| Contents |
| We focus in this work on optimal control problems of the following form
\begin{equation}
\min_{q \in \mathcal{M}_I} J(y) = \frac{1}{2}\| y - y_d \|_{L^2(\Omega,L^2(I))}^2 + \alpha \| q \|_{\mathcal{M}_I}
\label{cost}
\end{equation}
where $y$ is the solution of the nonlinear Korteweg de Vries equation with a time dependent measure valued source term acting as control
\begin{equation}\left\{\begin{aligned}
&\partial_t y +\partial_x y + \partial_{xxx} y + y\partial_x y = q \mbox{ in } \Omega,\\
&y(.,0) = y(.,L) = \partial_x y (.,L) = 0 \mbox{ in } \Gamma,\\
&y(0,.) = 0 \mbox{ on } \Omega.
\label{kdvcontrol}
\end{aligned}\right.\end{equation}
which is known to have traveling wave solutions. The control space $\mathcal M_I$ is either the Bochner space $L^2(I,\mathcal M(\Omega))$ or the space of vector measures $\mathcal M(\Omega.L^2(I))$ with values in $L^2(I)$. For both choices the controls are sparse in space and distributed in time. However, the first space allows for moving dirac measures while the second does not. We will tackle the following questions: well posedness of the KdV equation with a non-smooth source term, existence and characterization of an optimal control, algorithmic treatment of the problem by a semi-smooth Newton method in function space. In the end, we present some numerical examples that motivate our work: sparse stabilization of the KDV equation and sparse inverse source problems for the KDV equation (reconstruction of the topography and/or topography changes). |
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