21 December 2015 3 7K Report

Let T=[A B;C D] be a real (m+n)X(m+n) stable matrix with controllable pair (D,C). Choosing such matrices T uniformly at random:

What is the probability that the nonsymmetric algebraic Riccati equation

XCX-XD+AX-B=0 has solutions ?

Coveying a lot of computer experiments, I always came up with a solution to the above mensioned nonsymmetric algebraic Riccati equation.

On the other hand, for A=-I, D=-I,C=I, B=[0 1;0 0], the matrix T is stable (with eigenvalue equals to -1) and (D,C) is controllable, but there is no solution for

X^{2}=B.

I conjecture that the measure of the set of such matrices among all matrices T as above is 0, but I cant find any way to proove it. 

Can anyone suggest a good refference to such questions ?  

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