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My question is about an article by J.L. Lions 1, where he introduced the "Hilbert Uniqueness Method" (HUM) for finding a boundary control function (dirichlet action) to bring the system to rest within a finite time T.

Regard the initial value problem to the wave equation,

$$\partial_{tt}u(x,t)-\Delta u(x,t)=0\hspace{.2in}\text{in } \Omega\times(0,T)$$ with initial values $$u(x,0)=u_0,~ \partial_t u(x,0)=u_1$$ and boundary term $$u=v\hspace{.2in}\text{on }\partial\Omega\times (0,T).$$ The task is to find a control function $v$ on (parts of) the boundary $\partial \Omega$ so that the system stops until $t=T$, i.e. $u(x,T)=\partial_t u(x,T)=0$.

The HUM starts with writing down the 2 wave systems

$$\phi''-\Delta\phi=0\hspace{.2in}\text{in } \Omega\times(0,T),$$ $$\phi(0)=\phi_0,~\phi'(0)=\phi_1\hspace{.2in}\text{in } \Omega,$$ $$\phi=0 \hspace{.2in}\text{on } \partial\Omega\times(0,T),$$ and $$\psi''-\Delta\psi=0\hspace{.2in}\text{in } \Omega\times(0,T),$$ $$\psi(T)=\psi'(T)=0\hspace{.2in}\text{in } \Omega,$$ $$\psi=\partial \phi/\partial \nu\hspace{.2in}\text{on } \partial\Omega\times(0,T).$$ Now, Lions considers the map $$\Lambda\{\phi_0,\phi_1\}=\{\phi'(0),-\phi(0)\}.$$

The following argumentation is that if $\Lambda$ is invertible, the whole problem is solveable by considering $\Lambda\{\phi_0,\phi_1\}=\{u_1,-u_0\}$, since both systems obtain a unique solution. You then just solve for $\phi_0,\phi_1$, solve the first system, calculate the normal derivative of $\phi$ at the boundary for all times $t\in(0,T)$, insert in the second system and obtain a solution for a system that comes to rest at time $T$, so that the evaluation of $\psi$ at the boundary delivers the desired control.

My question: Why the choice $\psi|_{\partial\Omega}=\partial \phi/\partial \nu$? I mean, through a rather technical proof, he was able to show that with that choice, $\Lambda$ is invertible for sufficently large $T$. But I don't get the idea, intuitively.

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1 Answer

There's no intuition here! It is a classical Neumann Boundary value problem. Basically, this condition gives you information about the "angle" with which your solution hits the boundary. Recall that $\nu$ is the normal to the boundary itself.

The map $\Lambda$ is the Dirichlet to Neumann map.

Hope this help!

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