I'm studying Control of PDEs at the University from this site and these notes and there are some questions I am not capable of resolve. I've never studied EDPs...
So, there are two questions related to the optimatity problem of a capcitor in these notes I would like you to helped me. The problem we are studiyng is:
And the question I would like to solve and understand are:
- Interpretation if $a=0$ or $b=0$.
- Prove or disprove the convergence of this iterative method:
I am totally lost so any informations, include basic information will be very very apreciated. Thank you.
(1) $a=0$ implies that you do not care about the output $y$, in this case $u=0$ is the optimal control, which is boring.
$b=0$ means you do not penalize control cost. The problem is no longer coercive. If $U_{ad}$ is unbounded the problem might not be solvable.
(2) This iterative method is a fixed point method:
Given $u_{n-1}$ compute $y_n$ from (1.21), then $p_n$ from (1.22), then solve (1.23) for $u_n$. Since $b>0$ (1.23) is uniquely solvable. Its solution being the projection (in the $L^2$ sense) of $-\frac ba p_n$ onto $U_{ad}$.
This fixed point mapping is a contraction if $\frac ba$ is sufficiently large.