Summary
We investigate the following problem: To influence a heat conduction process in such a way that the conductor melts in a prescribed manner. Since we treat a linear auxiliary problem, it suffices to deal with a linear-quadratic defect minimization problem with linear restrictions, where we use splines or polynomials as approximation spaces. In case of exact controllability we derive various order of convergence estimates, which we discuss for some numerical examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beatson, R.K.: Restricted range approximation by splines and variational inequalities. SIAM J. Numer. Anal.19, 372–380 (1982)
Bonnerot, R., Jamet, P.: A second order finite element method for the one-dimensional Stefan problem. Int. J. Numer. Methods Eng.8, 811–820 (1974)
Budak, B.M., Vasil'eva, V.I.: The solution of the inverse Stefan problem. U.S.S.R. Comput. Math. Phys.13, 130–151 (1974)
Budak, B.M., Vasil'eva, V.I.: The solution of Stefan's converse problem. U.S.S.R. Comput. Math. Math. Phys.13, 82–95 (1974)
Cannon, J.R., Douglas, J.: The Cauchy problem for the heat equation. SIAM J. Numer. Anal.4, 317–336 (1967)
Cryer, C.W.: The solution of a quadratic programming problem using systematic overrelaxation. SIAM J. Control9, 385–392 (1971)
Fasano, A., Primicerio, M.: General free boundary problems for the heat equation I. J. Math. Anal. Appl.57, 694–723 (1977)
Jochum, P.: The numerical solution of the inverse Stefan problem. Numer. Math.34, 411–429 (1980)
Jochum, P.: The inverse Stefan problem as a problem of nonlinear approximation theory. J. Approximation Theory30, 81–98 (1980)
Knabner, P.: Stability theorems for general free boundary problems of the Stefan type and applications. Methoden Verfahren math. Phys.25, 95–116 (1983)
Knabner, P.: Fragen der Rekonstruktion und der Steuerung bei Stefan Problemen und ihre Behandlung über lineare Ersatzaufgaben. Dissertation. Universität Augsburg 1983
Knabner, P.: Regularization of the Cauchy problem for the heat equation by norm bounds. Appl. Anal.17, 295–312 (1984)
Lawson, C.J., Hanson, R.J.: Solving least squares problems. Englewood Cliffs, New Jersey: Prentice Hall 1974
Niezgódka, M.: Control of parabolic systems with free boundaries-application of inverse formulation. Control Cybern.8, 213–225 (1979)
Primicerio, M.: The occurence of pathologies in some Stefan-like problems. Int. Ser. Numer. Math.58, 233–244 (1982)
Reemtsen, R., Kirsch, A.: A method for the numerical solution of the one-dimensional inverse Stefan problem, part I, II. Preprint TH Darmstadt, Mathematik, Nr. 641, Nr. 652 (1982)
Schumaker, L.L.: Spline functions: basic theory. New York: Wiley 1981
Wang, P.K.C.: Control of a distributed parameter system with a free boundary. Int. J. Control5, 317–329 (1967)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Knabner, P. Control of stefan problems by means of linear-quadratic defect minimization. Numer. Math. 46, 429–442 (1985). https://doi.org/10.1007/BF01389495
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01389495