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

Summary: The authors consider the initial-boundary value problem for a strictly hyperbolic, genuinely nonlinear, Temple class system of conservation laws on the quarter t-x plane where $t,x geq 0$. For a class of initial and boundary data in $L^infty$ with possibly unbounded variation, they construct a flow of solutions that depend continuously, in the $L^1$ distance, both on the initial data and on the boundary data. Moreover, we show that each trajectory of such flow provides the unique weak solution of the corresponding initial-boundary value problem that satisfies an entropy condition of Oleinik type. Next, they study the initial-boundary value problem from the point of view of control theory, taking the initial data fixed and considering, in connection with a prescribed set of boundary data regarded as admissible controls, the set of attainable profiles at a fixed time $T>0$ and at a fixed point $x>0$, establishing closure and compactness of such sets in the $L^1$ topolog...