Compactness criteria for an operator constraint in the Arkin-Levin variational problem
Let (S,\({\mathcal E}_ s,\mu)\) and (T,\({\mathcal E}_ T,\nu)\) be measure spaces and let a triple of functions u: \(S\times T\times {\mathfrak R}^{\ell}\to {\mathfrak R}\), g: \(S\times T\times {\mathfrak R}^{\ell}\to \bar {\mathfrak R}^ k\) and \(\omega\) : \(T\to {\mathfrak R}^ k\) be given. Consider the well-known Arkin-Levin variational problem formulated as follows: \[ (P)\quad_{x}\int_{s\times T}u(s,t,x(s,t))d(\mu \otimes \nu) \] \[ subject\quad to\quad \int_{s}g(s,t,x(s,t))d\mu \leq \omega (t)\quad almost\quad everywhere. \] The existence of optimal solutions for (P) has been investigated by \textit{V. I. Arkin} and \textit{V. L. Levin} [Russ. Math. Surveys 27(1972), No.3, 21-85 (1973); translation from Usp. Mat. Nauk 27, No.3(165), 21-77 (1972; Zbl 0254.49006) ] and the author [Keio Econ. Stud. 18, 81-97 (1982)] where a special kind of infinite dimensional Lyapunov measure played a crucial role. In this paper, we present a more classical alternative approach to the existence problem, based upon the continuity theorem for nonlinear integral functionals due to \textit{A. D. Ioffe} [SIAM J. Control Optimization 15, 521-538 (1977; Zbl 0361.46037)] and the following compactness theorem. Theorem. Let (S,\({\mathcal E}_ s,\mu)\) and (T,\({\mathcal E}_ T,\nu)\) be finite measure spaces and f: \(S\times T\times {\mathfrak R}^{\ell}\to \bar {\mathfrak R}\) be (\({\mathcal E}_ s\otimes {\mathcal E}_ T\otimes {\mathcal B}({\mathfrak R}^{\ell})\), \({\mathcal B}(\bar {\mathfrak R}))\)-measurable, where \({\mathcal B}(\cdot)\) stands for the Borel \(\sigma\)-field on (\(\cdot)\). We denote by \(f^*(s,t,\cdot)\) the Young-Fenchel transform of \(x\mapsto f\) (s,t,x) for any fixed (s,t)\(\in S\times T\). If f satisfies the growth condition: \[ \int_{S\times T}f^*(s,t,y)d(\mu \otimes \nu)<\infty \text{ for all } y\in {\mathfrak R}^{\ell}, \] then the set \(F_ c=\{x\in L^ 1(S\times T,{\mathfrak R}^{\ell})| \int_{S}f(s,t,x(s,t))d\mu \leq c(t)\) a.e.\(\}\) is weakly relatively compact in \(L^ 1(S\times T,{\mathfrak R}^{\ell})\) for any \(c\in L^ 1(T,{\mathfrak R})\).
scientific article; zbMATH DE number 4151202