| Contents |
| We consider a scalar conservation law in one space dimension
\begin{equation}
\partial_t u + \partial_x f(u) = 0\,,
\qquad t\geq 0\,, ~x\in\mathbb{R}\,,
\end{equation}
with a smooth flux function $f$ that suffers a single inflection point. We augment
(1) with an initial datum
\begin{equation}
u(0,x)=
\begin{cases}
0\quad &\text{if $x\not\in [a,b]$}\\
z(x)&\text{if $x\in [a,b]$}
\end{cases}
\end{equation}
where $a$ and $b$ are given real numbers, and $z=z(x)$ is a bounded measurable function that
it is regarded as a control. We are interested in studying the set of attainable profiles
at a fixed time $T>0$, i.e. the set
\begin{multline*}
A(T) = \Big\{ v\in L^1(\mathbb{R}) : \exists z\in L^\infty(a,b) : v(x) = u(T,x)~\text{a.e.}\,,\\
~\text{$u=u(t,x)$ is the solution to (1)-(2)} \Big\}\,.
\end{multline*}
In literature several control problems for conservation laws
were studied, but in most cases only strictly convex flux functions were considered.
Indeed, the presence of an inflection point changes the structure of the waves
in a solution, allowing the presence of one-side contact discontinuities. The problem
may be of interest for applications to traffic flow. |
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