### On the Attainable Set for Scalar Non-linear Conservation Laws with Boundary Control

Fabio Ancona and Andrea Marson

Abstract: We consider the initial value problem with boundary control for a scalar non-linear conservation laws $$u_t [f(u)]_x=0, u(0,x)=0, u(\cdot, 0)= \bar u\in{\Cal U}$$ on the domain $\Omega=\{(t,x)\in{\bf R}^2: t\geq 0, x\geq 0\}$. here $u=u(t,x)$ is the state variable, ${\Cal U}$ is a set of bounded boundary data regarded as controls and $f$ is assumed to be strictly convex. We give a characterization of the set of attainable profiles at a fixed time $T>0$ and at a fixed $\bar x>0$. Moreover we prove that such attainable sets are compact subsets of ${\bf L}^1$ and ${\bf L}^1_{loc}$ respectively whenever ${\Cal U}$ is a set of controls which pointwise satisfy closed convex constraints, together with some additional integral inequalities.

Paper:
Available as PostScript
Title:
On the Attainable Set for Scalar Non-linear Conservation Laws with Boundary Control
Author(s):
Fabio Ancona, <ancona@ciram3.ing.unibo.it>
Andrea Marson, <marson@bsing.ing.unibs.it>
Publishing information:
To appear on SIAM Journal of Control and Optimization