behavior of q as a function of a. These results follow from the results developed in Sec-
tion 6 and some simple observations about the form of the elements of P.
From Eq. 4.14-4.16 and Eq. 4.22-26, all elements of the state transition matrix are
rational functions of a with denominator polynomial (1 + a)ML. Thus, for a > 0
(1 +aM P'l -I = I (1 + a)MLp-- (l + a)MII
=10,,-(1+00^ 11
where every element of Qn (and hence the value of I (1 + a)ML"I ) is a polynomial in
a. Further, since row m in Q- is zero, the polynomial value of the determinant includes
the factor (1 + a)ML. Consequently
(l + a)ML'- II P = (0) Eq. 7.1
for O,(a) some polynomial function of a. Proposition 7.1 below follows.
Proposition 7.1: For all a > 0, the value of the determinant Pj 1 is a rational function
of a with nonzero denominator polynomial (1 + oa)ML'-1I)
By applying Eq. 7.1 to Proposition 6.7, the components of q, can be written as
%5(00) e(a)
qf(m) = =O--(0) .(o) Eq. 7.2
IC XO e)((Xa) '
in S'
Hence, the q,(m) are rational functions of a, and since a rational function is continuous
everywhere its denominator polynomial is nonzero, application of Proposition 6.9 and
Eq. 7.1 (which together establish that E(a) = 8()a) 0) to Eq. 7.2 yields the following.
Proposition 7.2: For all a > 0, the components of q, are continuous rational functions of
the independent variable a.
Further, differentiation of Eq. 7.2 with respect to alpha yields a rational function of a