Minimizing Nonconvex, Simple Integrals of Product Type

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Abstract:

We consider the problem of minimizing simple integrals of product type, i.e. min ∫T0 g(x(t)) f(x′(t)) dt: xAC([0, T]), x(0)=x0,  x(T)=xT,where f:ℝ→[0, ∞] is a possibly nonconvex, lower semicontinuous function with either superlinear or slow growth at infinity. Assuming that the relaxed problem (**) obtained from () by replacing f with its convex envelope f** admits a solution, we prove attainment for () for every continuous, positively bounded below the coefficient g such that (i) every point t∈ℝ is squeezed between two intervals where g is monotone and (ii) g has no strict local minima. This shows in particular that, for those f such that the relaxed problem (**) has a solution, the class of coefficients g that yield existence to () is dense in the space of continuous, positive functions on ℝ. We discuss various instances of growth conditions on f that yield solutions to (**) and we present examples that show that the hypotheses on g considered above for attainment are essentially sharp. Copyright 2001 Academic Press.

Keywords: existence of solutions; nonconvex minimum problems; simple integrals

Document Type: Research Article

Affiliations: 1: Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, P.le Europa 1, Trieste, I-34127, Italy 2: Dipartimento di Matematica Pura ed Applicata “G. Vitali”, Università degli Studi di Modena e Reggio Emilia, Via Campi 213/B, Modena, I-41100, Italy

Publication date: March 1, 2001

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