| Popis: |
We consider the problem of minimizing simple integrals of product type, i.e. min ∫ T 0 g ( x ( t )) f ( x ′( t )) dt : x ∈ AC ([0, T ]), x (0)= x 0 , x ( T )= x T , ( P ) where f: R →[0, ∞] is a possibly nonconvex, lower semicontinuous function with either superlinear or slow growth at infinity. Assuming that the relaxed problem ( P **) obtained from ( P ) by replacing f with its convex envelope f** admits a solution, we prove attainment for ( P ) 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 ( P **) has a solution, the class of coefficients g that yield existence to ( P ) is dense in the space of continuous, positive functions on R . We discuss various instances of growth conditions on f that yield solutions to ( P **) and we present examples that show that the hypotheses on g considered above for attainment are essentially sharp. |
