Invariance method for extending Liapunov function to distributed parameter syste

Infante, E. F.

Slemrod, M.

English

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AN INVARIANCE PRDTCIPIX FOR DYPTAMICAL SYSTEMS IN HILBERT SPACES 
M. Slemod* 
and 
E. F. Infante** 
Center fo r  Dynamicd Systems 
Brown University 
Providence, Rhode Island 
y the National. Science Foundation, 
y the U. S. Army Research Office,  
1 Science Foundation, 
eronautics and Space 
d i n  pa r t  by the U. S. . DA-31-124-ARO-D-270, 
https://ntrs.nasa.gov/search.jsp?R=19690001002 2020-03-12T05:27:14+00:00Z
AN INVmIANCE PRINCIPLE FOR DYNAMICAL SYSTEMS I N  HILBERT SPACES 
In  recent years the extension of t he  second method of Liapunov 
to dist r ibuted parameter systems has gained much attenkion. Movchan 
[l] and- Zubov [2] have given r e su l t s  paral le l ing Liapunovts c l a s s i ca l  
theorems. I n  applications, however, these theorems are now always 
useful because of t h e i r  ra ther  s t r ingent  hypotheses. The approach 
taken to weaken the hypotheses of Liapunov's theorems for  autonomous 
systems in  f i n i t e  dimensions has been to s t a t e  an invariance principle 
[ 3 ] .  Analogously, it has become necessary to investigate 5.nvariance 
principles fo r  i n f i n i t e  dimensional problems i .e .  a study of abstract  
dynamical systems [ 4,5,6,7]. In eer ta in  mixed i n i t i a l  boundary value 
problems, d i f f i c u l t i e s  i n  applying recent resu l t s  have made new out- 
looks seem valuable, 
One such outlook i s  the approach taken i n  t h i s  paper. Specifi- 
ca l ly  the new concept of a weak dynamical system has been added to the  
concept of a dynamical system. 
h 
L e t  R+ denote the  in te rva l  [O,m) and 9 a Banach space 
with IIOII denoting the norm of an element @ 6 9. 
529 
Definition 1. We say u i s  a dynamical system on a Banach space 
+ 
i f  i) u i s  a mapping of R x in to  9, ii) u( t ,@)  i s  con- 
tinuous i n  (%,@), M i )  u(O,@) = @, and iv )  u(t+.c,@) = u(t ,u(a ,@))  
fo r  t,.c 2 0, @ E g.  The posit ive orb i t  Of(@) through @ E 9 
i s  defined as -I- O (@) = u u( t ,@) .  
t 2 O  
2 
Definition 2. We say u i s  a weak m a m i c a 1  system i n  a Banach 
space 9 i f  i) u i s  a mapping of R x -@ i n to  -@, ii) u(t ,@) I- 
i s  we&ly cm%inuous, i.e. i f  
u(tn,Qn) +u( t ,@) ,  iii) 
f o r  %,a 5 0, @ E 9. The posit ive orb i t  i s  as i n  Definition 1. 
It-tnl -+ 3 and @ 3 0 then n 
W 
u(O,@) = 9, and iv) u(t+z,@) = u(t,u(z,@)) 
Definition 3 .  A s e t  M i n  -@ i s  an invariant s e t  of the  weak 
dynamical system u if  fo r  each @ i n  M, there  ex is t s  a rUnction 
U ( t , @ ) ,  U(O,+) = 4, defined and i n  M for t E (-w,w), such t h a t  for  
any u C (-..,a), u(t,U(a,@)) = U ( t + a , @ )  f o r  t E R . (In particular,  
f o r  
-I- 
-I- t E: R , u( t ,@)  = U ( t , @ ) ,  @ E M). 
With t h i s  s t ructure  an invariance pr inciple  may be stated.  
L e t  58 and be separable Hilbert spaces and assume 
I: 9 3 9 i s  completely continuous. Let u: R+ x 9 3 9 be a 
weak dynamical system on 58. Let V be a continuous scalax func- 
t i ona l  i n  the larger  Hilbert  space 9 and define 
x 
Definition 4. We say a scalar functional. V i s  a Liapunov functional - 
on a s e t  G of 9 i f  2) V i s  a continuous functional ( in  the  .% 
topology), ii) V i s  bounded from below on G md, 'iii) ? ( c p )  5 0 
for all rp i n  G. 
Definition 5. Let R = {rp i n  9: $ ( c p )  = 0) and denote by M the 
3 
l a rges t  ;e+, i n  R 
tem. 
invariant with respect to the  weak dynamical sys- 
Then: 
Theorem 1 (Invarimce Principle): If i) v i s  a Liapun0.J function& 
3. i n  G c @ c g, ii) an o rb i t  0 ( c p )  belongs to G, G i s  a closed 
and bounded s e t  i n  -@, then weak d i s t  (u(t,cp),M) -+ 0 as t -+ 00, 
Here weak d i s t  (u(t,cp),M) e inf  p(u(t,cp),m), where p represents 
the metrized weak topology of closed, bounded se t s  i n  
53 
meM 
9. 
I n  appl-ications it i s  always assumed t h a t  the  system under con- 
sideration i s  both a dynamical system and a weak dynamical system on 
.@. 
proven. 
A s  motivation f o r  t h i s  assumption the follovitng theorem i s  
Theorem 2': The wave equation describes a weak dynamical system on 
h w2 (l)(n) x L2(R). 
The two major applications gLven i n  the paper are  
1. Tunnel diode system [ k ] ,  described by the system of equations 
L i  = -v - Ri3 t X 
O S X X l ,  t > O  
-Cvt = i 3- Gv, 
X 
and boundary conditions 
0 = E - V ( O , t )  - Roi (O, t ) ,  
-C,v(l,t) t = -i(I,t) -5 f(v(1,t)); 
where f (v)  has the  graph of the  fom. sl?_own i n  Figvre 1. 
Figure 1. 
I n  t h i s  example Theorem 1 i s  used to give suf f ic ien t  conriitions for  
complete s t a b i l i t y  . 
+? 
2. Asymptotic behavior of the equations of l i nea r  - themnoelasticity 1' 
i n  t h i s  case considered i n  t h e  form 
t > o  
R f i n i t e  n-dimensional domain 
5 
In t h i s  example Theorem 1 gives suff ic ient  conditions for  the  
system t o  approach the  equilibrium state 
as t -> 03. 
References 
c 1-1 
r21 
c31 
143 
P I  
h 
161 
r71 
181 
Movchan, A.A., The d i r ec t  method of Liapunov i n  s t a b i l i t y  
problems of e l a s t i c  systems. Prik. M a t .  Meh. 23 483-493 - -- - -2 
(1-959) 
Zubov, V.I.,  "The Methods of Liapunov and Their Application". 
Leningrad, 1957. U. S. Atomic Energy Cornmission, AEC-TR-4439. 
LaSaLle, J.P., I t A n  invariance principle i n  the theory of 
s tabi l i ty" ,  i n  International Symposium on Different ia l  Equa- - -- 
t ions and Dynamical Systems, ed, J. Hale and J.P. LaSalle 
(New York: 
-- 
Academic Press, 1967), p. 277. 
Brayton, R.K. and W.L. Miranker, A s t a b i l i t y  theory for Eon- 
l inear  mixed i n i t i d  value prcbiems. Arch,, Rat. Nech. 
Analysis 17, 338-376 (1964). 
-- -. 
Hale, J.K. and Infante, Eel?,, Extended dynamical systems and 
s t a b i l i t y  theory. Proc. PITat'l Acad. Sei., U.S.A. - 58 (1967)> - -
504-509. 
Hale, J.K., Sufficient conditions for  s t a b i l i t y  and in s t ab i l i t y  
of autonomous functional d i f fe ren t ia l  equations. J. Diff. - -  
Eqs, 1 (1965), 432-482. 
Hale, J.K., Dynamical systems and s t ab i l i t y .  
Journal of Math. Anal. and Applications.) 
Dafermos, C.M., On the existence and t he  asym-p'totic sta,bil i ty 
of so!.utions to the equattons of l i r e a r  thermoelasticity. 
(To appear, 
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Arch. Rat, Mech. Analysis 3 241-271 p968).  ---


An invariance principle for dynamical systems in Hilbert spaces

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