Jensen's inequality for the lower semicontinuous quasiconvex envelope and relaxation of multidimensional control problems

Assume that K⊂Rnm is a convex body with o∈int(K) and f:Rnm→R∪{+∞} is a function with f|K∈C0(K,R) and f|(Rnm∖K)≡+∞. We show that its lower semicontinuous quasiconvex envelopef(qc)(w)=sup{g(w)|g:Rnm→R∪{+∞} quasiconvex and lower semicontinuous,g(v)⩽f(v)∀v∈Rnm} obeys the Jensen's integral inequalityf(qc)(w)=f(qc)((∫Kv11dν(v)⋯∫Kv1mdν(v)⋮⋮∫Kvn1dν(v)⋯∫Kvnmdν(v)))⩽∫Kf(qc)((v11⋯v1m⋮⋮vn1⋯vnm))dν(v)∀ν∈S(qc)(w) for every w∈K where S(qc)(w) is a subset of probability measures. This result is then applied to multidimensional control problems of Dieudonné-Rashevsky type: Relaxation by replacement of the integrand by its lower semicontinuous envelope and relaxation by introduction of generalized controls lead to problems with identical minimal values.