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.
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