Abstract:We consider the initial-boundary value problem for a strictly hperbolic, genuinely nonlinear, Temple class system of conservation laws

u_t+f(u)_x=0, u\in {\mathbb R}^n, (1)on the domain $\Omega =\{(t,x)\in{\mathbb R}^2 : t\geq 0,\, x\geq 0\}.$ For a class of initial data $\overline u\in L^\infty(\R^+)$ and boundary data $\widetilde u\in L^\infty(\R^+)$ with possibly unbounded variation, we construct a flow of solutions $(\overline u,\,\widetilde u)\to u(t)\doteq E_t (\overline u,\,\widetilde u)$ that depend continuously, in the $L^1$ distance, both on the initial data and on the boundary data.Moreover, we show that each trajectory $t \mapsto E_t(\overline u,\,\widetilde u)$ provides the unique weak solution of the corresponding initial-boundary value problem that satisfies an entropy condition of Oleinik type.

Next, we study the initial-boundary value problem for (1) from the point of view of control theory taking the initial data $\overline u$ fixed and considering, in connection with a prescribed set ${\cal U}$ of boundary data regarded as admissible controls, the set of attainable profiles at a fixed time $T>0,$ and at a fixed point $\overline x>0$:

{\cal A}(T,\,{\cal U}) \doteq \big\{ E_T(\overline u,\,\widetilde u)(\cdot)~ ; ~ \widetilde u \in {\cal U} \big\} {\cal A}(\overline x,\,{\cal U}) \doteq \big\{ E_{(\cdot)}(\overline u,\,\widetilde u)(\overline x)~ ; ~ \widetilde u \in {\cal U} \big\}.We establish closure and compactness of the sets ${\cal A}(T,\,{\cal U}),$ ${\cal A}(\overline x,\,{\cal U})$ in the $L^1_{loc}$ topology, for a class ${\cal U}$ of admissible controls satisfying convex constraints.

**Paper:**- Available as PostScript (845 Kbytes) or gzipped PostScript (199 Kbytes; uncompress using gunzip).
**Author(s):**- Fabio Ancona, <ancona@ciram3.ing.unibo.it>
- Paola Goatin, <goatin@cmap.polytechnique.fr>
**Publishing information:**- Preprint n.14, Dipartimento di Matematica, Universit\`a di Bologna, 2000.
**Comments:****Submitted by:**- <ancona@ciram3.ing.unibo.it> December 7 2000.

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