In the context of robust control, I remember hearing that the two following problems are equivalent.

Find $P \succ 0$, such that $A P + P A^{\top} \prec 0$ for all $A \in \mathscr{A}$ where $$\mathscr{A} = \left\{ A+B \Delta C:\|\Delta\|_{\text{op}} \leqslant 1 \right\}$$

Find $P\succ0$ such that $$\left[\begin{array}{cc}A P+P A^{\top} & P B \\ B P & -C^{\top} C\end{array}\right] \preceq 0$$

There is another case where we require $$\mathscr{A} = \left\{ A+B \Delta C:\|\Delta\|_{\text{op}}\leq 1, \Delta \text{ is a diagonal matrix} \right\}$$ also known as diagonal norm bound.

**Question.** Are there any sources describing why these two formulations are equivalent, and also more information about their relation to the diagonal norm bound?

Despite searching on the internet for some time, I did not find anything relevant. This is also not in Boyd *et al*$^\color{magenta}{\dagger}$. I am pretty sure that this is an old result.

_{$\color{magenta}{\dagger}$ Stephen Boyd, Laurent El Ghaoui, Eric Feron, Venkataramanan Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Studies in Applied and Numerical Mathematics, SIAM, 1994.}