Numerical experimentsregarding the distributed control of semilinear parabolic problems

This work deals with some numerical experiments regarding the distributed control ofsemilinear parabolic equations of the type yt − yχχ + f (y) = uχω,in (0, 1) × (0, T), with Neumann and initial auxiliary conditions, where w is an open subset of (0, 1), f is a C1 nondecreasing real function, u is the output control and T > 0 is (arbitrarily) fixed. Given a target state YT we study the associated approximate controllability problem (given e > 0, find u E L2(0, T), such that y(T; u) −YT L2(0,1) ≤ by passing to the limit (when k → ∞) in the penalized optimal control problem (find uk as the minimum of Jk(u) = 1/2 u L(0,T)2 + (k/2) y(T; u) −YT L22L2(o,T) L2(0,1)). In the superlinear case (e.g., f (y) = y n−1y, n > 1) the existence of two obstruction functions Y±∞ shows that the approximate controllability is only possible if Y−∞ (x, T) < YT (x) < Y∞ (x, T) for a.e. x E (0,1). We carry out some numerical experiments showing that, for a fixed k, the “minimal cost” Jk(u) (and the norm of the optimal control uk) for a superlinear function f becomes much larger when this condition is not satisfied. We also compare the values of Jk(u) (and the norm of the optimal control uk) for a fixed YT associated with two nonlinearities: one sublinear and the other one superlinear.