Consider the initial-boundary value problem for a strictly hyperbolic, genuinely nonlinear, Temple class system of conservation laws [GRAPHICS] on the domain Omega = {(t, x) is an element of R-2:t >= 0, a <= x <= b}. We study the mixed problem (1) from the point of view of control theory, taking the initial data (U) over bar fixed and regarding the boundary data (u) over tilde (a), (u) over tilde (b) as control functions that vary in prescribed sets U-a, U-b, of L-infinity boundary controls. In particular, we consider the family of configurations A(T) ={u(T, center dot); u is a sol. to (1) (u) over tilde (a) is an element of U-a, (u) over tilde (b) is an element of U-b} that can be attained by the system at a given time T>0, and we give a description of the attainable set A(T) in terms of suitable Oleinik-type conditions. We also establish closure and compactness of the set A(T) in the L-1 topology.