We consider the one dimensional semilinear reaction-diffusion equation,
governed in Ω = (0,1) by controls, supported on any subinterval of
(0, 1), which are the functions of time only.
Using an asymptotic approach that we have previously introduced in [9],
we show that such a system is approximately controllable at any time in both
L
2(0,1)( and C
0[0,1], provided the nonlinear term f = f(x,t, u)
grows at infinity no faster than certain power of log |u|. The
latter depends on the regularity and structure of f (x, t, u) in x
and t and the choice of the space for controllability. We also show that
our results are well-posed in terms of the “actual steering” of the
system at hand, even in the case when it admits non-unique solutions.