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Convergence of minima of integral functionals, with applications to optimal control and stochastic optimization. (English) Zbl 0779.49030
Summary: Let \((S,{\mathcal A},\mu)\) denote a positive measure space, where \({\mathcal A}\) is the Borel \(\sigma\)-algebra of the Polish space \(S\) and where it is assumed that \(\mu\) is a bounded measure. The main result concerns \(\text{epi-}\lim_{\nu \to \infty}E^ \nu f=Ef\) under six different conditions, where \(f:X\times S \to \mathbb{R} \cup\{+\infty\}\), \(X\) a Banach space, \(Ef(x)=\int f(x,s)d \mu(s)\), \(E^ \nu f(x)=\int f(x,s)d \mu^ \nu(s)\), \(x \in X\), and where \(\mu^ \nu\) is a sequence of measures converging weakly to \(\mu\).

MSC:
49K27 Optimality conditions for problems in abstract spaces
49J45 Methods involving semicontinuity and convergence; relaxation
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