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 % $$ u_t+f(u)_x=0, \qquad\ \ u(0,x)=\ov u(x), \qquad \left\{\!\!\!\!\!\!\!\! \begin{array}{ll} &u(t,a)=\widetilde u_a(t), \\ \noalign{\smallskip} &u(t,b)=\widetilde u_b(t), \end{array} \right. \eqno(1) $$ % % % $$ % u_t+f(u)_x=0 % \qquad\quad u\in \R^n, % \eqno (1) % $$ 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 $\overline u$ 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 $$ \A(T\,) \doteq \big\{ u(T,\cdot)~; ~~ u \ {\rm \ is \ a \ sol. \ \ to} \ \ (1), \quad \widetilde u_a\in \U_a, \ \, \widetilde u_b \in \U_b %%%%%%%% % u(\cdot,0)=\overline u,\, % u(\cdot,a)=\widetilde u_a,\, % u(\cdot,b)=\widetilde u_b % \overline u,\,\widetilde u_a,\,\widetilde u_b,)(\cdot)~ \big\} $$ 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.