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

A systematic method for building smooth controls for smooth data

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.
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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