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Numerical experiments regarding the distributed control of semilinear parabolic problems

Díaz Díaz, Jesús Ildefonso and Ramos del Olmo, Ángel (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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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
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