ON THE ATTAINABLE SET FOR SCALAR NONLINEAR CONSERVATION-LAWS WITH BOUNDARY CONTROL

We consider the initial value problem with boundary control for a scal ar nonlinear conservation law () u(t) + [f(u)](x) = 0, u(0,x) = 0, u( .,0) = (u) over tilde is an element of U, on the domain Omega = {(t,x) is an element of R-2 : t greater than or equal to 0, x greater than o r equal to 0}. Here u = u(t,x) is the state variable, U is a set of bo unded boundary data regarded as controls, and f is assumed to be stric tly convex. We give a characterization of the set of attainable profil es at a fixed time T > 0 and at a fixed point (x) over bar > 0: [GRAPH ICS] Moreover we prove that A(T,U) and A ((x) over bar,U) are compact subsets of L-1 and L-loc(1), respectively, whenever U is a set of cont rols which pointwise U satisfy closed convex constraints, together wit h some additional integral inequalities.