On the strong controllability of the diffusion equation

Let (x,t)y (x,t),x[0, 1],t[0,T], be the solution of the diffusion equation in one spatial variable corresponding to zero initial conditions and boundary controluL₂(0,T). GivenfL₂(0, 1), it is not possible, in general, to find a controlu such thaty(·,T)=f. We extend the space of controls in such a manner thatL₂(0,T) can be considered to be a subset of a new spaceS of control elements; this space contains elements which do provide a solution to the problem of moments associated with the problem of makingy(·,T)=f inL₂(0, 1). We show then that the action of the elements ofS can be approximated by that of control functions inL₂(0,T) in a suitable manner.