Finding riccati solution of A*X+A'*X+X*W*X+Q

[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

 

[tomkeus]

Shouldn't hamiltonian be a hermite operator $$H=H^{\dagger}$$. Then you would have W=Q*.

 

[gs]

ya itis right but how it explains the existence of X

 

[tomkeus]

If you could rephrase the text I might help you more.

 

[gs]

my point is to if H has real parts of eigen values greater than zero;which may be due to either A+W*X is having eigen values greater than zero;ordue to -(A+W*X)in which
case A+W*X has negative eigenvalues .hence we cannot say whether X exists or not just by looking at the any eigen values on imaginary axis ;means this condition for existence of X is not sufficient ,which is my understanding but in text it stated is sufficient ,i want to know how can it.

 

[tomkeus]

Well, sorry I cannot help you with that.

 

[gs]

thing is the relation of eigen values of H and eigen values A+W*X is valid only for X Stable.hence it is sufficient