On the attainable set for temple class systems with boundary controls

Consider the initial-boundary value problem for a strictly hyperbolic, genuinely nonlinear, Temple class system of conservation laws $$ u_t+f(u)_x=0, u(0,x)=\bar u(x), \ \{ u(t,a)=\widetilde u_a(t), \\ u(t,b)=\widetilde u_b(t), \eqno(1) $$ on the domain $\Omega =\{(t,x)\in\R^2 : t\geq 0,\, a \le x\leq b\}.$ We study the mixed problem (1) from the point of view of control theory, taking the initial data $\bar u$ 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 $L^\infty$ boundary controls. In particular, we consider the family of configurations $$ A(T) \doteq \big\{ u(T,\cdot)~; ~ u iis a sol. to (1), \ \widetilde u_a\in \U_a, \ \, \widetilde u_b \in \U_b \big\} $$ 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 $L^1$~topology.