I have a question about LQR. 
I apply optimal controller by solving Ricatti equation based on normal plant. Suppose that I have one or few parameter variations in the plant that changes some values in matrix A, and let me name it Anew, then the cost will be different from normal value, could either increase or decrease the value of cost function 
                         J=trace(x0'*P*x0)
where P is solved from new Ricatti equation 
                      (Anew-BK)'P+P(Anew-BK)+Q+K'RK

Now my question is, if now I face the new plant Anew and I still keep the same optimal controller solved from normal plant, I want to know what level of variations, or what value of matrix Anew will make the new cost function achieve some certain value I specified J0.

Assume that I don't need to worry about stability, then P will be symmetric. But the problem is, with given initial condition x0, there are still infinite solutions about P from equation trace(x0'*P*x0)=J0. I know there are various numerical approach to solve this Lyapunov equation, but none of them specifies the target value of P.

To make it easier, you can assume only one or two elements in matrix A change from normal.

Any help will be appreciated!

Clark