Consider the initial-boundary value problem for a strictly hyperbolic, genuinely nonlinear, Temple class system of conservation laws % $ u_t+f(u)_x=0, \qquad\ \ u(0,x)=\ov u(x), \qquad \left\{\!\!\!\!\!\!\!\! \begin{array}{ll} &u(t,a)=\widetilde u_a(t), \\ \noalign{\smallskip} &u(t,b)=\widetilde u_b(t), \end{array} \right. \eqno(1) $ % % % $ % u_t+f(u)_x=0 % \qquad\quad u𝟄 \R^n, % \eqno (1) % $ on the domain $Ω =\{(t,x)𝟄\R^2 : t≥ 0,\, a \le x≤ b\}.$ We study the mixed problem (1) from the point of view of control theory, taking the initial data $\overline u$ fixed, and regarding the boundary data $\widetilde u_a, \, \widetilde u_b$ as control functions that vary in prescribed sets $\U_a,\, \U_b$, of $\li$ boundary controls. In particular, we consider the family of configurations $ \A(T\,) ≐ \big\{ u(T,⋅)~; ~ u \ {\rm \ is \ a \ sol. \ \ to} \ \ (1), \quad \widetilde u_a𝟄 \U_a, \ \, \widetilde u_b 𝟄 \U_b %%%%%%%% % u(⋅,0)=\overline u,\, % u(⋅,a)=\widetilde u_a,\, % u(⋅,b)=\widetilde u_b % \overline u,\,\widetilde u_a,\,\widetilde u_b,)(⋅)~ \big\} $ 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 $\lu$~topology.