# Boundary controllability of the heat equation and observation

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.