Some optimization problems with a monotone impulsive character. Approximation by means of structural discontinuities

This paper deals with a certain class of Bolza optimization problems (P) that can be represented by means of a differential manifold. A procedure is shown, which is useful for approximating and simplifying a regular problem of type (P) to an associated one, (˜P), by introducing some structural discontinuities. In problem (˜P), which is taken in a weak sense, the curvature involved is considered as a limit of a certain simple family, compatible with the “monotonicity” of (˜P)’s impulsive character. One determines an ordinary auxiliary problem (ˆP), whose solutions allow the possibility of solving (˜P), in the sense of determining both the weak infimum of the functional that has to be minimized and a minimizing sequence. The second aim of the paper is to show how to solve the original problem directly, without considering the auxiliary problem (ˆP). To this end, by introducing a so-called extended admissible process, independent of (ˆP), the authors give some existence theorems for the extended solution to (˜P). The work seems to be useful not only in physical (mechanical) applications but also in those with a nonphysical profile.