How to state necessary optimality conditions for control problems with deviating arguments ?
ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 2, pp. 381-409.

The aim of this paper is to give a general idea to state optimality conditions of control problems in the following form: $\underset{\left(u,v\right)\in {𝒰}_{ad}}{inf}{\int }_{0}^{1}f\left(t,u\left({\theta }_{v}\left(t\right)\right),{u}^{\text{'}}\left(t\right),v\left(t\right)\right)\mathrm{d}t$, (1) where ${𝒰}_{ad}$ is a set of admissible controls and ${\theta }_{v}$ is the solution of the following equation: $\left\{\frac{\mathrm{d}\theta \left(t\right)}{\mathrm{d}t}=g\left(t,\theta \left(t\right),v\left(t\right)\right),t\in \left[0,1\right]$ ; $\theta \left(0\right)={\theta }_{0},\theta \left(t\right)\in \left[0,1\right]\forall t$. (2). The results are nonlocal and new.

DOI: 10.1051/cocv:2007058
Classification: 49J15, 49J22, 49J25, 49J45, 49K15, 49K25, 49K22, 34K35, 47E05, 91B26, 91B28, 93C15
Keywords: functionals with deviating arguments, optimal control, Euler-Lagrange equation, financial market
@article{COCV_2008__14_2_381_0,
author = {Tahraoui, Rabah and Samassi, Lassana},
title = {How to state necessary optimality conditions for control problems with deviating arguments ?},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {381--409},
publisher = {EDP-Sciences},
volume = {14},
number = {2},
year = {2008},
doi = {10.1051/cocv:2007058},
mrnumber = {2394516},
zbl = {1133.49002},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv:2007058/}
}
TY  - JOUR
AU  - Tahraoui, Rabah
AU  - Samassi, Lassana
TI  - How to state necessary optimality conditions for control problems with deviating arguments ?
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2008
SP  - 381
EP  - 409
VL  - 14
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2007058/
DO  - 10.1051/cocv:2007058
LA  - en
ID  - COCV_2008__14_2_381_0
ER  - 
%0 Journal Article
%A Tahraoui, Rabah
%A Samassi, Lassana
%T How to state necessary optimality conditions for control problems with deviating arguments ?
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2008
%P 381-409
%V 14
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2007058/
%R 10.1051/cocv:2007058
%G en
%F COCV_2008__14_2_381_0
Tahraoui, Rabah; Samassi, Lassana. How to state necessary optimality conditions for control problems with deviating arguments ?. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 2, pp. 381-409. doi : 10.1051/cocv:2007058. http://archive.numdam.org/articles/10.1051/cocv:2007058/

[1] G. Carlier and R. Tahraoui, On some optimal control problems governed by a state equation with memory. ESAIM: COCV (to appear) | EuDML | Numdam | MR

[2] M. Drakhlin, On the variational problem in the space of absolutely continuous functions. Nonlin. Anal. TMA 23 (1994) 1345-1351. | MR | Zbl

[3] M. Drakhlin and E. Litsyn, On the variation problem for a family of functionals in the space of absolutly continuous functions. Nonlin. Anal. TMA 26 (1996) 463-468. | MR | Zbl

[4] M.E. Drakhlin and E. Stepanov, On weak lower semi-continuity for a class of functionals with deviating argument. Nonlin. Anal. TMA 28 (1997) 2005-2015. | MR | Zbl

[5] M.E. Drakhlin, E. Litsyn and E. Stepanov, Variational methods for a class of nonlocal functionals. Comput. Math. Appl 37 (1999) 79-100. | MR | Zbl

[6] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Inc. (1992). | MR | Zbl

[7] 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. | MR | Zbl

[8] A.A. Gruzdev and S.A. Gusarenko, On reduction of variational problems to extremal problems without constraints. Russians mathematics 38 (1994) 37-47. | MR | Zbl

[9] E. Jouini, P.F. Koehl and N. Touzi, Optimal investment with taxes: an optimal control problem with endogeneous delay. Nonlin. Anal. TMA 37 (1999) 31-56. | MR | Zbl

[10] E. Jouini, P.F. Koehl and N. Touzi, Optimal investment with taxes: an existence result. J. Math. Economics 33 (2000) 373-388. | MR | Zbl

[11] G.A. Kamenskii, Variational and boundary value problems with deviating argument. Diff. Equ 6 (1970) 1349-1358. | MR | Zbl

[12] G.A. Kamenskii, On some necessary conditions of functionals with deviating argument. Nonlin. Anal. TMA 17 (1991) 457-464. | MR | Zbl

[13] G.A. Kamenskii, Boundary value problems for differential-difference equations arising from variational problems. Nonlin. Anal. TMA 18 (1992) 801-813. | MR | Zbl

[14] P.L. Lions and B. Larrouturou, Optimisation et commande optimale, méthodes mathématiques pour l'ingénieur, cours de l'École Polytechnique, Palaiseau, France.

[15] L. Samassi, Calculus of variation for funtionals with deviating arguments. Ph.D. thesis, University Paris-Dauphine, France (2004).

[16] 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 Ser 338 (2004) 611-616. | MR | Zbl

[17] J.A. Wheeler and R.P. Feynman, Classical electrodynamics in term of direct interparticle actions. Rev. Modern Phys 21 (1949) 425-433. | MR | Zbl

Cited by Sources: