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Numerical experiments regarding the distributed control of semilinear parabolic problems

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Díaz Díaz, Jesús Ildefonso and Ramos del Olmo, Ángel Manuel (2004) Numerical experiments regarding the distributed control of semilinear parabolic problems. Computers & Mathematics with Applications, 48 (10-11). pp. 1575-1586. ISSN 0898-1221

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Official URL: http://www.sciencedirect.com/science/article/pii/S0898122104004110




Abstract

This work deals with some numerical experiments regarding the distributed control of semilinear parabolic equations of the type y(t) - y(xx) + f (y) = u(Xw), in (0, 1) x (0, T), with Neumann and initial auxiliary conditions, where w is an open subset of (0, 1), f is a C-1 nondecreasing real function, a is the output control and T > 0 is (arbitrarily) fixed. Given a target state y(T) we study the associated approximate controllability problem (given epsilon > 0, find u is an element of L-2(0, T), such that parallel toy(T; u) - y(T)parallel to(L2(0,1)) less than or equal to epsilon) by passing to the limit (when k --> infinity) in the penalized optimal control problem. (find u(k) as the minimum of J(k)(u) = 1/2 parallel touparallel to(L2)(2) ((0,T)) + (k/2)parallel toy(T; u) -y(T)parallel to(L2)(2) ((0,1))). In the superlinear case (e.g., f (y) = \y\(n-1)y, n > 1) the existence of two obstruction functions Y+/-infinity shows that the approximate controllability is only possible if Y-infinity (x,T) +/- y(T)(x) less than or equal to Y-infinity(x,T) for a.e. x is an element of (0, 1). We carry out some numerical experiments showing that, for a fixed k, the "minimal cost" J(k)(u) (and the norm of the optimal control u(k)) for a superlinear function f becomes much larger when this condition is not satisfied. We also compare the values of J(k)(u) (and the norm of the optimal control u(k)) for a fixed y(T) associated with two nonlinearities: one sublinear and the other one superlinear.


Item Type:Article
Uncontrolled Keywords:approximate controllability; equation; controllability; semilinear parabolic problem; numerical approximation; adjoint system; distributed control; implicit-scheme; large solutions
Subjects:Sciences > Mathematics > Differential equations
ID Code:15470
Deposited On:04 Jun 2012 08:38
Last Modified:12 Dec 2018 15:07

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