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

Abstract

Assume that K ⊂ R n m is a convex body with o ∈ int ( K ) and f : R n m → R ∪ { + ∞ } is a function with f | K ∈ C 0 ( K , R ) and f | ( R n m ∖ K ) ≡ + ∞ . We show that its lower semicontinuous quasiconvex envelope f ( q c ) ( w ) = sup { g ( w ) | g : R n m → R ∪ { + ∞ } quasiconvex and lower semicontinuous , g ( v ) ⩽ f ( v ) ∀ v ∈ R n m } obeys the Jensen's integral inequality f ( q c ) ( w ) = f ( q c ) ( ( ∫ K v 11 d ν ( v ) ⋯ ∫ K v 1 m d ν ( v ) ⋮ ⋮ ∫ K v n 1 d ν ( v ) ⋯ ∫ K v n m d ν ( v ) ) ) ⩽ ∫ K f ( q c ) ( ( v 11 ⋯ v 1 m ⋮ ⋮ v n 1 ⋯ v n m ) ) d ν ( v ) ∀ ν ∈ S ( q c ) ( w ) for every w ∈ K where S ( q c ) ( 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.