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 (1) ut + f (u)x = 0, u(0, x) = ū(x), {u(t, a) = ũa(t) , u(t, b) = ũb(t), on the domain Ω = {(t, x) ε ℝ2 : t ≥ 0, a ≤ x ≤ b}. We study the mixed problem (1) from the point of view of control theory, taking the initial data ū fixed and regarding the boundary data ūa, ūb as control functions that vary in prescribed sets Ua, Ub, of L∞ boundary controls. In particular, we consider the family of configurations A(T) = {u(T, ·); u is a sol. to (1), ūa ε Ua, ūb ε Ub} 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 L1 topology