Control processes with distributed parameters in unbounded sets. Approximate controllability with variable initial locus

We consider the following distributed parameter linear control system $$\displaylines {{\rm (E)}\hfil \hfil z_{xy}+A(x,y)z_x+B(x,y)z_y+C(x,y)z =F(x,y)U(x,y) \ . \qquad \hfil \cr}$$ Here $(x,y)$ ranges over the unbounded set $$L_{{}_{\scriptstyle I,J}} = \bigcup_{(u,v) \in I \times J}l(u,v)\ \ ,$$ where $$l(u,v) = ([u,+\infty[\times \{v\}) \cup (\{u\}\times [v,+\infty[)\ \ , \ \ \ (u,v) \in {\sym R}^2\ ,$$ and $I,J$ are two non-degenerate intervals of ${\sym R}\,$. The state vector function $z$ belongs to the Sobolev type functional space $$W^*_{p,{\rm loc}}(L_{{}_{\scriptstyle I,J}},{\sym R}^n) = \left\{ z \in L^p_{\rm loc}(L_{{}_{\scriptstyle I,J}},{\sym R}^n)\, : \, z_x,z_y,z_{xy} \in L^p_{\rm loc}(L_{{}_{\scriptstyle I,J}},{\sym R}^n)\right\}\, $$ and the control vector function $U$ is in $L^p_{\rm loc}(L_{{}_{\scriptstyle I,J}},{\sym R}^m)\,$. Moreover, for every $(u,v) \in I \times J\,$, the trace of $z$ on $l(u,v)\,$ is taken as the system state corresponding to the values $x=u$, $y=v$ of the parameters. All these traces belong to a functional space of Sobolev type, which does not depend on $(u,v)\,$. \par In this setting, given a point $(a,b) \in I\times J\,$, we study the controllability of system (E) from a given initial state, to be taken on the variable initial locus $l(a_0,b_0)\,$, $(a_0,b_0) \in I\times J\,$, $a_0 \le a\,$, $ b_0\le b\,$, to an arbitrary final state, to be taken on the fixed final locus $l(a,b)\,$. We get a characterization of the approximate controllability when the set of the available controls is the unit ball of $L^\infty(L_{{}_{\scriptstyle I,J}},{\sym R}^m)$.