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

In many situations, the null controllability of a given system is equivalent to the observability inequality of the adjoint backward system, which explain the presence of the normal derivative as the adjoint of the controllability map. To prove observability inequality we often use global Carleman estimate, proved by Fursikov and Imanuvilov, as a powerful tool to prove such inequalities. You can look for articles by Zuazua and Cara for the null controllability of heat equation.