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We consider the one dimensional semilinear reaction-diusion equation, governed in = (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 C0(0; 1), provided the nonlinear term f = f(x;t;u) grows at innity no faster than certain power of log j uj. The latter depends on the regularity and structure of f(x;t;u )i n xand 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. ut = u f (x;t;u;ru )+ v (x;t)!(x )i n Q T =(0;T ); u =0 in T = @T; u(x; 0) = u0(x )i n ; (S)

Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls