Preprint 2000-048

Uniqueness and Stability of $L^\infty$ Solutions for Temple Class Systems with Boundary and Properties of the Attainable Sets

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.