Existence and characterization of bv-curves for problems of calculus of variations

Measure Differential Inclusions: Existence Results and Minimum Problems

“…However, we would like to stress that our setting is much different: we work in the wider case of multivalued functions and G satisfies conditions involving only the bounded variation w.r.t. the Pompeiu excess and not the Pompeiu-Hausdorff distance (in contrast to [29,31,32], where it is a single-valued function continuous, uniformly Lipschitz w.r.t. x, having linear growth or to [28] where continuity in all arguments and local Lipschitz assumptions are imposed).…”

“…Remark 7 Finally, we notice that we are aware of the fact that Theorem 6 provides the existence of a solution to the minimization problem over a subset of measures and a specific subset of associated solutions to the measure differential inclusion (1.1), while in [29,31,32] or [28] the minimal pair (control, state) is obtained over all measures and all associated solutions of differential problem. However, we would like to stress that our setting is much different: we work in the wider case of multivalued functions and G satisfies conditions involving only the bounded variation w.r.t.…”

“…As an application of the compactness result, we deduce (Theorem 6) a condition for minimizing a functional This is an extension of a classical optimal control problem (which cannot be solved in the class of absolutely continuous functions for solutions, respectively of L 1 controls) to the space of functions of bounded variation for solutions, respectively of measures for controls (see [29,31,32] treating the single-valued case or [23,28]). For optimality conditions in the set-valued setting, but concerning different optimal control problems, we can refer to [39] (with convex velocity sets) or [14].…”