Simple LMI require properties

[mohmmed ahmed]

Hi

can you help me ... the question  in the attachment

Thanks

[Johan Löfberg]

They can not be equivalent. If A is Hurwitz, the second LMI immediately says P>0  (A hurwitz equivalent with A'P+PA <0 and P > 0). The first LMI though says P < 0.

[mohmmed ahmed]

iam sory the LMI >0

and P>0

in this case

are they equivalent

[Johan Löfberg]

Just look at the scalar case,in which the upper LMI, with switched sign and P>0, simplifies to a^2 < 1/4, i.e says -0.5 < a < 0.5. The second Lyapunov equation though says a < 0. They describe two completely different sets

[mohmmed ahmed]

I think there are equivalent if we change from > to < as shown in the attachment

could you see the new attachment     is it right

[Johan Löfberg]

Many mistakes

A'PA-P < 0

cannot be multiplied with inv(A) from right to obtain A*P-PA^-1 < 0, since the < operator only is applicable when both sides are symmetric and you must thus perform congruence transformations, hence the resulting constraint makes no sense. This is also evident in your final result which has A in the diagonal, which only makes sense if A is symetric

In the other LMI, you claim P-A^TPA>0 is equivalent to [P A';A P]>0. This is also wrong. You've missed an inverse in the Schur complement

[mohmmed ahmed]

Johan

this mean there is no way to convert   [P   A'P+PA; A'P+PA    P]  to

 [  A'P+PA      P;  P   A'P+PA   ] 

is this right??

[Johan Löfberg]

How many different proofs do you require?

Here is another one. Let A=0 and consider scalar case. The first condition simplifies to P<0, and the second condition is infeasible for any P