L1L^1-Minimization for Mechanical Systems

Second order systems whose drift is defined by the gradient of a given potential are considered, and minimization of the $L^1$-norm of the control is addressed. An analysis of the extremal flow emphasizes the role of singular trajectories of order two [25,29]; the case of the two-body potential is treated in detail. In $L^1$-minimization, regular extremals are associated with controls whose norm is bang-bang; in order to assess their optimality properties, sufficient conditions are given for broken extremals and related to the no-fold conditions of [20]. An example of numerical verification of these conditions is proposed on a problem coming from space mechanics

Time-scale analysis non-local diffusion systems, applied to disease models

The objective of the present paper is to use the well known Ross-Macdonald models as a prototype, incorporating spatial movements, identifying different times scales and proving a singular perturbation result using a system of local and non-local diffusion. This results can be applied to the prototype model, where the vector has a fast dynamics, local in space, and the host has a slow dynamics, non-local in space