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

Abstract

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.