ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV)

ESAIM: COCV, Vol. 14, N°2, pp. 381-409
DOI: 10.1051/cocv:2007058

How to state necessary optimality conditions for control problems with deviating arguments?

Lassana Samassi and Rabah Tahraoui

Ceremade, Université Paris IX-Dauphine, France; samassi@ceremade.dauphine.fr; tahraoui@ceremade.dauphine.fr

(Received June 13, 2006. Published online November 21, 2007.)

Abstract
The aim of this paper is to give a general idea to state optimality conditions of control problems in the following form: ${\displaystyle\inf_{{\displaystyle(u,v)\in {\cal U}_{ad}}} \int_ f\left(t, u(\theta_v(t)),u^{\prime}(t),v(t)\right){\rm d}t}$ , (1) where ${\cal U}_{ad}$ is a set of admissible controls and $\theta_v$ is the solution of the following equation: $\{ \frac{{\rm d}\theta(t)}{{\rm d}t}=g(t,\theta(t),v(t)), t\in [0,1]$ ; $\displaystyle\theta(0)=\theta_0, \theta(t)\in [0,1] \forall t$ . (2). The results are nonlocal and new.

Mathematics Subject Classification. 49J15, 49J22, 49J25, 49J45, 49K15, 49K25, 49K22, 34K35, 47E05, 91B26, 91B28, 93C15

Key words: Functionals with deviating arguments, optimal control, Euler-Lagrange equation, financial market

© EDP Sciences, SMAI 2008