Boundary controllability of the heat equation and observation

Question

I'm studying the boundary controllability of the heat equation

\begin{array}{c} y_{t}=\Delta y\text{ in }\Omega \times (0,T), \\ y=\mathbf{1}_{\Gamma }u\text{ on }\partial \Omega \times (0,T), \\ u(0)=u_{0}\text{ .}% \end{array}

where $\Omega $ is an open in $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n},$ $\Gamma $ ia a portion of $\partial \Omega ,$ and $\mathbf{1}_{\Gamma }$ is the usual characteristic function and $u\in L^{2}(\Gamma \times (0,T))$ is the control.

In order to prove the null controllability of the above system, we prove the following observability inequality

$$\left\Vert \varphi (0)\right\Vert _{L^{2}(\Omega )}\leq C\int_{\Gamma }\left\vert \frac{\partial \varphi }{\partial n}\right\vert ^{2}d\Gamma$$ where $\varphi$ is solution of the backward heat system.

I want to know how can we obtain this observability inequality? From where the normal derevative comes? are there any articles or books which deal with these kinds of stuff? Thanks.

Answer 1

1. This equivalence between the null-controllability for a given system and the observability inequality for its adjoint backward system comes from a functional analysis theorem which states: given two linear operators $F$ and $G$ between Banach spaces into $Z$ then we have $$\mathrm{Im}\,F \subset \mathrm{Im}\, G \Leftrightarrow \exists C>0 \; \|F^* z\| \leq C \|G^* z\|, \forall z\in Z.$$ In your case, the normal derivative comes from the adjoint of controllability operator.

2. To prove the observability inequality we usually use Carleman estimate.

3. You can find sevral references in control theory, I recommend this one: Controllability of Partial Differential Equations by E. Zuazua.