I'm lost, I don't even know what the question to ask! I'm studying a 1D wave equation, and I found the exact solution. using the D'alembert formula so far so good. But now, I'm feeling stuck I don't know what I can do with it! We usually try to get the controllability or observability (which are generally equivalent in one sense). What I am asking you is the following: do you have an idea, a comment or a reference in which they have used the D'alembert formula's solution to prove either controllability, stability, or observability? I know I have to read and understand the solution that I have, to be able to extract something, but I need an example for inspiration.
Thank you, Best regards.