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Journal of Convex Analysis 09 (2002), No. 2, 439--462
Copyright Heldermann Verlag 2002

On Limits of Variational Problems. The Case of a Non-Coercive Functional

Lorenzo Freddi
Dip. di Matematica e Informatica, UniversitÓ di Udine, via delle Scienze 206, 33100 Udine, Italy

Alexander D. Ioffe
Dept. of Mathematics, Technion, Haifa 32000, Israel

Typical convergence theorems for value functions and solutions of (parametric families of) optimization problems based on Gamma-convergence of the corresponding functionals usually rely on equi-coercivity assumptions. Without them the connection between the Gamma-limit of the functionals and values and/or solutions of the problems may be completely broken. The question to be discussed is whether it is possible, even in the absence of a coercivity-type assumption, to find limiting optimization problems (parametrized in a similar way and determined by functionals which may differ from the Gamma-limits of the functionals of the sequence) such that the value functions and solutions of the problems of the sequence converge in a certain sense to those of the limiting problems. A positive answer to the question is given to a class of variational problems (containing optimal control problems with linear dynamics).

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