On the Attainable Set for Scalar Nonlinear Conservation Laws with Boundary Control

Abstract

We consider the initial value problem with boundary control for a scalar nonlinear conservation law \begin{equation*} u_t+[f(u)]_x=0,\qquad\qquad u(0,x)=0,\qquad u(\cdot,0)= \tilde u\in{\cal U},\tag{$\ast$} \end{equation*} on the domain $\Omega=\{(t,x)\in\real^2: t\geq 0, x\geq 0\}$. 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 point $\bar x>0$: \begin{equation*} \begin{aligned} \rag &=\{u(T,\cdot): u \hbox{is a solution of} \ragx &=\{u(\cdot,\bar x): u \hbox{is a solution of} (\ast)\}, \end{aligned} \qquad\quad{\cal U}= \ellein({\Bbb R}^+). \end{equation*} Moreover we prove that $\rag$ and $\ragx$ are compact subsets of $\elleuno$ and $\elleuno_{loc}$, respectively, whenever ${\cal U}$ is a set of controls which pointwise satisfy closed convex constraints, together with some additional integral inequalities.