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November  2010, 14(4): 1375-1401. doi: 10.3934/dcdsb.2010.14.1375

A systematic method for building smooth controls for smooth data

Sylvain Ervedoza 1, and  Enrique Zuazua 2,

1. 

Université de Toulouse, UPS, Institut de Mathématiques de Toulouse and CNRS, 118 route de Narbonne, F-31062 Toulouse Cedex 9, France
2. 

IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao - Basque Country, Spain

Received  October 2009 Revised  February 2010 Published  August 2010

We prove a regularity result for an abstract control problem $z' =A z + Bv$ with initial datum $z(0) = z_0$ in which the goal is to determine a control $v$ such that $z(T)=0$. Under standard admissibility and observability assumptions on the adjoint system, when $A$ generates a $C^0$ group, we develop a method to compute algorithmically a control function $v$ that inherits the regularity of the initial datum to be controlled. In particular, the controlled equation is satisfied in a strong sense when the initial datum is smooth. In this way, the controlled trajectory is smooth as well. Our method applies mainly to time-reversible infinite-dimensional systems and, in particular, to the wave equation, but fails to be valid in the parabolic frame.
Keywords: Observability, Time-dependent weights., Regularity of controls, Controllability, Linear systems.
Mathematics Subject Classification: Primary: 93B05, 93B07; Secondary: 93C0.
Citation: Sylvain Ervedoza, Enrique Zuazua. A systematic method for building smooth controls for smooth data. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1375-1401. doi: 10.3934/dcdsb.2010.14.1375
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