Consider the quadratic form $x^T(B^TP + PB)x$ where is $B=DA$, where $D$ is diagonal with positive or nonnegative entries, and $A$ is Hurwitz. Now consider a symmetric positive definite matrix $Q$ and that $P$ is the (positive definite) solution to the Lyapunov equation $A^TP + PA=-Q$. It is clear that if $D=I$ (the identity matrix), then $x^T(B^TP + PB)x = -x^TQx \le -\lambda_{min}(Q) \|x\|^2$. Is there any way to give some bounds probably depending on $D$ if this is not the case? i.e. I want to find some function of $D$ such that $x^T(B^TP + PB)x \le -f(D)\lambda_{min}(Q)\|x\|^2$. The matrix $A$ may even have some particular structure.
Quadratic bounds
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linear-algebra
control-theory