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From: ylfink@water.UUCP
Newsgroups: ont.events,uw.talks
Subject: Continuous vs. Discrete Reachability Codes.
Message-ID: <1262@water.waterloo.edu>
Date: Fri, 13-Nov-87 13:09:20 EST
Article-I.D.: water.1262
Posted: Fri Nov 13 13:09:20 1987
Date-Received: Sun, 15-Nov-87 04:02:07 EST
Distribution: ont
Organization: U of Waterloo, Ontario
Lines: 64
Keywords: Dr. R. Stern, Thurs., Nov. 26/87, 4:00 PM, MC 5097.
Xref: utgpu ont.events:758 junk:6361



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