤② ♦♣❡r❛t♦r ❢♦r ❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥ ✭✶✳✷✮✳ ❚❤❡r❡
❡①✐sts ❛ ✉♥✐q✉❡ ♠❛tr✐① ❢✉♥❝t✐♦♥ ❞❡♥♦t❡❞ ❜②

(Φ(n, m))n

m s❛t✐s❢②✐♥❣

Pm−1 (Φ(m, m) − I) = 0.

En Φ(n + 1, m) = An Φ(n, m),
❚❤❡ ♠❛tr✐① ❢✉♥❝t✐♦♥

(Φ(n, m))n

m ✐s ❝❛❧❧❡❞ t❤❡ ❈❛✉❝❤② ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞

✇✐t❤ ▲❙❉❊ ✭✶✳✷✮✳ ❇② ✉s✐♥❣ t❤❡ ❞❡❝♦✉♣❧❡❞ s②st❡♠ ✭✶✳✶✷✮✱ t❤❛t ✐s ❝♦♥str✉❝t❡❞
✇✐t❤ ❝❛♥♦♥✐❝❛❧ ♣r♦❥❡❝t♦r ♣❛✐r

P n , Qn ✱

✇❡ ♦❜t❛✐♥

m

Pk G−1
k Ak ; n > m

Φ(n, m) =

n0 ,

❛♥❞

Φ(m, m) = Pm−1 .

k=n−1
❉✉❡ t♦ t❤❡ ✜rst ❡q✉❛❧✐t② ♦❢ ✭✶✳✹✮✱ t❤❡ ❡q✉❛❧✐t②
❢♦r ❛❧❧

n

m

n0 ✳

❈❧❡❛r❧②✱

(Φ(n, m))n

m s❛t✐s✜❡s t❤❡ r❡❧❛t✐♦♥

Φ(n, m) = Φ(n, k)Φ(k, m)

✶✹

Φ(n, m) = Φ(n, m)Φ(m, m) ❤♦❧❞s

❢♦r ❛♥②

n

k

m.