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

Abstract

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The aim of this paper is to give a general idea to state optimality conditions of control problems in the following form: inf ( u , v ) ∈ 𝒰 a d ∫ 0 1 f t , u ( θ v ( t ) ) , u ' ( t ) , v ( t ) d t , (1) where 𝒰 a d is a set of admissible controls and θ v is the solution of the following equation: { d θ ( t ) d t = g ( t , θ ( t ) , v ( t ) ) , t ∈ [ 0 , 1 ] ; θ ( 0 ) = θ 0 , θ ( t ) ∈ [ 0 , 1 ] ∀ t . (2). The results are nonlocal and new.

How to cite

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Samassi, 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

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  1. G. Carlier and R. Tahraoui, On some optimal control problems governed by a state equation with memory. ESAIM: COCV (to appear) 
  2. M. Drakhlin, On the variational problem in the space of absolutely continuous functions. Nonlin. Anal. TMA23 (1994) 1345–1351.  
  3. 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.  
  4. 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.  
  5. M.E. Drakhlin, E. Litsyn and E. Stepanov, Variational methods for a class of nonlocal functionals. Comput. Math. Appl37 (1999) 79–100.  
  6. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Inc. (1992).  
  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.  
  8. A.A. Gruzdev and S.A. Gusarenko, On reduction of variational problems to extremal problems without constraints. Russians mathematics38 (1994) 37–47.  
  9. 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.  
  10. E. Jouini, P.F. Koehl and N.Touzi, Optimal investment with taxes: an existence result. J. Math. Economics33 (2000) 373–388.  
  11. G.A. Kamenskii, Variational and boundary value problems with deviating argument. Diff. Equ6 (1970) 1349–1358.  
  12. G.A. Kamenskii, On some necessary conditions of functionals with deviating argument. Nonlin. Anal. TMA17 (1991) 457–464.  
  13. G.A. Kamenskii, Boundary value problems for differential-difference equations arising from variational problems. Nonlin. Anal. TMA18 (1992) 801–813.  
  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 Ser338 (2004) 611–616.  
  17. J.A. Wheeler and R.P. Feynman, Classical electrodynamics in term of direct interparticle actions. Rev. Modern Phys21 (1949) 425–433.  

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