We study the controllability with variable initial locus of the following distributed parameter linear control system $$ z_{xy}+A(x,y)z_x+B(x,y)z_y+C(x,y)z \ = \ F(x,y)U(x,y) \ .\leqno {\indent\tm (E)}$$ Here $(x,y)$ ranges over the following unbounded subset of ${\bb R}^2$: $$L\abbassa2pt{I,J} \ = \ \bigcup_{(u,v) \in I \times J} l(u,v)\ ,$$ where $I,J$ are two non-degenerate intervals of ${\bb R}$ and $$l(u,v) \ = \ ([u,+\infty[\times \{v\}) \cup (\{u\}\times [v,+\infty[)\ , \ \ \ (u,v) \in {\bb R}^2\ .$$ The state vector function $z$ belongs to the Sobolev type space $$W^\ast_{p,{\tm loc}}(L\abbassa2pt{I,J},{\bb R}^n) \ = \ \Big \{z \in L^p_{\tm loc} (L\abbassa2pt{I,J},{\bb R}^n) \ : \ z_x,z_y,z_{xy} \in L^p_{\tm loc}(L\abbassa2pt{I,J},{\bb R}^n)\Big \} \ $$ and the control vector function $U$ is in $L^p_{\tm loc}(L\abbassa2pt{I,J},{\bb 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 attained on the variable initial locus $l(a_0,b_0)$, $(a_0,b_0) \in I\times J $, $a_0 \leq a$, $ b_0\leq b $, to an arbitrary final state, to be attained on the fixed final locus $l(a,b) $. \par We consider both exact and approximate controllability. We get a characterization of the approximate controllability when the set of the available controls is the unit ball of $L^q (L\abbassa2pt{I,J}, {\bb R}^m)$, $q \geq p$. Also, assuming a suitable invertibility property for the matrix $F$, we show a necessary and sufficient condition for the exact controllability when a larger set of available controls is considered. Finally, we study the permanence of both controllability properties with respect to changes of the fixed final locus $l(a,b) $.

Processi di controllo con parametri distribuiti in insiemi non limitati. Controllabilità con luogo iniziale variabile

VILLANI, Alfonso
2005-01-01

Abstract

We study the controllability with variable initial locus of the following distributed parameter linear control system $$ z_{xy}+A(x,y)z_x+B(x,y)z_y+C(x,y)z \ = \ F(x,y)U(x,y) \ .\leqno {\indent\tm (E)}$$ Here $(x,y)$ ranges over the following unbounded subset of ${\bb R}^2$: $$L\abbassa2pt{I,J} \ = \ \bigcup_{(u,v) \in I \times J} l(u,v)\ ,$$ where $I,J$ are two non-degenerate intervals of ${\bb R}$ and $$l(u,v) \ = \ ([u,+\infty[\times \{v\}) \cup (\{u\}\times [v,+\infty[)\ , \ \ \ (u,v) \in {\bb R}^2\ .$$ The state vector function $z$ belongs to the Sobolev type space $$W^\ast_{p,{\tm loc}}(L\abbassa2pt{I,J},{\bb R}^n) \ = \ \Big \{z \in L^p_{\tm loc} (L\abbassa2pt{I,J},{\bb R}^n) \ : \ z_x,z_y,z_{xy} \in L^p_{\tm loc}(L\abbassa2pt{I,J},{\bb R}^n)\Big \} \ $$ and the control vector function $U$ is in $L^p_{\tm loc}(L\abbassa2pt{I,J},{\bb 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 attained on the variable initial locus $l(a_0,b_0)$, $(a_0,b_0) \in I\times J $, $a_0 \leq a$, $ b_0\leq b $, to an arbitrary final state, to be attained on the fixed final locus $l(a,b) $. \par We consider both exact and approximate controllability. We get a characterization of the approximate controllability when the set of the available controls is the unit ball of $L^q (L\abbassa2pt{I,J}, {\bb R}^m)$, $q \geq p$. Also, assuming a suitable invertibility property for the matrix $F$, we show a necessary and sufficient condition for the exact controllability when a larger set of available controls is considered. Finally, we study the permanence of both controllability properties with respect to changes of the fixed final locus $l(a,b) $.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11769/25609
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