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

Typical convergence theorems for value functions and solutions of (parametric families of) optimization problems based on È-convergence of the corresponding functionals usually rely on equi-coercivity assumptions. Without them the connection between the È-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 ßnd limiting optimization problems (parametrized in a similar way and determined by functionals which may diÞer from the È-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).