sax

PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28

AL

(1=1, ..., n).

That is, the characteristic numbers of the whole system appear
in pairs with their reciprocals.
We now make the following assumption which is the joker
allowing us to avoid cyclic exceptions to the Turnpike
Theorem; that is, all characteristic numbers X; other than
\i=2X,.1 have absolute value different from unity. We also
assume, for simplicity, that 1 is the sole characteristic number
which is multiple. It can be shown that the solutions to (22)
and (29) may be written in the following forms:

b, = Ty + XA + +a, A
(32)
z,=[I+D+ … + D] «+o, + vee +a, kA, +
+BEnn1 + BaEn rad + ee FB.E A,

where

D=(1+7) [1 - (A0 + B°+C°)]-"(1 - d) (BOY,

and ©; ({=1,...,n) and §, (¢=1, .., 2n) are n-dimensional
column vectors; in particular, €, ; is the eigen-vector of D associated
 with X,,;  Scalars, &;, > Xp» By» ---> By Can be determined
 bv 2# equations:

zy =k + ob + + ok + BE + Bb t oo +86.

wrt

ZT >

1+D+...+DT Ja,ËE, + ab AT +. +E ATH
+BE,1 + BaEnnad 142 + FBLA

9] Morishima - pag. 16