I do now know how to solve it with Julia are there similar solved problems with Julia that I can study?
Here both P\geq0 and lambda are unknown and \Pi \geq0 matrices are tunable.

So, you do now know or do not know? But since you are not showing the code, I guess that the latter holds.

Let me link here your previous post where you asked for a Julia code for what is obviously a previous version of the problem: Linear Matrix Inequality. Honestly, I think it was quite appropriate to continue there because this is really still the same story, isn’t it? I could then direct you to my previous responses in that thread. Namely: 1) your constraints are still not linear (well, you can do some simple search for lambda in the outer loop), 2) for general matrices A and B, the terms that you are constraining to be positive or negative definite will not be symmetric and not even square, are the expressions correct?, 3) your LMI constraint on a huge block diagonal matrix can be rewritten as a bunch of smaller constraints, leave it up to the parser to assemble these, your code will be simpler. But this I have already advised before.

Where do the matrices A, B, C, Q, R come from? What do we know about them? This particular choice of symbols reminds me of some LQ-optimal control stuff. The reason I am asking is that if, say, B matrix is tall, the term PB is tall too and what sense does it make to impose the (semi)definiteness constraint? Similarly, the term 2PA. For a general A, the product will not be symmetric. The term reminds me of some Lyapunov stuff, are you sure that you want 2PA? Isn’t it rather something like A’P+PA, which only reduces to 2PA in the scalar case?

Yes it is because P is a symmetric matrix, A, B, C are systems associated matrice, and R,Q, S are matrices required in the proof of the stability and all of them are square matrices.