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

Lassana Samassi; Rabah Tahraoui

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- Volume: 14, Issue: 2, page 381-409
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topSamassi, Lassana, and Tahraoui, Rabah. "How to state necessary optimality conditions for control problems with deviating arguments?." ESAIM: Control, Optimisation and Calculus of Variations 14.2 (2008): 381-409. <http://eudml.org/doc/250383>.

@article{Samassi2008,

abstract = {
The aim of this paper is to give a general idea to state optimality conditions of control problems in the following form:
$\{\displaystyle\inf_\{\{\displaystyle(u,v)\in \{\cal U\}_\{ad\}\}\} \int_\{0\}^\{1\} f\left(t, u(\theta_v(t)),u^\{\prime\}(t),v(t)\right)\{\rm d\}t\}$, (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.
},

author = {Samassi, Lassana, Tahraoui, Rabah},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Functionals with deviating arguments; optimal control; Euler-Lagrange equation; financial market; functionals with deviating arguments},

language = {eng},

month = {3},

number = {2},

pages = {381-409},

publisher = {EDP Sciences},

title = {How to state necessary optimality conditions for control problems with deviating arguments?},

url = {http://eudml.org/doc/250383},

volume = {14},

year = {2008},

}

TY - JOUR

AU - Samassi, Lassana

AU - Tahraoui, Rabah

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2008/3//

PB - EDP Sciences

VL - 14

IS - 2

SP - 381

EP - 409

AB -
The aim of this paper is to give a general idea to state optimality conditions of control problems in the following form:
${\displaystyle\inf_{{\displaystyle(u,v)\in {\cal U}_{ad}}} \int_{0}^{1} f\left(t, u(\theta_v(t)),u^{\prime}(t),v(t)\right){\rm d}t}$, (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.

LA - eng

KW - Functionals with deviating arguments; optimal control; Euler-Lagrange equation; financial market; functionals with deviating arguments

UR - http://eudml.org/doc/250383

ER -

## References

top- G. Carlier and R. Tahraoui, On some optimal control problems governed by a state equation with memory. ESAIM: COCV (to appear)
- M. Drakhlin, On the variational problem in the space of absolutely continuous functions. Nonlin. Anal. TMA23 (1994) 1345–1351.
- M. Drakhlin and E. Litsyn, On the variation problem for a family of functionals in the space of absolutly continuous functions. Nonlin. Anal. TMA26 (1996) 463–468.
- M.E. Drakhlin and E. Stepanov, On weak lower semi-continuity for a class of functionals with deviating argument. Nonlin. Anal. TMA28 (1997) 2005–2015.
- M.E. Drakhlin, E. Litsyn and E. Stepanov, Variational methods for a class of nonlocal functionals. Comput. Math. Appl37 (1999) 79–100.
- L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Inc. (1992).
- L. Freddi, Limits of control problems with weakly converging nonlocal input operators. Calculus of variations and optimal control (Haifa, 1998), Math. 411, Chapman Hall/CRC, Boca Raton, FL (2000) 117–140.
- A.A. Gruzdev and S.A. Gusarenko, On reduction of variational problems to extremal problems without constraints. Russians mathematics38 (1994) 37–47.
- E. Jouini, P.F. Koehl and N. Touzi, Optimal investment with taxes: an optimal control problem with endogeneous delay. Nonlin. Anal. TMA37 (1999) 31–56.
- E. Jouini, P.F. Koehl and N.Touzi, Optimal investment with taxes: an existence result. J. Math. Economics33 (2000) 373–388.
- G.A. Kamenskii, Variational and boundary value problems with deviating argument. Diff. Equ6 (1970) 1349–1358.
- G.A. Kamenskii, On some necessary conditions of functionals with deviating argument. Nonlin. Anal. TMA17 (1991) 457–464.
- G.A. Kamenskii, Boundary value problems for differential-difference equations arising from variational problems. Nonlin. Anal. TMA18 (1992) 801–813.
- P.L. Lions and B. Larrouturou, Optimisation et commande optimale, méthodes mathématiques pour l'ingénieur, cours de l'École Polytechnique, Palaiseau, France.
- L. Samassi, Calculus of variation for funtionals with deviating arguments. Ph.D. thesis, University Paris-Dauphine, France (2004).
- L. Samassi and R. Tahraoui, Comment établir des conditions nécessaires d'optimalité dans les problèmes de contrôle dont certains arguments sont déviés ? C.R. Acad. Sci. Paris Ser338 (2004) 611–616.
- J.A. Wheeler and R.P. Feynman, Classical electrodynamics in term of direct interparticle actions. Rev. Modern Phys21 (1949) 425–433.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.