Numerical Experiments Regarding the Distributed Control of Semilinear Parabolic Problems∗

Abstract

This work deals with some numerical experiments regarding the distributed control of semilinear parabolic equations of the type yt − yxx + f(y) = uχω in (0, 1)× (0, T ), with Neumann and initial auxiliary conditions, where ω 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 2 > 0, find u ∈ L2(0, T ) such that ‖y(T ; u)− yT ‖L2(0,1) ≤ 2) by passing to the limit (when k → ∞) in the penalized optimal control problem (find uk as the minimum of Jk(u) = 12 ‖u‖L2(0,T ) + k2 ‖y(T ; u)− yT ‖L2(0,1)). In the superlinear case (e.g. f(y) = |y|n−1 y, 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 ∈ (0, 1). We carry out some numerical experiences 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.

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Cite this paper

@inproceedings{Ildefonso2004NumericalER, title={Numerical Experiments Regarding the Distributed Control of Semilinear Parabolic Problems∗}, author={Jes{\'u}s Ildefonso and Angel Manuel Ramos}, year={2004} }