ECE6554: Homework #1
Due: Jan. 22, 2016
Problem 1. [6 pts] Using the sample code for using ode45 put up online, modify it to implement the simple
adaptive controller from the notes (beginning chapter):
p(t) = Lp (ap(t) + (t)
(t) = k(t)p(t)
k(t) = p2 (t)
U
ECE6554: Homework #2
Due: Feb. 5, 2016
Problem 1. [6 pts]
Find the Lipschitz constant on the region x [1, ) or prove none for f (x) = 1/x.
Problem 2. [6 pts]
Show that f1 + f2 is locally Lipschitz, when f1 and f2 are locally Lipschitz.
Problem 3. [6 pts]
ECE6554: Homework #3
Due: Feb. 19, 2016
T
Problem 1. [8 pts] A gradient system is a dynamical system where x = grad V (x) for grad V (x) [DV (x)]
and V : D Rn R is C 2 (D; R).
1. Show that V (x) 0, x D, and V (x) = 0 if and only if x is an equilibrium poi
12
Lipschitz Continuity
Calculus required continuity, and continuity was supposed to require
the innitely little, but nobody could discover what the innitely
little might be. (Russell)
12.1 Introduction
When we graph a function f (x) of a rational variabl
Homework 4 Solutions
1. For each of the following initial value problems y + p(t)y : g(t)7 y(t0) 2 yo use the existence and unique-
ness theorem for linear lst order ODEs to determine the largest time interval containing to on which the
initial value prob