Liapunov functions and exact differential equatio

Chen, C.-S.

Kinnen, E.

English

NASA Technical Reports Server

Kinnen, Chen 
Lyapunov Fn. 
LYAPUNOV FUNCTIONS AND THE . 
EXACT DIFFERENTIAL EQUATION 
ABSTRACT 
Lyapunov f u n c t i o n s  are considered from t h e  viewpoint 
of exac tness  of  t h e  d i f f e r e n t i a l  equat ions .  It i s  
shown tha t  t h e  search f o r  p o s s i b l e  Lyapunov func t ions  can 
be restated i n  terms of cond i t ions  f o r  exac tness  and s ign-  
d e f i n i t e n e s s .  T h i s  procedure al lows a g e n e r a l i z a t i o n  of 
many well-known techniques t h a t  have been descr ibed  i n  
t h e  l i t e r a t u r e .  
Manuscript r ece ived  December 2 ,  1966. The work 
r epor t ed  i n  t h i s  pape r  was supported by N A S A ,  Office' of 
Grants and Research Cont rac ts ,  under Grant NsG-574/33-019-014. 
.__ "I _l____-- 
J %%., -4.4 *n' 
https://ntrs.nasa.gov/search.jsp?R=19680017928 2020-03-12T10:46:01+00:00Z
LYAPUNOV FUNCTIONS AND THE 
EXACT DIFFERENTIAL EQUATION 
Edwin Kinnen 
Chiou-Shiun Chen 
Department of E l e c t r i c a l  Engineer ing 
Un ive r s i ty  of Rochester 
Rochester ,  New York 14627 
T e l :  GR 3-3000 
L 
Kinnen, Chen 2, 
Ly.apunov Fn. . 
LYAPUNOV F U N C T I O N S  AND T H E  
. 
EXACT D I F F E R E N T I A L  EQUATION 
I. I n t r o d u c t i o n  
If a d i f f e r e n t i a l  equat ion  i s  the time d e r i v a t i v e .  
, of a func t ion  o f  one or more t i m e  dependent v a r i a b l e s ,  i t  
can be sa id  t o  be an exac t  d i f f e r e n t i a l  equat ion;  t h e  
f u n c t i o n  i s  cal led the  f irst  i n t e g r a l  o f  t h e  d i f f e r e n t i a l  
' equa t ion  [l]. Consider a s y s t e m  descr ibed  by a set  o f  f irst  
o r d e r  autonomous d i f f e r e n t i a l  equat ions  
where x and - f are n dimensional v e c t o r s  and - 
fi. = fi  (x1,x2, ... , xn). 
t h e  ex i s t ence  of i t s  first p a r t i a l  d e r i v a t i v e s ,  unique s o l u t i o n s  
t o  eq.  (11, and an equal ibr ium p o i n t  a t  - -  x=O. 
Assume: t h e  d e f i n i t i o n  of  f i  and 
Defining a new vec to r  g such t h a t  
+ f  + ... + fn ,  fi-l . i + 3 _  gi = -fl - f* 0 . .  - 
Kinnen, Chen 
Ly.apunov Fn. . 
3 
. 
. the sum of giGi f o r  a l l  i can be shown to be zero, ,  so  tha t  
eq. (1) can be r e w r i t t e n  as a s c a l a r  i n n e r  product  
- 
Suppose tha t  there e x i s t s  a f u n c t i o n  h (x )  - - such tha t  
the a d d i t i o n  o f  < h , b  - - = <h, f>  - -  t o  eq. ( 2 )  w i l l  r e s u l t  i n  an 
exac t  d i f f e r e n t i a l  equat ion  
e 
A first  i n t e g r a l ,  V(x), can then b e  found such t h a t  
Genera l iz ing  a procedure f o r  s o l v i n g  an e x a c t  d i f f e r e n t i a l  
equat ion  f o r  n = 2 [l], i t  can be shown, f o r  example, t h a t  
> 
\ 
Therefore ,  i f  w e  are s u c c e s s f u l  i n  f i n d i n g  an h - which i n s u r e s :  
(a) t h e  exac tness  o f  the l e f t  hand s ide  of eq. ( 3 ) ,  
(b) t h e  nega t ive  semide f in i t eness  of t h e  r i g h t  hand 
s i d e  of eq. ( 3 1 ,  and 
( c )  t h e  p o s i t i v e  d e f i n i t e n e s s  of  the f i r s t  i n t e g r a l ,  
e q -  (51, 
then V(x) - i s  a Lyapunov func t ion  w i t h  r e s p e c t  t o  eq. (l), 
by d e f i n i t i o n  [3]. Using an argument similar t h e  one found 
i n  [l], it fo l lows  t h a t  t h e  necessary and s u f f i c i e n t  cond i t ion  
for (a)  i s  that :  
(a ' )  the ma t r ix  
be symmetric. 
Kinnen, Chen 
Lyapunov . Fn . 
. 
Note t ha t  these three cond i t ions  r e p r e s e n t  s u f f i c i e n t  
s t a t emen t s  f o r  t h e  development o f  a Lyapunov func t ion .  
11. C a l c u l a t i o n  o f  h -
The problem of f i n d i n g  a Lyapunov func t ion  f o r  
eq. (1 )  has been restated as a problem i n  c a l c u l a t i n g  the  
components of  a v e c t o r  h - such t h a t  cond i t ions  (a)  or ( a ) )  - ( c )  
are satisfied. A v a r i e t y  of methods can be  considered t o  
a i d  t h e  sea rch  f o r  a s u i t a b l e  form f o r  h. Some of  these are 
summarized below and are r e a d i l y  i d e n t i f i e d .  w i t h  f-amiliar 
. .  
- 
techniques f o r  developing Lyapunov f u n c t i o n s .  
i) The s i m p l e s t  ca se  e x i s t s  when a l l  hi = 0 sat isf ies  ( a ) .  
T h i s  imp l i e s  tha t  eq. ( 2 )  i s  e x a c t .  Condition ( b )  i s  
sat isf ied and only ( c )  needs t o  be considered.  I f  the  
a p p l i c a t i o n  o f  eq. (5) r e s u l t s  i n  a func t ion  t h a t  sa t isf ies  
( c ) ,  i t  i s  a Lyapunov func t ion .  T h i s  procedure has been 
called the method of f irst  i n t e g r a l s  [2,3]. 
ii) If eq. ( 2 )  i s  not  e x a c t ,  a l l  hi cannot be s e t  t o  zero ,  
as i n  (i). Without t r y i n g  t o  d i f f e r e n t i a t e  between g - and h ,  
we  could proceed, f o r  example, by t r y i n g  t o  s e l e c t  a - VV 
which sa t i s f ies  ( a ) )  and ( b ) .  I f  t h i s  i s  accomplished, 
( c )  can then  be considered.  I f  ( c )  i s  not  sa t isf ied,  t h e  
. 
procedure might be repeated f o r  ano the r  - VV s a t i s f y i n g  ( a '  ) 
and (b), e t c .  
iii) 
represen ted  by l i n e a r  and n o n l i n e a r  terms o f  t h e  dependent 
For a g iven  d i f f e r e n t i a l  equat ion  (2), each gi i s  
variables, such t h a t  w e  could d e f i n e  
I 
where ga  con ta ins  only t h e  l i n e a r  terms. 
i s  s i m i l a r l y  chosen, t h e  procedure i n  (ii) could be followed 
If h - = ha  - + hn 
by  s e l e c t i n g  a l i n e a r  and n o n l i n e a r  p a r t  of - VV. Equation (5)  
can then b e  w r i t t e n  as 
The first of t he  two i n t e g r a l s  r e so lves  i n t o  a q u a d r a t i c  form 
i n  - x, and eq. ( 7 )  i s  seen t o  be t h e  familiar q u a d r a t i c  p l u s  
i n t e g r a l  form [SI .  If t h e  n o n l i n e a r i t y  of t h e  system i s  
known a n a l y t i c a l l y ,  i t  may a l s o  be p o s s i b l e  t o  eva lua te  t h e  
Kinnen, Chen 7 
” Eyapunov Fn, 
. 
second i n t e g r a l .  Otherwise t h e  s a t i s f a c t i o n  of ( c )  can be 
considered,  i n  p a r t ,  d i r e c t l y  through t h e  i n t e g r a l  c h a r a c t e r i s t i c s  
of t h e  n o n l i n e a r i t y  [5].  
i v )  Rather than  e f f e c t i v e l y  s o l v i n g  f o r  g, i n  eq. ( 6 )  as i n  
-(iii), but  t o  improve t h e  l i k e l i h o o d  o f  s a t i s f y i n g  ( c ) ,  w e  
might write, f o r  example, 
- allxl + a12x2 + ... + alnxn 
gl * hl 
t hn - anlxl + an2x2 + . + 2xn, 
gn 
where t h e  a have a cons tan t  par t  p l u s  a func t ion  of 
i j  
e.g., a = aijk t aijv. The aii can 
(x  1 - a i i v  i 
n-1 13  x1,x2 e.., x 
- f u r t h e r  be l i m i t e d  such t h a t  aiik>O and aiiv 
t o  f a c i l i t a t e  s a t i s f y i n g  ( c ) ,  e t c .  Proceed by s e l e c t i n g  t h e  
undetermined cons t an t s  i n  eq. ( 8 )  t o  f i r s t  s a t i s f y  ( b )  and 
then  t h e  remaining terms i n  t h e  aij t o  s a t i s fy  ( a ) ,  L a s t l y  
' Kinnen, Chen 8 
I ,  - Lyapunov Fn. 
cond i t ion  ( c )  would be checked fo l lowing  t h e  use  of eq . (5 ) .  
T h i s  corresponds to a v a r i a b l e  g r a d i e n t  method [6]. 
v) 
parts hl - t 91, where. Ql i s  selected s p e c i f i c a l l y  t o  s a t i s f y  
Al t e rna te ly  cons ide r  h - i n  eq. ( 3 )  sepa ra t ed  i n t o  two 
(a )  only. h2 - could then  b e  chosen w i t h  r e s p e c t  to cond i t ion  t 
( b ) ,  and f i n a l l y  cond i t ion  ( c )  would be examined. I f  ( c )  cannot 
. be s a t i s f i ed ,  an a l t e r n a t e  h2 - may be  considered.  T h i s  procedure 
i s  a g e n e r a l i z a t i o n  of a method desc r ibed  i n  the  l i t e r a t u r e  
by I n f a n t e  and by Walker, a l though t h e i r  o b j e c t i v e s  are l e s s  
apparent  [4,71. I n  t h e i r  terminology t h e  "new modified 
s y s t e m "  would b e  equ iva len t  t o  eq. ( 3 ) ,  where 
v i )  If one i n i t i a l l y  s e l e c t s  a func t ion  as a f irst  i n t e g r a l  
such as t o  s a t i s f y  ( c )  and i f  t h e  second pa r t i a l  d e r i v a t i v e s  
e x i s t ,  t hen  ( a 1 )  i s  a l s o  satisfied,  and cond i t ion  ( b )  need 
only be considered.  Another first i n t e g r a l  can be t r i e d  i f  
cond i t ion  ( b )  i s n ' t ' s a t i s f i e d .  While t h i s  corresponds t o  t h e  
method o f  p rognos t i ca t ion  which so o f t e n  confuses  t h e  neoph i t e ,  
it i s  a l s o  t h e  basis o f  t n e  more s o p h i s t i c a t e d  method of  squa r ing  - 
proposed by Krasovski i  [ 31. 
- I , Kinnen, Chen ' 9 .  Ly-apunov Fn - # *. 
0 
v i i )  Using a somewhat d i f f e r e n t  technique  from those  mentioned 
above, w e  migh t  beg in  i n i t i a l l y  by s e l e c t i n g  a func t ion  
Q, = eh , f> ,  - -  t h e  r i g h t  s ide  o f  eq. ( 3 ) ,  t o  s a t i s f y  ( b )  bu t  a l s o  
to a l low a s o l u t i o n  f o r  V d i r e c t l y  from eq. (4) without  
appea l ing  t o  (a) and eq. ( 5 ) .  If t h i s  can be done and ( c )  
i s  sat isf ied,  a Lyapunov func t ion  has been found and (a )  has 
been au tomat i ca l ly  sat isf ied (assuming s u f f i c i e n t  con t inu i ty  
cond i t ions ) .  If ( c )  cannot b e  s a t i s f i ed ,  a more s e l e c t i v e  
choice  o f  $ may be sought .  T h i s  i s  t h e  approach proposed by 
Zubov [3,8].  I n  e f f e c t ,  one attempts t o  circumvent a d i r e c t  
cons ide ra t ion  o f  cond i t ion  (a )  and eq. ( 5 )  by s o l v i n g  t h e  
par t ia l  d i f f e r e n t i a l  equat ion  ( 4 ) .  
I11 Summary 
The cons t ruc t ion  o f  Lyapunov func t ions  f o r  t h e  s tudy  
of  t h e  s t a b i l i t y  p r o p e r t i e s  o f  non l inea r  d i f f e r e n t i a l  equa t ion  
s o l u t i o n s  has been cons idered  from a un i fy ing  pe r spec t ive .  The 
exac t  d i f f e r e n t i a l  equa t ion  appears t o  provide  the  common base 
f o r  many of  t h e  seemingly un re l a t ed  methods sugges ted  i n  t h e  
l i t e r a t u r e .  But i n  a d d i t i o n  i t  i s  now a l s o  p o s s i b l e  t o  
beg in  i n v e s t i g a t i n g  new techniques  more sys t ema t i ca l ly .  
Kinnen, Chen 1 0  
~ Lyapunov Fn. 
"*" ,* I 
> e 
J 
IV Reference 
. .  
[l] Cohen, A., A n  Elementary Treatise on D i f f e r e n t i a l  Equation, 
D.C. Heath and Company, N.Y., 1933. 
[2] Drake, R.L., Methods f o r  Systematic  Generation o f  Lyapunov 
Funct ion,  Pq r t  I and 11, NASA N66-10626 & 7 (1965) 
[3] Hahn, W. ,  Theory and Applicat ion of Lyapunov's D i rec t  
Method. Prent ice-Hal l ,  Englewood C l i f f s ,  N . J . ,  1963. 
[ 4 )  I n f a n t e ,  E.F., A N e w  Approach t o  the Determination of  the 
Domain o f  S t a b i l i t y  of  Nonlinear Autonomous Second Order 
System. Doctoral  Thes is ,  Univers i ty  of Texas, 1962. 
[ 5 ]  Lure, A . I . ,  Some Nonlinear Problems i n  t h e  Theory of  
Automatic Cont ro l ,  Her Majesty's S t a t i o n e r y  Of f i ce ,  London, 1957. 
1[6J Schutz ,  D.G. and JOE'. Gibson., The v a r i a b l e  g r a d i e n t  
method f o r  g e n e r a t i n g  Lyapunov f u n c t i o n s .  Trans.  Am. I n s t .  
of E l e c t r i c a l  Engineers,  N.Y. 81:203-210 (1962). 
[7] Walker, J . A . ,  An I n t e g r a l  Method of Lyapunov Funct ion 
Generation for Nonlinear Autonomous Sys tems.  Doctoral  
Thesis ,  Univers i ty  o f  Texas, 1964 . 
[8] Zubov, V . I . ,  Methods o f  A.M. Lyapunov and T h e i r  Appl ica t ion .  
(Engl i sh  T r a n s l a t i o n )  P. Noordhoff, Groningen, The Nether lands,  
1964. 


Lyapunov functions and the exact differential equation

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server