Good day!
We solve an optimal control problem $$ J(u) = \|y - y_d\|^2 \to \inf $$ where $y$ is a solution of the PDE $$ \frac{dy}{dt} + Ay = Bu. $$ $A$ is a nonlinear operator, $(Bu, v) = \int\limits_\Gamma \gamma u^4 v d\Gamma$,
$y \in L^2(0,T;V)$, $u \in L^\infty(\Gamma)$.
The set of admissible controls is $$ U_{ad} = \{ u \,\, \colon \, 0 \leq u \leq M \}. $$
Our goal is proving the solvability of the solution of the optimal control problem.
We consider a minimizing sequence $u_k$: $J(u_k) \to j = \inf J$.
Sinse $U_{ad}$ isn't compact we should use weak convergence. Here we may suppose that $u \in L^2(\Gamma)$ because $L^\infty(\Gamma)$ isn't reflexive. Thus, there exists a subsequence $u_k \in U_{ad}$ that converges weakly to $u \in U_{ad}$.
Next we should to pass to the limit in the equation so we need that $u_k^4 \to u^4$ weakly. But in the general case it isn't so.
How to prove the existence of optimal control?
Thanks for your help!
You can interprete $u^4$ as a new control, $\tilde u$, with constraints $0\le \tilde u \le M^{1/4}$. Then the operator $B$ becomes linear, $$ B(u,v) =\int_\Gamma \gamma u^4 vd\gamma = \int_\Gamma\gamma\tilde u vd\gamma, $$ which makes passing to the weak limit an easy task.