Finding Riccati Solution of A*X+A'*X+X*W*X+Q: Hamiltonian Matrix H

[gs]

in finding riccati solution of

A*X+A'*X+X*W*X+Q that is

X which stabilises A+W*X(real parts of eigen values are <0) ,it’s existence can
Found out by
Eigen values of Hamiltonian matrix H given by


H MATRIX=
!A W!
!Q -A!
because we have the relation

EIGEN VALUE OF H ARE GIVEN BY= EIGENVALUES OF (A+W*x)& - (A+W*x);

In text it is stated as if there is no eigen values of H are on imaginary axis then X exists

Means it can have in real parts of ( eigen values can be >0)

This can be possible
If A+W*x has negative real parts

And also A+W*x has positive real parts in which it is un stable

If it is so how can we say that just H matrix not having eigen values on imaginary axis is
Sufficient for X toexist
Can anyone explain me about this
Thanking you

 

[gs]

this relation of eigen values of h and (a,w)is valid for x stable hence it is sufficient

 

[quantumdude]

in finding riccati solution of

A*X+A'*X+X*W*X+Q that is

I'm a little confused. What are the dimensions of these quantities? Are they matrices? Vectors? Scalars?

EIGEN VALUE OF H ARE GIVEN BY= EIGENVALUES OF (A+W*x)& - (A+W*x);

Is there some significance to the symbols & and ; here?