##
**Optimal control of a fine structure.**
*(English)*
Zbl 0812.49009

Author’s abstract: “An optimal control problem for a multivalued system governed by a nonconvex variational problem, involving a regularization parameter \(\varepsilon>0\), is proposed and studied. The solution to the variational problem exhibits typically rapid oscillations (a so-called fine structure) corresponding to a multiphase state of the material. We want to control only this fine structure. Existence of an optimal control is proved. Its convergence with \(\varepsilon\to 0\) is studied by means of an optimal control problem for a relaxed variational problem involving (suitably generalized) Young measures. The uniqueness of the solution to the relaxed variational problem, which is nontrivial but is very important in the context of optimal control, is studied in special cases. A finite-element approximation is proposed”.

Reviewer: O.Cârjá (Iaşi)

### MSC:

49J27 | Existence theories for problems in abstract spaces |

35B25 | Singular perturbations in context of PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

### Keywords:

optimal control; nonconvex variational problem; fine structure; relaxed variational problem; finite-element approximation
PDF
BibTeX
XML
Cite

\textit{K. H. Hoffmann} and \textit{T. Roubíček}, Appl. Math. Optim. 30, No. 2, 113--126 (1994; Zbl 0812.49009)

Full Text:
DOI

### References:

[1] | Chipot M (1991) Numerical analysis of oscillations in conconvex problems. Numer Math 59:747–767 · Zbl 0737.65054 |

[2] | Ekeland I, Temam R (1976) Convex Analysis and Variational Problems. North-Holland, Amsterdam · Zbl 0322.90046 |

[3] | Friesecke G (to appear) A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems. Proc Roy Soc Edinburgh · Zbl 0809.49017 |

[4] | Kinderlehrer D, Pedregal P (1992) Weak convergence of integrands and the Young measure representation. SIAM J Math Anal 23:1–19 · Zbl 0757.49014 |

[5] | Kinderlehrer D, Pedregal P (1992) Remarks about gradient Young measures generated by sequences in Sobolev spaces. Research Report No. 92-NA-007, Carnegie-Mellon · Zbl 0833.49012 |

[6] | Lions J-L (1981) Some Methods in the Mathematical Analysis of Systems and Their Control. Science Press, Beijing; Gordon and Breach, New York · Zbl 0542.93034 |

[7] | Meeks SW, Auld BA, Maccagno P, Miller A (1983) Interaction of acoustic waves and ferroelastic domain walls. Ferroelectrics 50:245–250 |

[8] | Müller S (1989) Minimizing sequences for nonconvex functionals, phase transitions and singular perturbations. In: Problems Involving Change of Type (K Kirchgässner, ed). Lecture Notes in Physics, Vol 359, Springer-Verlag, Berlin, pp 31–44 |

[9] | Roubíček T (1991) Evolution of a microstructure: a convexified model. Math Methods Appl Sci |

[10] | Roubíček T (1992) Optimality conditions for nonconvex variational problems relaxed in terms of Young measures. DFG Report No. 375, Technische Universität München |

[11] | Roubíček T (to appear) Finite element approximation of a microstructure evolution. Math Methods Appl Sci |

[12] | Swart PJ, Holmes PJ (1992) Energy minimization and the formation of microstructure in dynamic anti-plane shear. Arch Rational Mech Anal 121:37–85 · Zbl 0786.73066 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.