Electrical

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473

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I{onlinear Systems: Homework ff2

Professor Wei Lin Dept. of Eiectrical Eng. & Computer Science Case Western Reserve UniversitY Cleveland, Ohio 44106, USA e-mail: linwei@case.edu : http / /nonlinear. case. edu/ -1i1*"i7

Due Feb. 17

1. Solve the following ODE by the Picard's iteration method.

(a) r: Ar, r(0) : so' r(1) :rl2; (b) r:lnlsinrf, (c) Does the nonlinear differential equation above have a unique solution on a neighborhood of (to, ro) : (7,r l2)? 2. Consider the non-lipschitz continuous system

* : r7/3, r(0) :

0.

(a) Find the two solutions for the ODE above; (b) Use the two solutions thus obtained to construct infinite-many soIutions foom the origin r(0) : 0, thus showing that a nonlinear differential equation may have infinite many solutions from the same initial condition, if a local Lipschitz condition is unsatisfied. 3. Problem 3.1

( and (6) 4. Problem 3.2 - )

5. Problem 3.7

(5),(7),(8)

6. Problem 3.8

7. Problem 3.31 (hint: define z(t) : r(a)+ Bellman inequalitv to do the problem)

fi f $,y(s))ds

and use Grownwall-

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il6

F";

dr*

fl*"dt

;

fio+ A*',(

2

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A ti(.+Atroir)

=

fi^ +At"^

'?(,

,,#

IeA'-t [(+)= il, (e"-D un;flt*

* = lrt

goln*tr;qa.

(t) +

lq,;"n

1u7=

Ltrlhn*

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dt :

r,+

o

olt = 1,+l

= 3+{

C:o

r:(t)=

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bla"rf*Q"ld+=

+l^r eto+

:!+c

zI.

=il'

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n-* r{r= da

id

r-3 :

o(r

l{L)

=

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*

d"r.,$

?rrr.t [g r,) ,.;{L x. +o T1,*

d4

oDe.

i

= zt

I

, Xtolr" I"qf in$nitll

i

3LA +o{ a1i,)o.l,o. A B,q $nr rn} $'rr7= j C"*lt eoeot;el. d,e oee *t [,h^" ;+'.\.rJ hn^{ srrutu"s

ll&t,'o -fd,x;l;-. Lllr-x[

=+

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