On the properness of an impulsive control extension of dynamic optimization problems

¹ UNESP – Universidade Estadual Paulista, Department of Applied Mathematics, São José do Rio Preto, SP, Brazil
dmitrykaramzin@mail.ru, antunes@ibilce.unesp.br & gsilva@ibilce.unesp.br
² Computing Center of the Russian Academy of Sciences, Moscow, Russia
³ SYSTEC, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal
flp@fe.up.pt

Received: 27 October 2013
Revised: 16 June 2014

Abstract

This article investigates the properness, or well-posedness, of impulsive extension of a conventional optimal control problem. This includes both well-posedness of the solution to impulsive control systems arising as result of an impulsive extension of ordinary differential systems, and existence theorems. Well-posedness in the classic Cauchy sense is proved. Approximation lemmas that guarantee sensitivity to small perturbations in control variables are obtained. Filippov type existence theorems are established. A model example is provided to show the relevance of the impulsive controls problems which are under study.