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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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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

C. Fabre, J.P. Puel, E. Zuazua. Contrôlabilit approchée de l'équation de la chaleur semi-linéaire C.R. Acad. Sci. Paris, 315 (Serie I) (1992), pp. 807–812

J.I. Díaz, A.M. Ramos. Positive and negative approximate controllability results for semilinear parabolic equations Rev. Real Academia de Ciencias Exactas, Físicas y Naturales, de Mardrid, Tomo LXXXIX (1995), pp. 11–30

J. Henry. Etude de la contrôlabilité de certaines équations paraboliques Thèse d'EtatUniversité Paris VI, New York (1978)

J.I. Diaz. Mathematical analysis of some diffusive energy balance models in climatology J.I. Diaz, J.L. Lions (Eds.), Mathematics, Climate and Environment, Masson (1993), pp. 28–56

J.L. Lions. Remarques sur la contrôlabilité approchée. Proceedings of Jornadas Hispano-Francesas sobre control de sistemas distribuidos, Univ. de Mâlaga (1991), pp. 77–87

L.A. Fernández, E. Zuazua. Approximate controllability of the semilinear heat equation via optimal control Preprint de la Univ. de Cantabria (1996)

J.I. Diaz. Controllability and obstruction for some nonlinear parabolic problems in climatology Modelado de sistemas en Oceanografia, Climatologfa y Ciencias Medio-Ambientales, Univ. de Mâlaga (1994), pp. 43–58

E. Fernández-Cars. Null Controllability for Semilinear Heat Equation, ESSAIM: Control Optimization and Calculus of Variations (1997), pp. 87–103

A.Y. Khapalov. Global controllability properties for the semilinear heat equation with superlinear terms Revista Matemática Complutense 1.2 (1999), pp. 511–535

J.M. Coron, E. Trélat. Global steady-state controllability of 1-D semilinear heat equations Preprint (2003)

R. Glowinski, J.L. Lions. Exact and approximate controllability for distributed parameter systems, Part II. Acta Numerica (1995), pp. 159–333

J.I. Diaz, A.M. Ramos. Numerical experiments regarding the localized control of semilineax parabolic problems CD-ROM format Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2000), Barcelona, 84-89925-70-4 (September 11–14, 2000)

H. Brezis, F. Browder. Strongly nonlinear parabolic initial-boundary value problems Proc. Nat. Acad. Sci., 76 (1979), pp. 38–40

J.L Lions, E. Magenes. Problèmes aux Limites Non Homogènes et Applications, Volumes 1 and 2Dunod (1968)

A.M. Ramos, R. Glowinski, J. Periaux. Nash Equilibria for the Multi-Objective Control of Linear Partial Differential Equations. Journal of Optimization, Theory and Applications, 112 (3) (2002), pp. 457–498

M. Berggren, R. Glowinski, J.L. Lions. A computational approach to controllability issues for flow-related models. (I): Pointwise control of the viscous Burgers equation. International Journal of Computational Fluid Dynamics, 7 (1996), pp. 237–252

D.C. Liu, J. Nocedal. On the limited memory BFGS method for large scale optimization Mathematical Programming, 45 (1989), pp. 503–528

R. Glowinski, A.M. Ramos. A numerical approach to the Neumann control of the Cahn-Hilliard equation. ISBN: 4-7625-0425-4 ,in: R. Glowinski, H. Kawarada, J. Periaux (Eds.), Computational Methods for Control and Applications, Gakuto International Series: Mathematical Sciences and Applications, Volume 16, Gakkotosho Co., Paris (2002), pp. 111–155

J.I. Diaz, J.L. Lions. On the Approximate Controllability for some Explosive Parabolic Problems International Series of Numerical Mathematics, Volume 133 (1999), pp. 115–132

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