A systematic method for building smooth controls for smooth data

Abstract

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.