Optimal control of partial differential equations

Remarks

Example: let $\Omega \subseteq \R^d$ be a bounded open set, let $y_d \in L^2(\Omega)$, $\alpha > 0$ and $a < b$. Minimize $\frac12 \norm{y - y_d}_{L^2(\Omega)}^2 + \frac{\alpha}{2} \norm{u}_{L^2(\Omega)}^2$ over $u \in L^2(\Omega)$ with $a \le u \le b$ almost everywhere, where $y \in H^1_0(\Omega)$ is the weak solution of $-\Delta y = u$ in $\Omega$, $y = 0$ on $\partial \Omega$. This fits the definition with $Y = H^1_0(\Omega)$, $V = L^2(\Omega)$, $Z = H^{-1}(\Omega)$ and $e(y, u) = -\Delta y - u$. Here the problem has a unique solution, because the control-to-state map is linear and bounded. So the reduced cost is strictly convex, continuous and coercive on the closed convex set $V_{\mathrm{ad}}$. Non-example: if the state equation is an ordinary differential equation in $\R^n$, the problem is an ordinary optimal control problem, not one for a partial differential equation. When $e(y, u) = 0$ has a unique solution $y = S(u)$ for each $u$, the problem reduces to minimizing $f(u) = J(S(u), u)$ over $V_{\mathrm{ad}}$.