On the Attainable set for Temple Class Systems with Boundary Controls

Details On the Attainable set for Temple Class Systems with Boundary Controls On the Attainable set for Temple Class Systems with Boundary Controls

2002-05-15

arxiv.org/abs/math/0205167v1

Mathematics

Analysis of PDEs

26 pages, 2 figures

Scientific paper

Consider the initial-boundary value problem for a strictly hyperbolic, genuinely nonlinear, Temple class system of conservation laws % $$ u_t+f(u)_x=0, \qquad u(0,x)=\ov u(x), \qquad {{array}{ll} &u(t,a)=\widetilde u_a(t), \noalign{\smallskip} &u(t,b)=\widetilde u_b(t), {array}. \eqno(1) $$ 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 $\li$ boundary controls. In particular, we consider the family of configurations $$ \A(T) \doteq \big\{u(T,\cdot); ~ u {\rm is a sol. to} (1), \quad \widetilde u_a\in \U_a, \widetilde u_b \in \U_b \big\} $$ 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.

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