ylfink@water.UUCP (11/13/87)
DEPARTMENT OF COMPUTER SCIENCE
UNIVERSITY OF WATERLOO
SEMINAR ACTIVITIES
SCIENTIFIC COMPUTATION SEMINAR
- Thursday, November 26, 1987
Dr. Ron Stern, of the Concordia University, will speak
on
``Continuous vs. Discrete Reachability Codes''.
TIME: 4:00 PM
ROOM: MC 5097
ABSTRACT
nxn
Consider the linear o.d.e. x(t) = Ax(t), where A - R
is assumed to be essentially nonnegative; that is,
tA
a > 0 for i / j. (This is equivalent to e > 0
ij
n
t > 0.) The R -reachability cone X(A) is defined to be
t
the set of all initial states x(0) such that the
tA
trajectory x(t) = e x(0) eventually enters (and
remains in) the nonnegative orthant. Specifically,
n tA -tA n
X(A) = {x - R : e x > 0 for some t > 0) = U e R .
t>0 t
Now consider the discrete-time approximation of the
o.d.e. given by the Cauchy-Euler scheme with increment
h > 0:
x((k+1)h) - x(kh) = Ax(kh); k = 0,1,2,....
------------------
h
- 2 -
n
Let X(A,h) denote the associated discrete-time R -
t
reachability cone; this is the set of all initial
states x(0) such that the discrete trajectory x(kh)
n
enters R . Intuition suggests that as h | 0, the cones
t
X(A,h) in some sense approximate X(A). We can prove
that in fact more is true: There exists h > 0 such that
X(A,h) = X(A) whenever 0 < j < h. This result yields a
simple and stable numerical method for testing a given
point for containment in X(A).
November 13, 1987